 Okay, so now I get to the second part of the talk, and I will actually mostly in the beginning it will be very kind of basic introduction about turbulence. And then I will mostly talk about the properties of turbulence in the solar wind and the solar wind well is an example of or to study MHG actually, considering the plasma is one fluid. And so, and I apologize if the introduction is very really basic but maybe I would have repeated something that Sergio has said but it's always good to repeat things. So as you all know so turbulence is ubiquitous and the it's ubiquitous phenomenon in the universe. And we basically, and basically at all the scales turbulence with is characterized by the presence of eddies and vertices, and from the microscopic scales up to the topical one. And it's a very important mechanisms because it's basically the phenomena that will controls actually the transport processes much more rapidly if only molecular diffusion processes were involved. Now, but how. So how do we study turbulence and what is turbulent so let's go really back to the beginning. I mean like everything that we know basically turbulence was discovered by Leonardo da Vinci I mean he's maybe discovered many things. And among this is turbulence and Leonardo da Vinci was really obsessed actually or passionate let's say observing the motion of the flow of water. And, and in one of his text actually this was a translation to English I found. He wrote that so observe the motion of the surface of the water, which resembles that of here and here I put here and read you will know what, and has two motions of which one goes on with the flow of the surface the other forms the eddies. That's the water forms adding were pulls one part of which are due to the impetus of the principal current and the other to the incidental motion and return flow. So basically, since, like, since the beginning so you know, Leonardo da Vinci he could characterize this this ordered motion as a laminar flow by describing it as a hair, I put in red, and this webinar flow will breaks into this ordered eddy like motion. And this motion doesn't stay stable. And also I mean this phenomena was also observed by the naked eye by a van Gogh in his very famous painting here, which he could observe as well the transition from a laminar flow to a turbine and flow like in the atmosphere even without any kind of instrument or nothing. Actually, it was is we cannot say that it's only a disordered motion turbulence is disordered motion on a multi scale level. So here I'm showing you an example. This is, I think one of the biggest numerical simulations and higher dynamics that was done. If I zoom in in this square so you see there's lots of kind of fluctuations and kind of a mess inside. If I zoom in in this square, you will still see this kind of density fluctuations or disorder. And if I zoom even more, you will always see this kind of disorder, and so on. So turbulence is not only a disorder but it's a multi scale disorder disorder. And, as you know, one of the characteristics of turbulence is that in different environments so in laboratories or experiments, these turbulence behavior is characterized by this so called famous minus five third power law, the Mogorov power law. So in experiments in the solar wind as well, we observe this power law, and as well in the interstellar medium and this is a very famous so this is the power spectral density of. So this is here the density fluctuations actually. And here we have at least 10 orders of magnitude. And this was measured in the interstellar medium and it's also characterized by this minus five third power law. So now why turbulence is actually multi scale and why do we have a power law. Now the answer here I wrote would be in the way in which the nonlinear system actually will process the energy which is injected to it. So let's take the very basic so the Navier-Stokes equation. And you know so here we have so different terms, the rate of change of the velocity of the fluids, the moving so the second term is kind of the third term, which represent the moving fluid element which affect each other. And this is equal to the pressure gradient of viscosity term which represent the dissipation, and we have an external force, which have been the injection of energy, which would be like the spoon, which is steering the coffee that Sergio mentioned yesterday. So if we take the total so the kinetic energy of the system, which is written here. So, so we integrated so over the volume. And if we want to measure actually the rate of change of this kinetic energy. We obtain basically this formula, which says that the rate of change of the kinetic energy is equal to the power which is injected minus the power which is dissipated. And to calculate to obtain this formula, you, you probably have done it all but you basically, you multiply this Navier-Stokes equation by the mass density times the velocity and then you would obtain this Navier-Stokes equation here. And if we consider a steady state. So the rate of change of the energy would be equal to the power which is injected and equal to the power which is dissipated. Yes. Yeah, yeah. Yes. Okay, and if we do like a simple dimensional analysis, you can find that the power which is injected. It's actually can be written as a function of the mass density times the velocity cube over the scale L represent the injection scale, and the power which is dissipated as well we can write it in this form here. Let's compare the power which is injected to the power which is dissipated. If we take the ratio of these two expressions, you will find actually with the simple dimensional analysis, the so called Reynolds number. And as you know, which so the Reynolds number is, so it's u times the scale L over the viscosity. And it happened that for all turbulent flows the Reynolds number is much bigger than one. So, which means it's an imbalance system. And as you all know, nature does not like imbalance system does not like non-equilibrium. So it has to correct for this non-equilibrium. So how does nature correct for this equilibrium, a non-imbalanced system. Can you tell me how. So it has to decrease actually this Reynolds number. And how does it do that? So we are increasing this dissipation, the power which is dissipated. And how do we increase the power which is dissipated if we look at this term is by increasing the wave number k. And so basically what we will have, we have large scales where the energy is injected L. We have a very small scales where the energy starts to dissipates. And we have so a range of scales that we call the inertia range. And the energy is transmitted from the large space to this very small space by cascading and that's what we call the energy cascade rate. Okay, so this was in the physical space this was a kind of illustrated or described by Richardson who described it as collection of eddies at the large case they will interact with each other. Then they will decompose into smaller and smaller ones until they reach the very small space where the energy dissipates. Okay, and I think, yeah, I need to go maybe fast, but so this the first exact law actually a fully developed incompressible turbulence was derived by Kolmogorov in 1941. And this is this is known as the fourth fish law and assuming collection of hypotheses actually the first one universality, there is no dependence on like we don't consider any special system. Homogeneity as well. There is no, we don't consider any special location in space and isotropy there is no special direction and locality as well in the scale space. And he could derive this exact law that relates excellent here is the energy cascade rate, which is equivalent as well to the energy dissipation rate, which is directly related to the third order structure function here that you see. And this third order structure function is written as a function as you see here delta is the increments of the velocity. What does it mean is just takes the differences if you take for instance, in space we have a varying let's say a flow we take one measurement at this point here. And then we take another measurements at this point separated by scale at so it's a difference in the velocity separated by scale and. And actually, also doing a dimension dimensional analysis we you can find that the energy spectrum is actually equal or is direct or equal to K to the minus to the power minus five. Okay, so this was turbulence in fluids. Now turbulence is in plasma. So, on sufficiently large scales, as you know we can treat the plasma as one fluid and that's what is like need to hydrodynamics, and there are at least two hypotheses and an HD very slow variations and very large scales large scales, much larger than the characteristic iron, for instance, larma radius and the byline of the electrons, and the very slow variations match lower or than the characteristic of timescale of the irons for the iron gyro frequency. And as a consequences of these slow variations. So there are two main consequences quasi neutrality so we have always equal number of positively charged particles and negatively charged. So, and also zero current. Now the key difference to neutral fluid is the presence of the magnetic field and this would break the isotropy that Sergio talked about yesterday. Okay, so now if we take the energy equations, we can also. Okay, we can express actually the and if we express the magnetic field into a mean field plus a fluctuating field, we can express the momentum equation as a function when we can rewrite it in this way. So here, we wrote it as a function of this debt variable which is called the el satir variables that couples actually the velocity fields plus the magnetic field normalized to the mass density. And so it's equal, but it couples basically the velocity fields plus the alvein. And also equivalently to the fluid turbulence week. Well, 1998 polytano and pokey they derived the exact law for incompressible m hd turbulence that relates the energy cascade rate here to a third order structure function but here we have an increments of the el satir variables that couple the magnetic fields with the velocity field. So, and since this is written as a function of the velocity the magnetic field the mass density, we can measure these values in situ and so we can actually estimate this energy cascade rate in the solar way. Okay, now. Yes. I just found the point of something go back. Yes. So, and such variables, they seem to be kind of nice and clean, but they are fundamentally wrong. This quantity is made from a vector view, and an actual vector view. Putting them together is not portion because the given quantity has no limits in variance property. So, in fact, if you are really doing this very correctly, as I said, we need you to non covariant system. The purpose of this calculation is to evaluate the law for the increments. Yeah. Actually, because the point is, if you write down the law, the rate of change, which is one equation, and then there is an equivalent. So, all of them have changed exactly the same thing, which law, but any, any law that you like, it has to be sorry, you know, it depends, it depends on the law, it's a conservation law, it's on a scale. No, no, no, no, you don't listen to the point. Every conservation law is a full dialogue. So, in order to be able to maintain a property conservation law, you can learn some variance. These variables are not useful. That doesn't mean that some of the intermediate results isn't okay. They use this variable just to obtain the law, which of course, as you can see is a linear combination. Yeah, you can go back to the primitive. Right. Just trying to point out. Okay. For instance, for instance, for instance, if you really think in this term, there's a holistic observation. No, no, it's there. It's going to be the difference of these two. No, what's the point of introducing that. Because you can write it down. No, no, no, that's okay, but I'm trying to say that. It's very useful for a non-covariant formulation. But if you at all are going to get to a very important issue, it should be the final law of nature. These are not the best. Okay, okay, that's good. Yeah, yeah, yeah. You know, magic, which is not, it's not because we are here. That's okay. I'm just pointing out for the general audience to understand. Yeah, yeah. Okay, so I just wanted to mention here just for what is, well, for fluid turbulence, we have this phenomenology of Eddie's interacting with each other, but how does phenomenology becomes an MHD turbulence. Now in the 60th, the two scientists irashnikov and crack, like they proposed actually independently physical image of this MHD turbulence knows as the irashnikov knock-knock phenomenology, which is based on the nonlinear interaction interaction in this inertial range by counter oops counter propagating is it here. Yeah, counter propagating with packets that are moving at the alvein speed. So that's kind of the comparison between fluid and MHD turbulence. Okay, so now more recently so now I showed you the exact law that relates the incompressible cascade rate to the third order suction function influence. In incompressible MHD turbulence, and then now more recently. Actually my colleagues, they have derived the exact law, well for either thermal compressible MHD turbulence, and this law relates now the compressible cascade rate. So we don't consider that the density is constant anymore, which is equal to a sum of different terms. So the first two terms are flux terms, and are written as well as a function of the increments of the velocity fields with times the mass density. So this first flux terms, it just can be seen as the generalization to the compressible case, actually of the incompressible fluxer. And then we have a second flux term here, which is written as a function of the increments of the mass density, the internal energy and the increments of the velocity so this second term is kind of the new and additional purely compressible. So other terms which are non flux terms, and these are written as a function of the divergence actually of the velocity fields. So this basically term. There are kind of the source terms. So we, I will not show it here but we could estimate these terms, either using the American simulations or the data with MMS spacecraft specially, and we could show that they are sub dominance, compared to these terms. So now we obtain. So here I just for to summarize we have a compressible exact load that relates the compressible cascade rate to the sum of two flux terms. And here we have this exact law for incompressible MHD that also relates the incompressible cascade rate to a floxter. So this is important actually to to to be able to estimate this compressible cascade rate. The solar wind is nearly incompressible density fluctuation is about maybe 20% or 10% maximum is is really like incompressible media. So to other environments such as the magneto sheath plasma it's highly compressible, you cannot estimate the cascade rate using incompressible MHD it's not valid anymore in magneto sheath also astrophysical medium are highly compressible if we can get an idea of the incompressible cascade rate for instance in a compressible medium and the solar wind maybe I will show later on we could extrapolate to have an idea of this cascade rate for astrophysical medium where we cannot really measure in situ yet the cascade rate. I still have time. Okay, should I if, should I, maybe I can continue. It's a different. So in the next part after the break, I will talk about more so MHD properties of the turbulence in the solar winds and in the magneto sheath incompressible and compressible medium. Thank you. You were talking about iron calculation and observation and maybe you are familiar or not I don't know that in Japan and in Iowa people have been doing experiments to produce pure ionic plasma and they say there are no electrons this is on the temperature is low and this is on the negative ones right. So experimentally I found that it's not easy. But do you think that in nature, some system like in the ceremonial or planetary atmosphere, where there are many kinds of the electron density can decrease to extend that electron plasma oscillation frequency become much lesser than. So we have a system which is not, not neutral not. I don't know if it's possible. Okay. I don't think, I mean, yeah, on Earth, yes, but Yeah, Yeah. a very nice presentation and also I like the fact that sometimes this third order law pops up because it is the only regular law of pure words while former Europe is some more like an engineer approach where you do some qualitative, I mean something. No, it's very, there's a lot of interpretation, of course, but really you start from the equation and you obtain the law. There are many other terms you didn't speak of. No, no, I did. That's only the inertia range. This is very theoretical and it's nice that you can obtain a law from the accepting from the equation, while Komodorov does not. No, yeah, yeah, yeah. The whole idea that Komodorov-Majanism prevailed almost in all departments. Except in the post-reliant age, when you have energy being generated at some particular location, there are unstable modes in the system which do not correspond to the frequency. Then obviously you cannot have Komodorov like that. Like even in MSU, when you remove the illicit ingredients, you can have inverse cash. Yes. And in addition to that, every other thing that comes into the system. But Komodorov-Law does remain on the basis of certain direction. And the question is, there may be cases in which the inertia range is very small. Exactly. That includes the size. Good. Thank you. Thank you. Any more questions? So let's have a coffee break. So a quick announcement before we start. I don't know if you know, but it's supposed to be a reception. It's being moved for tomorrow, for Wednesday at 7 p.m. on Saturday. You know? No, no. Oh, nobody talks to me. You know it was today. Me, too. In the program, it said the reception. So let's go on with the lecture. Yes, so let's continue the second lectures on turbulence. So, well, I have one hour and I was 55 minutes and I was planning to do, well, now it's a fourth lecture on ionosphere. So I hope at least I can start the lecture on the ionosphere in the second half of this hour. And maybe in the afternoon, we can continue informally the discussion about planetary ion spheres and Saturn's ionosphere. So I will not rush to finish this presentation. I will continue as if I was completely on time. Okay, so an example of an MHD turbulent plasma is the solar wind. So as you all know, the sun emits continuously a non-collision. Yes, a non-collision plasma, which is the solar wind and the solar wind propagates in space. Now, the solar wind is a non-collision plasma. It's well about 96% consist of protons. The speed of the solar wind varies between about 300 kilometers per seconds up to 800, 700 kilometers per seconds and that's why sometimes you would hear talking about slow solar wind or fast solar wind. The density is few particles per cc, temperature is about 150V, and of course in our solar system, the parameters of the solar wind would vary at different locations. And actually the solar wind is the only collisional, a non-collisional plasma that we can sample directly. And that's why we say that solar wind is a kind of our laboratory actually to study turbulence in situ. And as you all know as well, so the planetary magnetosphere, they are not opaque to the solar wind. So the planetary magnetosphere would present two kind of interfaces with the solar wind. The external interface, which is the ionosphere, which is basically formed by photo ionization due to the EUV radiations, and we have an external interface, which is the magnetopause. And the magnetopause is basically the boundary of the magnetic bubble or the planetary magnetic field. And the magnetopause, so this is kind of the external interface, it's separated from the solar wind by a key region here that you see, which we call the magnetosheets. And basically, so the magnetosheets is just the transition region between the solar wind and the planetary magnetosphere. Magnetosheets is a very particular plasma because it's made of compressed, so it's basically the compressed solar wind due to its interaction with the ball shock here that you see. So the solar wind would decelerate, it would be compressed and it would be heated. I will come back to the differences between the magnetosheets and the solar wind later on. And yes, and ionosphere, which I just mentioned. So, and turbulence, as I mentioned, it plays a key role actually for a much more efficient energy and transport of the matter from the solar wind into the planetary magnetosphere through the region called the magnetosheets. And so this plot was shown yesterday by Sergio and it actually shows, so this is the proton temperature as a function of the distance from the sun. And it shows actually the decrease in the temperature of the solar wind as a function of the distance of the sun in comparison to an adiabatic expansion model. And actually this figure shows that the solar wind plasma is cooling down much more slowly than what we expect than what is predicted from an adiabatic expansion model, which means that there is a source, which is heating the solar wind. I mean, you would imagine that the solar wind, the temperature would decrease as we go away from the sun, but this is not the case, it's actually decreasing much more slowly. And, and one of the mechanisms, there are many mechanisms that could heat the solar wind or add heating to the solar wind, but one of these mechanisms is turbulence. One of the mechanisms is so to play a key role in heating the solar wind, and, and being able actually to measure the energy cascade rate, which is equivalent to the energy dissipation rate. And it can give kind of a direct quantification of this heating of the plasma in the solar wind. So that's why it's important to be able to measure this excellent in the solar wind but also in, which is an incompressible medium but also in other compressible medium. Interplanetary shocks can play a role in heating the solar wind as well. Reconnection as well. So, and turbulence. Now, there are other, you can talk about turbulence in reconnection or reconnection in turbulence. And of course all of them play a role in heating or adding some heat into the solar wind turbulence. And of course this coupling between the sun and the earth magnetosphere is also the same with all the magnetosphere is in our solar system. This is just an illustration here, and it's very interesting actually to do comparative studies of turbulence at different radial distances, because, for instance, as a Saturn and I will show that later on the amplitude of the magnetic pattern is much lower than at Earth, for instance, or at Mercury is different. And so, for instance, the plasma beat us can be different from Earth Mercury and Saturn and so we can study turbulence in situ in different kind of conditions and compare them, compare these properties at different condition. And also we can extrapolate this interaction between the sun and the planets with the interaction of the stellar winds with the whole genius here. Actually, and even more to exoplanets as well. And, and that's what's really makes the solar winds I would close this laboratory to study turbulence in situ. And I will show you in the next slide some observations of. I will start with the spectral properties of the turbulence in the solar wind, compared to the magnitude, using in situ spacecraft data I will show you observations from the tennis Artemis NASA mission. This is around one a you also from the cluster mission. I will show you observations using the Cassini spacecraft mission so around Saturn so at 10 you much further away from the sun. And also I will show you data from the Maven. Yes, I will Maven mission closer to Mars or at 1.5 a you and from the Parker solar probe mission which is a recent mission that will that is getting really close to the sun. So at 0.2 a you. So I will show you different kind of properties at different radio distances from the sun. Okay there is. Let me check something. Maybe yes, if I put it. I don't know if it why it is like this, but basically let me keep it like this. So this is so and I will show you. So I will start with the spectral properties by comparing actually solar wind observations with the magneto sheet observation. In the beginning the solar wind is a nearly incompressible medium, but the magneto sheet is very particular region. Why, first, it's a bounded region. On one side, you have the ball shock, and the other side you have the magneto close, we don't have these boundaries and the solar wind. It's a much more complex medium because we have lots of temperature and I saw trophies and the solar in the magneto sheet, and this would induce the formation of kinetic instability. And also at the boundaries of the magneto pose we have some large scales in homogeneities like the Kelvin Helmholtz instability as well that could affect as well the properties of turbulence. And let me see. And also, okay the reference here is is not shown, but also the, we have different kind of properties of these turbulent fluctuations depending on the location where we are inside the magneto sheet, because of the curved natures of the ball shock, we can define different kind of geometries, which is defined with respect to the normal to the ball shock, you can hear talking about quasi perpendicular ball shock. When the magnetic field is quasi perpendicular to the normal to the ball shock, and quasi parallel ball shock where the magnetic field, the angle between the interplanetary magnetic field, and the normal to the shock is quasi parallel. Okay, so let's start with specs spectral properties of turbulence in the magneto sheets of Saturn. And so just here I will show you data from the magnetometer. Now that you know how it works on board the Cassini, and also actually, what I will not show it here but this is just the Cassini plasma spectrometer which was the father let's say of MSA the mass spectrum analyzer on board the picolombo. I don't know why the, let's do it like this. And so we are in the magneto sheets of Saturn. So this region here. And here I'm showing you an example of a time series taken selected inside from the magneto sheet. And the, the y axis it shows the magnetic field as a function of the time. And you see the difference in the amplitude and the magnetic field going from the solar wind which is very low, and then you observe a clear gradients here which is which reflects the crossing of the boundary the bow shock, and then inside the magneto sheets the fluctuations becomes much more important, because it's, well it's a very compressed medium, and then the spacecraft goes back in the solar wind. So we from this data here, we will compute the power spectral densities from the different components actually of the magnetic fields. And that's what we obtain. Is it going to be. Okay, that's fine. So that's what we obtain. So this is an example of a power spectral densities computed inside the magneto sheet of Saturn. And you see that it is characterized in, well, let me tell you something before. Actually, well, you see that it covers different frequencies above the iron gyro frequencies of the ions. So what happens is that at earth, if you want to cover this whole range of frequencies, you need to have the DC flux gate magnetometer, and then you need to add to these measurements measurements from the search code magnetometer so you can cover the energy scales and the sub ions case. But because at Mercury, the magnetic fields amplitude is much lower than at earth closer to the sun, all the characteristics frequencies are shifted to lower frequencies. And so with only one instrument, which is the flux gate magnetometer, we can cover this whole range of frequencies. And so you can see that they are, well, the noise level here is not shown, but we are way above the noise level, and you can see the presence of two different kind of ranges. The first one it is characterized by a power law of minus one. And the set and then there is a break in the spectrum around the iron gyro frequency and then the spectrum becomes deeper and here we add to the sub ion scale. The power law is much deeper and we have it's about after the minus 2.6. Now if we compare this power spectral density to the typical power spectral densities observed in the solar wind. Here, this is an example of a very typical power spectral densities measured in the solar wind at one EU, and it is computed by combining different data from different instruments actually so we can cover the full range of frequencies. And you see that the typical power spectral density in the solar wind it is characterized by three different ranges. First, we have this range characterized by a power law of F minus one. And actually, the existence or the origin of this power law of F minus one is still highly debated in the community. Some people say that this actually originate from the photosphere in the sun. Some other work from simulation they showed that this could be related to inverse cascade actually an incompressible MHD. Well, and it's also showed that it could be due to nonlinear interaction between counter propagating advanced waves as well, or uncorrelated fluctuations as well. So there are different kind of mechanism that could explain this F minus one, the formation of this F minus one, and is known to be called the energy containing scale. This is the scales that contains really the energy. Then after this we reach the correlation scale here you see we have this formation of the so called the minus five third power low the initial range, and then the spectrum becomes deeper again and below the iron. At the sub iron scales, we have a power law of about minus 2.7 which. So if we compare this power spectral density that we compute in the magnitude to the solar wind, you would directly see that inside the magnitude shift so we have this formation of the F minus one, we have the sub iron scales but we don't have the formation of this minus five third power low. So, why, and this is the fundamental difference actually with the solar wind observations is the absence of this minus five third combinator flow. Now, if you do this on, because this was one example but if you do this statistically on many, many samples. And if you fit the spectrum at the m hd scale, so everything which is below the iron gyro frequency, you would see that most of the values actually are, while some of them they are around the minus five third, but most of them, they are around minus one or so. The spectrum is kind of broad. And, and if you do the same for the sub iron scale you see that the history, I said spectrum but it's the histogram density, it's kind of peaks around minus between minus 2.5 and minus three. And, and actually, in the solar wind, I didn't put it here, but the, if we compute the power, the slope at the sub iron scale in the solar wind, we obtain a very similar results for the sub iron scales, but very different results for the m hd. Yes, I will come to this. So why, why we don't have this minus. So then, we do, we can do it even much because at Saturn. Well, or let's say it in a different way on earth we have many spacecrafts orbiting in the magnetosphere of Earth. And so we have much more data and much more coverage of the magneto sheets of Earth, especially with the cluster spacecraft actually. So, we can do this on a much larger statistical samples. And here it shows actually. So this is the bow shock and the magneto pose model. Of course, the color bar here it represents the slope at the m hd scales. Okay. And the second figure here it shows the slope at the sub iron scales. And you can see that for the m hd scales, most well. And okay, the, the color which is around which is blue. It reflects it's about minus 1.5. So it's the called Mogorov kind of power low, and green and yellow is not. So you see that behind the bow shock. We don't see this power low, but we observe it only further away from the bush up and closer to the magneto pose. And so we clearly see in this case a kind of a dependence in the location. Yes. Yes, yes, below the iron. Okay, this case. And, however, at the sub iron scales, there is no dependence on the location within the magneto sheets. Actually, most we see clearly no dependence and most of the values is about minus 2.5 or minus 2.7. I will show you another plot. So let's take a trajectory, for instance, this, an example of spacecraft crossing this way from the solar wind, crossing the bow shock and getting away from the bow shock closer to the magneto pose. In this panel, the first one it shows the magnetic field components from the solar wind into the magneto sheet and getting closer to the magneto board. And then here is the local variation of this slope for each power spectral density computed at each time. The red line it shows the minus one slope. The green one is the minus five thirds. And you see in the solar wind, the value is about minus five thirds. Once we cross the bow shock. It's, well, it's, it deviates and it becomes minus one and only away from the bush up closer to the magneto pose it's minus five thirds. So why do we have this variation. I didn't put all the plots here, but one of the explanation of this is that the shock because, because of the shock all the pre existent correlations in the solar wind would be destroyed and closer to the bow shock here. So these correlations are destroyed and turbulence needs some time to be developed. Do you think that there is no connection with the correlation, the magneto sheet, and before the encounter with the shock. It's everything is destroyed by crossing the bow shock. You disagree. So, okay, so, yeah, of course, actually this is the problem. I mean, I'm not working on this anymore but I'm still interested in with, because it also so what we did as well. So this turbulence is kind of needs time to be developed and is, we kind of observe this comogorov spectrum only further away from the bow shock. So the scale they changed. Yes, so it's becoming much smaller. Yes. So, but something else is happening. So what we did, what we did as well, we computed the core, we compared the scales. So the correlation scales to the characteristic iron scales, and we see that for the cases where we don't have the comogorov range there's no separation than scales actually. Yeah. Yes, we don't. We, we, well, we don't have enough separation between the scales so that turbulence would be developed. In order to develop the comogorov, you need large separation between the scales. So when you say correlation, then, do you mean the tidal? Yes, yeah, yeah. And that could explain why I would say this, this is the real reason why we don't have the formation of this comogorov range, but also it could be due, I don't know, and this would be interesting to investigate further in detail. Maybe the, because at the magnetoboes we have lots of Kelvin-Helmholtz instabilities actually. Maybe the presence of the Kelvin-Helmholtz instability could play a role in the formation of the energy cascade range. I don't know how, but it would be interesting actually to investigate this and to take, to take much larger samples only in the present where we are sure that we have the Kelvin-Helmholtz instability and see if there is any connection between the formation of this turbulence cascade with the Kelvin-Helmholtz instabilities actually. You don't have data downstream along the magneto-sheet. No, yes. Here, you mean below? Yes. Yeah, yeah, yeah, you have more, yes. Okay, so this is an example how, so one of the properties, the energy properties of turbulence, at least in the magneto-sheet, the dominant features that we don't have this formation of the comogorov, it depends where you are in the located inside the magneto-sheet. We can characterize the nature of the waves actually that dominates in the plasma, in the solar wind, or in the magneto-sheet. One of the quantities or one of the kind of the diagnostics to know what kind of waves dominate is given by this quantity that we call the magnetic compressibility. The magnetic compressibility is given by the power spectral densities of the parallel fluctuations of the, with respect to the background magnetic fields over the total power. And here I'm showing you actually in blue, green, red, the theoretical magnetic compressibility for the three nonlinear MHD modes. In green mode, the incompressible mode, which you see the profile of the magnetic compressibility is kind of rising like this. And then the red and the green are the profile of the compressible, the magneto-sheet modes, the fast and the slow. And you see that the profile is very different, they are highly compressible, and this is shown, it means that the parallel components really dominates, and that's why you have this kind of profile. So we take an example of the magnetic compressibility computed in the solar wind. These are observations, the blue and the red observations in the solar wind, I think using cluster data. Magnetic compressibility is a function of the frequency, and you see in the solar wind it has a rising like profile, like this, similar to the, well, to the theoretical profile of the theoretical LVAN mode. And this shows that in the solar wind, highly dominated actually, or at least for this case, by the presence of the incompressible like fluctuation. Now if we take an example from the magneto-sheet, you see that the profile is not rising like, it's like more constant, and this shows the dominance of the parallel fluctuations. And so the dominance of the magnetosonic like compressible like modes. So this was an example, but we can do this on a much larger statistical sample, and we, so you can do compute this magnetic compressibility on all the samples in the magneto-sheet, and when, we could obtain different kind of profiles. Some profiles, they have, the red curve is like the average of all these cases. Some profiles, they have a rising like profile like this, and these cases are dominated by LVAN like fluctuation. The other ones, they are more kind constant or, well, much higher value at the MHD scales of the magnetic compressibility. And these are actually the magnetosonic like modes, most probably slow like propagation. And so this also shows, which is a main difference from the solar wind, the dominance of the magnetosonic like modes at the MHD scales in the magneto-sheet in comparison to the solar wind where the incompressible like alveonic fluctuations will dominate. Okay, so this was kind of examples about the MHD properties, spectral properties, the nature of the propagation modes in the solar wind and the magneto-sheet. And as you have seen in these two examples, so the magneto-sheet is highly different from the solar winds and is highly compressible and dominated by compressible mode. Now how about the estimation of this energy cascade rate or equivalently the dissipation rate of the turbulence inside the magneto-sheet, inside in the solar winds and the magneto-sheet. In the next slides, I will show you actually to the application of these two exact laws. The first one I repeated again is the exact law for incompressible MHD. Again, it relates the incompressible cascade rate to the third order structure function that we can estimate in situ. And the second law is the equivalence of this one, but for a compressible MHD that relates the compressible cascade rate to a sum of different flux terms actually that also we can estimate in situ. And both is a function of the increments, and this is really the difference in the fluctuations, yes, separated with the scale. Yes. I agree. And also it's isothermal. Yes, I agree. And there are other, I think, that this compressible, actually I'm not sure for the incompressible one. In the magneto-sheet, the example for being probably. It's more. Both solar winds is very much. Yeah, yeah, I agree. But I think there is actually I'm not sure, but this, there was kind of how to say, works that also generalize this anisotropic as well, MHD. Sorry. So, but I don't, because now I'm not really, I have very bad memory. And that's why actually to compute this we have, you have to select. Well, some cases that are relatively stationary, you don't have sharp gradients and the, so it's really tricky actually the magneto-sheet. And yeah, actually I need to look back at this, the anisotropic law because I'm not sure if we can use the data. Maybe you need multiple spacecraft. Okay. And so we first, well, this incompressible cascade rate has been estimated in the solar wind, long time ago actually it's not the first time that we estimate the energy cascade rate. And I'll show you just the estimation of this energy cascade rate in the solar wind at different radial distances using Parker solar probe around 0.2 AU using tennis spacecraft data around one EU and using a Maven spacecraft data at Mars around 1.5. Well, this is just to show you an example of time series where this is around at 1 AU you have from the first the magnetic components of the magnetic field the velocities the densities and so on. So here is the computation actually of this compressible this isn't the solar wind. Okay, the y axis is the compressible cascade rate as a function of the incompressible cascade rate. This dotted line here is just a slope equal to one actually. Using Maven data in circles tennis data in squares and triangles PSP. So, yes, okay in this tub figures here it shows the velocity fluctuations and the magnetic field fluctuations at different distances. And basically before we look at this line. If we look at this inserted figures, you see that the velocity fluctuations and the magnetic field fluctuations are much more important closer to the sun. For PSP data, rather than around Earth or Mars. And what this plot basically shows is that closer to the sun with PSP there is very moderate increase of the total compressible cascade rate closer to the sun with respect to the further away from the solar wind at all in the solar wind we don't have much differences actually between the compressible cascade rate and the incompressible calculate it. And this, this is expected because the solar wind is again nearly incompressible medium, and the compressible model should actually converge to the incompressible one. And also to the sun we slightly see some increases. That means that the cues that we showed in the previous slide are sort of negligible. Exactly. I will not. Yes, they are negligible we could estimate them using MMS data and the magnitude because here we have multi spacecraft data with with the measurements we showed that it's sub dominant. Okay, then also what you can do you can do a bit much more detailed analysis, it's unfortunate it's not very clear. I will go maybe slowly here but you can study the contribution of the different terms in this compressible exact law with respect to the incompressible flux term. The first panel is show the data closer to the sun and and further and further away from the sun. Okay, the y axis is the ratio of this purely compressible term over the incompressible one. The access is the, the, the, the kind of the generalization of the compressible term over the incompressible one. Okay. And what you see, and the, the color bar it represents the density fluctuation, or it reflects the compressibility. And you see that when the density fluctuations are small. So, about this purple color here, or blue. And this one here is the compressible young like component. So, this one. And this one, they coincide with each other. This is value equal to one here you see. And it's a bit complicated this term. Yes. So is. So the values are here along this line one. And if you look at the y axis, the purely compressible term sub dominates over the incompressible one. The one value is here. So most of the cases are below one. This shows the really sub dominance of this purely compressible term as well. But only when the fluctuations becomes important so up to about 25% the yellow color here is here you there. It's start to see some kind of competition actually between the compressible. So you see the more yellowish colors here which is the dominance over the compressible term over the incompressible one, and a kind of increase as well of the purely compressible term. So when the density fluctuations start to increase, there's a kind of competition between these terms, which one will dominate the purely compressible one, or the compressible. So this. Okay, so and then also combining the data we can see if there any relation actually between. Because when we say heating of the plasma we should. Yes. For which case for this one here. Yes, it. Okay, so they are really few points. And I don't know if it's really statistically valid or not but it shows, at least for these few cases, which are highly well highly relatively highly compressible, you see that the compressible purely compressible term would dominate over the incompressible one. And this is due to the compressibility effect and so which mean that we should use the model which is related to the compressible MHD. Yes, yes, this means that this this terms, this is the one that dominates. Okay. Yeah. So when we say heating of the plasma, you would expect an increase of the temperature, actually, and if we if we plot the temperature here of the solar wind as a function of the energy compressible energy cascade rate, you could see there is a kind of overall correlation between these points. Well, let me explain to you better. The circles are maven. So this is further away from the sun triangles are Parker sort of close to the sun. Okay, and the squares here is around earth at tennis. So all the tendencies that the larger is the compressible cascade rates, the larger is the temperature that we measure in the solar wind as well. Except for these points and I think for this point there was a large error bars in the temperature estimation at least at earth, around earth by the tennis spacecraft. So this was about the estimation of this compressible cascade rate and the solar winds. Okay, so, and as you show we don't see any kind of large differences between the compressible exact low and the incompressible one. Yes. Yes, should be lower. So, because I think there was probably there was a large error bars in the estimation of the temperature from the distribution function of the iron. Okay, so how about a purely compressible environment like the magnetoshis. So we, you can also estimate this compressible cascade rate and the magnetoshis of earth. And here we have lots of data actually orbit from many spacecraft orbiting in the magnitude sheath of earth. At least, you can use the class data from the cluster spacecraft and the tennis spacecraft. You can estimate this energy cascade rate inside the magnetoshis, but as I showed you earlier. And the magnetoshis most of the cases they don't show the presence of this energy cascade rate. So first you should select the case studies that actually are characterized by the formation of the magnetoshis and also you need to make sure about the stationarity kind of condition. And what you can do. You can compare the behavior of this energy cascade rate or two different type of fluctuations, we can compare these compressible cascade rate for alveonic light fluctuations and many to sonic like fluctuations. And to do this, you can select these two different types of fluctuations by computing this magnetic compressibility, which I showed earlier, and all these green line. The profile is rising like and this could reflect the alveonic like fluctuations. Well the red one are clearly very different than the green one and this reflects the many to sonic like fluctuations, the green one are the cases which is in between so we don't really consider them. So to make sure that we really compare two different things. We compare so the alveonic like and the red one and the green one. So here I will show you two examples of this compressible cascade rate. The first example is kind of a computer on an alveonic like event. The second one is a much more compressible if money to sonic like events. The y-axis is the absolute value of the energy cascade rate. Here it's a function of the time lag actually which is using the Taylor hypothesis, you can come back to the frequency actually frame. And the black curve represents the incompressible cascade rate. The red one is the compressible cascade rate. And the red one is that, at least for both events. Okay, in the inside the magneto sheet, the compressible model, it gives a higher value than the incompressible one. So this means that inside the magneto sheet which is much more compressible medium than the solar wind one should use a compressible MHD model and not an incompressible one. But also if you come when comparing both for the money to sonic like event which are characterized by compressible like fluctuations, the, the energy cascade rate is amplified by at least two orders of magnitude with respect to the alveonic like fluctuation. I need to check. Usually the law must go down. Yes, yes. At least we've one of the conditions as well is to take into account the correlation. I think for at least with all this time series. I mean, in time it was at least we took. No, no, we took 30 I mean 30 minutes to calculate the spectrum to make sure that. But I need to check in the papers actually. Also, so this shows the importance of the compressibility actually on increasing and enhancing the energy cascade rate in the magneto sheet. Now, what we could observe as well. So by plotting this energy cascade rate so this is also in the magneto sheet as a function of the turbulence Mach number. In some cases, the turbulent Mach number is kind of subsonic actually is 0.1, but we, you can see that there is a kind of a power law, or kind of a clear correlation between these both. And this is kind of interesting because if maybe we can validate this kind of correlation for different values of turbulent Mach number higher values and lower values. So this could be extrapolated and applied for other regimes astrophysical regimes. So for if you have, for example, a medium with a, you know, so in which you know the turbulent Mach number, we can get an order of magnitude let's say of the heating rate of the plasma in this astrophysical Okay, something the last thing I want, I think it's the last thing I wanted to talk about here for God, but the role of the kinetic instability is as well. So, here, this plot is kind of the Brazil plot degraded. They're just showed yesterday though the y axis it shows the temperature and I saw trophy, and the x axis is the beta, and the color bar represents the compressible casting great again for the alveonic like events and for the money to sonic like event. And you can see that if you look at the, and then the dotted line or the instabilities different kind of instability thresholds actually for the alveonic like events. Maybe most of the cases here the blue cases, where the compressible cascade rate is lower, they lie around the stability kind of condition, where the temperature ratio is equal to one. However, if we look at the money to sonic like events, the highest compressible cascade rate in yellow here, you see they lie around the mirror instability. And this, this result could imply actually, or could show the role of the instabilities or the mirror instabilities in injecting the energy actually in the background fluctuations and enhancing the energy cascade. Okay, actually it's not my last slide. I forgot about this one, but in all these observation that I showed you and studies, we make a very strong assumption which is the Taylor hypothesis that Sergio talked about yesterday. And again so the Taylor hypothesis it consists in assuming that the measurements taken on board the spacecraft, they just correspond to one dimensional spatial sample. So this is the general formula actually relating the frequency of the wave. Okay. And the plasma rest frame to the one measured on board the spacecraft. And this is the formula. And so in case if we have so the face, the face speed of the waves, if it's negligible with respect to the flow speed. If we have omega over K is much lower than V, then we can use or do an assumption, and this is what is called the Taylor flow pros and inflow assumption, and we can directly go from the time domain or the frequency domain into the spatial domain, assuming this hypothesis. Now in the solar winds at the energy skates, usually, so the face speed, well it's equal to the album speed is about 50 kilometers per second 50 kilometer per seconds. Well, let's say the average solar wind speed is 500 kilometers per second, let's say, and so the face is about 10 times lower than the flow speed. And so the Taylor hypothesis is almost always valid in the solar wind. Now, in the magnetosheets data hypothesis here I put it in bold, it is thought to be violated. Why, because the face speed where let's say is equal to the album speed, or also the sound speed is about 100 200 kilometers per second, which is almost the same value as the flow speed and the magnetosheets, the solar wind is decelerated in the magnetosheets, and the speed is about also 200 kilometers per second. And so, so maybe you would think that this condition is not valid anymore. However, it can be valid in some different condition. The second condition, if we have strongly an isotropic turbulence is k parallel is much lower than k perp and that's what we usually observe it's well widely known, and the solar wind but also in the magnetosheet. And if you do the calculations actually you can show that the Taylor hypothesis is still valid for strongly isotropic turbulence, or also if we have stationary fluctuations for the frequency of the wave is equal almost to zero, but also, like the slow mode. There are kind of stationary fluctuations, and we know that the magnetosheets are also dominated by slow modes. And so, the Taylor hypothesis can be valid in certain conditions in the magnetosheet as well. Now, of course, these conditions are not enough. Yes, in time. Yeah. And, and I was saying that these conditions are not enough to validate or not validate the Taylor hypothesis to precisely validate or unvalidate the Taylor hypothesis, you need to have the exact measurement of the phase speed of the wave, but also of the direction and the orientation of the wave vector. And for that you need to have multi spacecraft measurements. So with the one spacecraft measurements you all you always need to do some assumptions you cannot calculate the wave vector and the phase speed of the wave. The last slide, I think, is I wanted just to talk about some coordination. So in the, in the heliosphere we have lots of spacecraft missions, okay, covering different radial distances from the sun. So, we have spacecraft orbiting around Mars, around Earth we have many spacecraft missions around one EU, Venus, Mercury as well with that the Colombo and recently with the launch of the solar missions, Parker solar probe and solar orbiter we can cover even closer from the sun. So now what we are trying to do is try to coordinate observations between these different, different probes, at least Parker solar probes solar orbiter and Bepi colombo to try actually to study the different radial distances, different phenomena, including turbulence, and one of the challenges actually is try to measure the turbulence properties in one plasma parcel, let's say closer to the sun, and try to detect the same plasma measured away from the sun, let's say by Bepi colombo for instance or solar orbiter and study the properties of the turbulence for the same plasma parcel at different radial distances. And so, to be able to see how the turbulence properties evolve as a function of the radial distance from the sun that also related to the expansion of the solar wind. And this is, I mean there are many studies that are done using this coordinate observations from the different spacecraft, but I would say it's a very big challenge to prove. I mean, if the, what we are observing is exactly the same plasma parcel that we observe closer to the sun, and away from the sun. Yes, this is something different. This is like large structures exactly, exactly, but the turbulence, the inherent properties, I mean it's very difficult. And actually it was it's interesting to to study I mean I'm interested to do this but I don't know how to do it actually, I haven't really spent time to to work on this, but to see if the turbulence properties, we observe that it's different, further away from the sun, to be able to show in situ the role of the expansion of the solar wind on the very varying the properties of the turbulence. So, are these differences due to the variation. So is it really interesting to the turbulence fluctuations, or is it due to the expansion of the solar wind. Yes, so I think this was my last slide. I think I keep it at an open question. I think it's a contribution of course of both, but it would be really interesting in the data to see if we can distinguish between both. And I think we need to write in simulations as well and so on but yes, that was the end of this second part. I'm really sorry I wasn't expecting that I didn't so I don't think I will have time to start the third part which is about. I was going to say perfect timing. Sorry. The worst timing. So there was the ionosphere. I can. Yes. Also we have this afternoon. Yes. And so. Yes, so what I was just to say very quickly I was planning to talk about planetary ion spheres. The properties like, as I did for turbulence, very basic introduction for planetary ion spheres and then show you in detail in situ for the first time we could measure in situ the ionosphere of Saturn. And I wanted to show you how it interacts actually with the rings around Saturn. And this is very particular for Saturn actually. But, well, I don't have time I can present you this. I'll be in the second and the afternoon. Yes. And in any case I will upload the presentation. This, I mean all the presentation will be uploaded so. Of course, yes. I hope it was clear. But you can ask me. Yes, sure. I was going to say that in compressible energy also in compressible turbulence. The cascade idea is because of a local transfer local transfer of energy in case that the energy goes from this case, the neighboring case. I mean, in compressible turbulence that. No, yes, because of that maybe not all the flow of energy alone case. Can be written divergence version so you have because of that yes. Yeah, yeah, yeah, because of that. Yes. You don't need to get kind of. I mean we don't. Exactly. Yes. Exactly. Exactly. Exactly. Yes. There's a nice example of this. I did it. No, yeah. Thank you. So this third part today, I'm going to speak about some, we did with some some properties of particles in a turbulent field. So, since we have both codes, we have the both the model that which is oil area and both the kinetic one, the Lagrangian one. We can look at the aspects about particle acceleration, which is very relevant for astrophysical plasmas. Okay, but accelerated particle are observed like it pretty much in an ubiquitous way. So, first I will speak about particles then I will go back to turbulence a shocks which today is the hot topic. And then we will do experiments with our, the code we built together yesterday, I will send the turbulence against a shock and see what happens. Look at the tracing the properties of the shock and if there is an energy, an energy station of particles but also we will build the theory, each one of the step we will have like observation experiment numerically and handwork. Okay, really, is the way pretty much a close like way, a typical physicist approach, I will say experiments simulations and theory. So, at the end, we will like a little bit dream far away, what is happening in a compact near compact objects or supernova remnants or the things that are more far away. We're exporting what we have learned so far in the first two parts of the course. So, of course, now I will like to you to to make a mental movie of this. So there is a corneral max ejection. And of course there are, you can imagine to follow little particles that come starts very close. So if you have a turbulent field, you, you observe a turbulent field like, I don't know, anywhere in a fluid, and you imagine two pieces of plot of the fluid that are two very close, they will diverge. Okay, is a, it will be natural to imagine this divergence in space. So, with this picture in mind, we go back to our experiments. So we repeat the experiments of turbulence. But with a particle in cell called people of you, like you, many of you are familiar with this. La grande like simulations, they are very noisy. Okay, there is a lot of noise, as we said yesterday, because of the numerical reasons. Okay, now we know why, but we can suppress this noise by increasing the number of particles thanks to to super computer and we know how to make parallel calls now, but thanks to super computer we can increase the number of particles, so that we really tends to solve exactly the blast of a question without no noise here you can see the patterns are pretty similar to the other calls that are more sophisticated but what do we observe the same analogies here it's a typical power spectrum in the solar wind. The green line is the typical electric field spectrum, which is a little bit higher here at high frequency, because of like whole m hd effects, for example, or dispersive waves. So, when you go to model smaller scales we observe the same with our picots here is there's a little bump here. This pile up is typical of the noise of particles. Please forgive me for this. Okay, but we must have it. And also the other relations we can you can verify that some typical characteristic scales but now what is interesting to do, and is to follow the particles of turbulence itself. Okay, there is a if you check like particles in turbulence and acceleration of particles. There is a really large variety of works. Okay, well, they usually basically they use like the test particle approach test particle is very popular. You have a field assigned and you follow particles integrating that's the first exercise all of you should do it if you're interested in this topic. What we do it here again is the opposite. We integrate particle integrate the blast of a question and follow now the real particles from lasso. At that time when we did that was pretty new this thing because most of the people were working with test particles. Now once you have a code which is simulating turbulence, and you have also trillion particles. Now we thought it's not easy, even easy to extrapolate a few particles from such a huge code for the, there is an architecture to follow and to extract info, very small piece of information because particles are flying in your processors in different pieces so you need to collect them and you need really good students to do that, not me and I know how to do that actually, but I pay people for doing this. So you extract some particles and that's what happens in turbulence. This is a to this simulation where we are varying beta. So you can see that particles do this erratic motion, like, during people do this actually tool, because it's a typical of diffusive processes, you can see here that is really unpredictable, and also the spiral, of course the spiraling can be really completely unpredictable by looking at the force acting on the particles, but what you see from these cases we are bearing the plasma beta the compressibility and the typical alpha velocities. If you go to very small plasma beta it becomes more, more crazy the motion but you observe a smaller normal radius of course with smaller beta, you are exploring other regimes other energies. The normal radius is less visible is there, but it's just smaller. Okay, this is the main difference. Yes. Yeah, if you believe in the simulation is plasma particles. If you trust it. Okay. So really, what you should do. The first thing the first classical problem of physics, thanks also to Albert Einstein will, who will come back later is a is a study of Brownian motion. If you imagine this particle is doing an erratic motion here again, and it will keep memory of the field, which is under is driving the particle. So I can teach time I can measure the distance. If I start from here and I do my erratic motion at each time T I can measure the distance from my position to the original position. Okay, and I can square it. And the typical and the classical theory if you if it the most is really the motion is really ergodic, then it's going to go in a diffusive process like this mean square displacement is going to be proportional to time with a coefficient, which has all the significance. This is very, very, especially in talking about plasma says very, very important to evaluate for example the bomb diffusion coefficient at the boundaries because it tells you how much the plasma is spreading from the center of the column toward the wall. It's the same, except we don't have a such a nice theory yet because our plasma is less well behaved in a talk about talk about is more magnetized. Maybe we know much more about laboratories done in in space. You can measure this with pieces of plasmas and of course you can then you have to be in to see if there is difference by being in energies. So there are particles that are low energy mean average energy, high energy, and as a function of energy time as a function of energies, you get these slopes and all of this law at very large time go as like time, that would be power one. So it means that for very large times if you follow this particles, you are undergoing the Brownian Brownian motion, a diffusive process. Which depends on turbulence. So the main question is like, since this diffusive process is a nice law and tells us how much the spreading goes as time, how much is this value D, the diffusive coefficient and how it depends on turbulence that's the main question about So we will study this and there is a theory that was being developed much before this on test particle has been very well tested by be better Matthews and roof follow is like is a quasi linear theory where you the diffusive coefficient depends with some course in hypothesis depends on the properties of the field. If you want to understand the diffusion of particles, you have to use the spectrum of the field, which is driving your particles and is in is embedded here is pretty is a little bit complicated to solve this a question not for this case but generally this Q of K is the power spectrum, and here there are some other here you see what is the difficulty. Many of you will solve this in mathematics, immediately or with math lab is that the confusion coefficient is both on the left side and as a on the right side so you have to do it numerically. You can approximate we did it numerically but if you approximate with for very small correlation times you can obtain some sort of an average diffusion coefficient, and in here is this line that you cannot see, and also there's a line here, we can interpret the results by means of this quasi linear theory, where we use the spectrum that we can observe. So, really, this is interesting for far away objects, because if you look at some pieces of plasma that spread away, and they're very far away, we do, we do the other way around. We look at the spreading we cover calculate the diffusion coefficient, and probably we can extract information on the properties of turbulence think about that. You use the experiment the other way around if we can establish this link between diffusion and turbulence then we can, we have a telescope. Okay, so what happens to single particles and what and today energy. Well, this is the journey of a particle that starts from here. Then, and this is the islands that are the populate turbulence. Generally, some of this particle are really trapped in this driving force like in a washing machine. They start to travel around them and while they travel around them. There is, there are sites of magnetic connection, which is an ingredient of turbulence like it or not. And when it will hit the x points when they are engaged with I will take a kick. Take a kick. And when they hear you see there is a reconnection rate when the particles during the journey hits a current sheet. It gets against a lot of energy. So particles are like trapped, and they can take a lot of kicks, while they are trapped into in this in this closed surfaces, until they will hit a current sheet that boom will give them a little blast in the z direction and they will be accelerated at the end. If I start from a distribution which is Maxwellian, I will start to have tails. So this is the main process. The mechanism is that particles which have a lot more radius, enter to the other region, and I will have some sort of strange resonance. So this is a reconnection region. Basically it's not on this slide here and here there's a current sheet, you see, with an X point this current sheet as a typical characteristic length. Okay, we call it lambda. And if I am particles that are already large energy. I cannot see this current sheet. But if I have a particle, which is like thermal and as a alarm or radius which is on the order, if there are more radius is on the order of the current layer, then they resonate with this. They get a lot of kicks locally, and they have been accelerated. This is our model, which is popular also for test particles and think we is the way that can accelerate particles. It's very basic. As everything that works must be very basic. Yes, yes, exactly. It's very basic. And this can also also hit the plasma. But remember that these regions here are very small volume filling. There are many of them but very small. So it's more efficient to accelerate than eating. Eating can involve maybe other processes. It's an open question though. This is very good point. So this is what happens to particles. Now we study paths of particles. So a bouquet of particles that start from our piece. You imagine you color your paint some particles all blue in a very small region and we have an infinite number of particles and then you follow it in time. What happens is like really there is a very explosive spreading of particles. Okay, so that's what happens if you zoom in here. You see this is interesting that at the beginning they stay very close then they do a really exponential dispersion. Yes, and here with this is the next the next slide. So that's what first we look at the experiment. We see that you see there is like an exponential like divergence probably power law we will see it. And then these are the gyro centers. Imagine the particle is flying for people that like gyro kinetics. This blue spots is the gyro are the gyro centers. Okay, we plot them because are very good. We don't really we are not really interested into the gyro motion can be a little bit boring right it's just a disc right and so in time is diverging. So modeling, modeling, we went back to the theory before we were speaking about single displacement that is a particle and I look at the particle with respect to the original position. Now, we do use the statistics of pairs, a couple, a couple of dancer like in, in a tarantella which is like a typical dance which is popular in the south of Italy, close to Sicily where they come from people do this is very is very similar to observations, because people like male and a female they spin around, and sometimes the male leaves the female goes to another one actually happens, right, or vice versa. And then they change up exchange couples, and there is a spreading the but the circular motion is our gyro center motion. And the geocentric displacement is the center of this of each circle so so they spin and they depart. And if you measure the distance between blue red male female in time, this spreading is not like a power law, like time. It's much higher power, it's a power law with time to square on time even time t cube t power three in three dimensions in two dimensions a little bit reduced. But why is that it's a super spreading it's called like super diffusion. This is we didn't discover that we didn't discover many of the things I'm speaking about this is very ancient, and it goes back to Richardson. So Richardson was a brilliant scientist. He was a experimentalist theorists. He was extremely good in doing calculations, but he was really attached to nature so one day he was looking at the volcano emitting chimneys. And he was looking at the dust always spreads going up into the volcano and taking it was taking like data, just by I okay now the dust is like few miles, 10 miles, 20 miles. He was looking at the spreading of this dust as a function of time, and he said, hmm, the probability that two pieces of dust at a time are, and at a time t and a distance are must obey the fusion of a question, which is, it has a kernel here a diffusive coefficient, which is not an independent of our if the spreading is so fast it means that the motion is not diffusive yet. If you if it's diffusive here that depends that this guy is going to be my diffusion coefficient and the solution to this fucker plank a question is going to be a Maxwellian. And I will have classical diffusion. So if I ever the motion without memory for very large scales will be classical diffusion, and I will have a coefficient here that does not depend on the scale. But if we are in turbulence, there is a steering motion, probably there is the field that gives memory to the particle of this coherence and accelerate them. And it gets into adding here a coefficient diffusion which depends on the scale, and it has the information of the spectrum. Okay, you see the difference. This is very elegant, but it's amazing, in my opinion the work of, of, of Richard's on because this was pre prior to come over three years before come over. And he used turbulence as characteristic scaling laws, and he used this relation here where gamma is two thirds from Kolmogorov by accident. He said, I don't know why here there is a diffusion coefficient that goes like R to two to four thirds. I don't know why I don't know why three years later Kolmogorov was listening to the paper, and he published this first paper where he gave the recipe, but he anticipated the result he used it accidentally, and he's Okay, but because I think it was a genius. This is the kind of solution to this pair diffusion and it does it's essentially a distribution of particles with items, the items are the accelerated particles. And so it's like a stretch exponential you can give names to this, this kind of solution probably there is some. There's a lot of botanical on this but I don't know it, the recently really the name of this complex division function so really we can interpret the results. And this brings me now to move from my particle study to another subject that we will look at now is that the interaction of turbulence and shocks. I mean, in the literature shocks are ubiquitous as much as turbulence. Okay, everywhere here there are some examples. I'm going to conclude it is a chandra. Supernova remnants. And here, okay, okay, the typical example of turbulence that we have spoken so far. The question is now what happens when I have such an interaction between a shock and turbulence, like in the our bow shock, for example. We want to do this from different perspective but of course, since we are thinking about astrophysical plasmas we want to look at the collision less interaction. So, this is a cartoon, what is happening here how do we imagine this interaction with a cartoon. So, this idea goes back to important works by Zank and collaborators Gary Gary Zank, but he did produce a lot of these cartoons, he never did the simulation. We did, but, but the idea is very it comes from from him, where he imagines that I have turbines with a finite length. So, what happens doesn't know about the shock. It doesn't is already is fully developed is nicely going from the sun with Kolmogorov spectrum, for example, and then at some point encounters the shock. Okay, so we want to repeat this cartoon now numerically. The shock moves fast. One way is like some people are tempted to do this. I can send a kinetic shock against some random motion. This is what I will do first, but if the shock moves fast and the field is random randomness is not turbulence turbulence it has structures. Okay, so I want to do it better this I want to send fully developed turbulence already in a fully turbulent regime against a shock with coherent structures already. So I don't have a current sheet and the connection maybe it's just a good starting point but it's not so so close to reality, we want to do it as much as realistic as we can. So that's what we did. We have a fully kinetic simulation with particle in cells, we sent a shock. And this is the case in which the shock we are sending from the left turbulence like solar wind and the shock is moving from the right to left. So it's a system which is sitting like this there is an observer here and there's a shock coming from left and a turbulence coming from right. Okay, both of them they moved. They really is a squash of some time and here you can see, we prepare turbulence by with by using an image dissimulation, we use an image dissimulation, we run it. We have nice coherent vortices, we extract all the fields from the image dissimulation and copy here on the left side inject into the kinetic simulation. So this is so we will have really interaction between the image the kind of cascade with a kinetic shock. The thing we are in mind. And what happens between here, here we have like the case with the shock with no turbulence really doesn't look like anything exciting there is a shock moving. That's it. Here we have like 40% turbulence here like 80% turbulence this is like solar wind like, and here we have like 200% turbulence where I mean be zero is the normal to the shock we are studying oblique shocks. And just to have all the possible angles there is a mean field here, and we can set up a be zero and the amount of turbulence, we are keeping it we are providing a recipe with varying the level of turbulence. One can be useful for a CME in the corona low beta. Another one can be useful for the magnitude. Another one can be useful for the supernova remnants, where probably the mean field is very tiny in the interstellar medium and you have a large amount of magnetic field fluctuation not really a mean field but a lot of fluctuations. Okay, so here what happens to particles. Now, here delta be zero over zero is this distribution. Actually it has two peaks. Why two peaks. Many of you that to like lesser plasma interaction they know that if I have a shock, I form a shock. The shock is going to accelerate particles and create it as a thermal population and create a beam in front of the shock is reasonable right. If there is a shock there is a electric field which has accelerating locally plus mass. And so if the shock is moving while it moves here a particle accelerated. And there is a beam of particle accelerating in front of the shock. So the picture is very clean and is this this one is the thermal population this is the beam, which is produced with the red one. But if you increase the level of turbulence, all the population they spread, they diffuse, you have higher energy, high level of energy, much more like a either order order magnitude. So turbulent the interaction of turbulent with shocks is spreading this population mixing and is accelerating. So why because shocks gives a kick to particles particles receive other kicks from turbulence and therefore you have a excellent acceleration mechanism with the shock and turbulence that interact. Okay, that's how we see this observations. Yes, what we observe like if you integrate all over the volume. This is the case with the very small amount of turbulence to population. This is because there are the particle accelerated by turbulence and the particle of the thermal motion coming from the solar wind. But if you increase the number, this is a trajectory of a particle in the velocity space, difficult to imagine a particle in the velocity spaces is erratic. It's like a sub space we never think about. If you take a particle in the beam in the in the case of zero turbulence, it will stay in the beam. Once it's accelerated doesn't care anymore. But if you increase the level of turbulence then particle can migrate between the two population. Okay, you see that is really again a cartoon, you have more communication about these two populations because turbulence is spreading and is accelerating the system. So, here, this is just average it over the volume but if we look locally your pictures where the data really look like MMS data in each little piece here, the distribution function as this crescent like distribution this beam, and it has multiple beams. There is a lot of fun going on there locally. Yes, that will be our first order the shock second order when particle gets more key. That's how the Zanke collaborators proposed this except now we observe it now and then, and also we, I will say that we didn't stop at just reproducing the idea and measuring the difference. I will say that the first thing that pops up from the analysis we are showing is that really we have diffusion, but we have diffusion in space because particle can be accelerated and they move with following regions. Yes. Why did not. Oh yes. Excellent point, because indeed the distribution before there was a even lower energy turbulence can accelerate. It can be a good paper we can write. Okay, because particles are concentrated can be can cheese their, their energy, they leave their energy to to to fields and can therefore can lose can be become more quiet. This is another mechanism. We are always excited about acceleration but this is another interesting point of view. In any case, there is a spread in space and in the velocity space. So there is a face space diffusion. Yes. Acceleration or negative acceleration, which is the deceleration. Okay, it's not that you get older. No, no. Wings and also you also have some local eating so you're also broad and a little bit your distribution. Okay. You have just a wider. Accelerate but yes, you put the energy to absolutely. Yes. So this is the idea. You have spreading. Okay, do you agree that we are spreading energy both in space in velocity space. And if when is this kind of diffusion. Now it's a new kind of diffusion. It's a multiple multiphase diffusion in space and velocity space. So we really want to go back to the good hydrodynamics and use coarse graining. I can, I can take a picture of a bunny here, and I have an image, and then I want to know how the, and how the image smears. So this solute, how do you say that you see the solve. While I look at them probably with glasses or without glasses I'm coarse graining. When I take out my glasses that I forgot it today. I'm course again in the image and become more smooth. It's like a diffusion in this in this I'm changing the grid of this that's what, similarly to the process we are going to describe so. It becomes a sphere. If you go to grain too much. If you're really blind, then you see a sphere here. But here, if you, you see this is an isotropic one it changes while I change the scale of course graining course graining is very powerful in hydrodynamics, but now we have to invent it in a, in a glass of war. So I'm coarse grain here in space in this in each point of the space. I have my velocity distribution function and I can average the particles in a little box of Radius R. So velocity space, I do this in the physical space in the velocity space I can build some classes circles around the origin. So of my distribution function, my distribution function will have the beam and the core. And so I would like to course grain by integrating I want to separate the two population beam and core, because I want to know how much of the blue particle that are called go to the beam that is red. And I was also in integrating space by integrating the, the, the changing the length, how much well in the initial range. So I can scan my initial range of turbulence, right. So, with a little bit since during the pandemic didn't have too much to do at home. So, you can take the glass of a question, you can use a filtered velocity distribution function which is something that can really experimentalist may like it, because you can the bdf in each point, and you can just integrate by shells is easy to do. I have channels as Lena was showing before I have channels of energy, I can just integrate from zero to a channel and leave everything outside it. And I can integrate in space. I love it is because then they're all the uncertainties the Taylor hypothesis problem, all the technical problems will be integrated will be reduced suppressed statistically. So, we multiply this velocity distribution functional by a kernel, this is more refined, which is like a, like a Gaussian. Suppose it's a Gaussian by default. Then, once I write the course granted lots of a question, I can integrate from zero to energy, which separates the two populations. Okay, so I can compute. We are going back from the, the first exercises you were doing what you when you were a young plasmist, where taking the blast of a question and obtain the fluid moments. Okay, that's the first exercise ever obtained the match the question starting from lasso when you do that the blackboard there is a teacher that waiting for you to give you grades. Now we did the same except we do it by filtering at a scale L and giving energy boundary. At the end, you will obtain this equation is a continuity equation for the number of particles at a particular scale and inside that sphere. So this is a conservation law. And here, this term is like is the rest of the course graining is small scale effect. So there is a time derivative. There is a divergence of a flux like in the conservation law. There is a term which is proportional to the E and V cross B force, which is accelerating particles that's the one we were looking at, and some small scale rest. So this is the, like it is like in a course grained hydrodynamics. Yes, it's here. There is a divergence then I use the Gauss theorem. Okay, and then I have that divergence on the shell. Yes. That is the picture. And then what happens when this shop interact with this and move a little bit the distribution function behind the shop. Oh yeah, this is a yes it is but we are studying just what is happening before the shock ahead of shock in the upstream in the downstream we will see in a little this is everything that the region, the near interaction of the shock with turbulence just in front of the the things change, of course, we will see. Now this is just a theorem. And, but now we can do a mosaic of two mosaic of the turbulence, we do a mosaic where you can measure this terms in each in you once you we have did we have separated our physical space into a grid of size, maybe in the correlation or my smaller than the correlation in the national range. Things do not change in the national range from this is interesting means that is very valid to do this analysis. So, with green is regions where I have positive divergence, the plasma is diffusing. The orange one is the, the regions she was speaking about when plasma is compressing. So there are compressive effects in this of this thermal more of this thermal part, but the most exciting thing is the velocity space divergence, because we computed very coarse grained. So, in the here that is the shock in the upstream region the shock is interacting with turbulence and I have region of acceleration and this iteration. I can measure now blue region where there is the thermal particle are becoming crazy, or where crazy particles are becoming like going to the thermal mode. So we can see it and measure it, and this is a strong correlation with the, this is the parallel electric field. Of course, when there is parallel electric field can make the two, when there is a parallel electric field between the thermal and the beam, then there is particle flowing because electric field accelerate particles. Incoming field and then there is a cooling. Okay, so we have this mosaic and once we didn't do yet this on the experiments but we are planning to this is very recent. But the question now is what happens to structures when it's coming from the solar wind and entering the shock from the experimental point of view. This is like two points. It's difficult to connect the plasma of course we don't know if the plasma is the same. It's the same partial at one a you are inside the magneto sheets we can never really be sure we can look at the angle a little bit we studied some angle that are aligned with this magneto sheets region but we are not crazy. We have the properties of turbulence upstream connected to downstream after the shock, and we want to see what happens statistically to coherent structures when they pass the shock. Okay, since we have the experiment. So, first thing we can apply the technique we introduced yesterday to measure like sort of like intermittency and the reconnection events in both cases. Before recently, we invented a local analysis of magnetic electricity. Okay, if I have a swirl. This will have a magnetic field lines that are like tide, and this kind of magnetic flux rope. It has magnetic electricity. Okay, by definition, if there is a three components of the velocity field of the magnetic field. So we invented a one day measurement of this magnetic electricity that is can run is like a spectrum of magnetic electricity. So, in the upstream wind, there are a lot of spikes for the PBI for the magnetic field strength intermittency, there is a lot of reconnection and spikes. Also, there are, there is some an amount of this magnetic electricity, there are some, like, coherent structure that are very, very tiny. There are some flux tubes in the solar wind are there is there were very well measured the reason we are not discovering this. But if you apply the same analysis in the magneto sheet, I mean you still have a lot of intermittency and your magnetic electricity increases. This has been found by other authors even before us so the wind comes as the intermittency turbulence after interacts with the shock, it acquires electricity. Okay, the structures are changing as Lena was saying, from the wind to the to the to the magneto sheet. Yes. Yeah, we have we have done a series of it is a day they change, of course, with quasi parallel and this is another, and we're just starting the oblique case to remain like 50 50. Okay, but, but now, what we've done is just we have our experiment, we send to the shock against turbulence, and now we can measure the electricity in the simulation. And what happens when turbulence is this are a little piece of turbulence which is hitting about the against the shock which is on the right side. When the turbulence is hitting against the shock. That's what happens. It happens that the magnetic electricity is in is increasing. This can be written. I didn't add the, the, the slide where it's. I'm going to write it. At least it is like, HM nice group is a lot be. Okay, it's a density of electricity. This is an ensemble average. Okay, is the alignment within the bottom a magnetic potential. So I have be is equal name and democratic field itself. If you from MHD from compressible MHD you write down the equation for you have the question for db dt equal blah blah blah. Let's write some some of this blah blah blah, because I otherwise I will run out of time. And then this will have you got grad B minus be dot plus you got grad B. Plus be divergence of you. Because now is no more is no more the case is no more an invariant. There is a term related to the divergent why I'm so excited about this divergence of you because of the shock. We have divergence of you. It is observed. It's sort of like the pile up. You observed. Okay, so, so, so we hear we have like the a d t is equal to you cost be. And if you with the, again, during the second part of the pandemic, you do the age dot be d t is going to be like there is a term which is proportional to grad you. So, they'll Yes. We have a purely MHD, but then the kinetic scales change the here we cannot see we don't have a larger system but as she was saying if you have a larger system, the correlation length reduces because the system, but we are not we are interested only to the coherency. Why, because each one of this particular solution brings together an announcement in an isotropy. It then also the kinetic effects are announced the magnetoships. And this can also be obtained by a law similar to this. So interaction, but you cannot this only if you make your shock interacting with turbulence, where you have like a Maxwellian plasma with small electricity. And the shock becomes more electrical and more kinetic. Okay, this was selected as a decor page this one, one go like thing it is. It reminds you that the starry night a little bit as a cover page of pins. We don't know. We don't know the with this kind of model we are just very delta B over B. That's the only thing we do but we have to do it more relativistic with you go far away or other regimes. Absolutely. That's. Yes, yes, yes, yes. It's part of the inspiration of this work. Okay, what do you observe at the, at the heliopause. Right. Okay. Yeah, there is a big community working on that of course. So, okay, now we are moving away far away. We want to understand if we can export some of these things to the behind the oil sphere. So, supernova remnants. So there are many people excited in my department of supernova remnants. They may basically do a lot of simulations of these explosions with the mh decompressible god of like methods that are very dissipative but but good enough for large scales. They studied this blast waves. And now they propagate. So there is a initial condition with a really high density plasma and I very high pressure, you leave it numerically and pros with this blast waves. What's exciting is because if you look at the x rays of Chandra here, you see there is an anomaly, anomaly, the at the density, pretty much is at the higher at the pole here, and here. Okay. So, why is that is plasma facings. Now we can use simulation to probe something which is very far away. First of all, if I do supernova remnants with nothing around. I just share a boring shock that doesn't look like this. Nice picture here. But here is a imbalance between this region and this region here. It can be explained as following here we did the simulation with a shock interacting with turbulence, a meeting which is turbulent because the interstellar video must be turbulent. So the Delta be over be, but the main field is out of the plane. If the main field is a simple to the simulation and it's good that this today, because it helps us to understand this with the main field out of the plane, the shock propagates, and it will be pretty much symmetric. But if you flip the main field into the plane, then you can move this mean field until you obtain a correlation with the image. And actually, we measured you see now that they meet with the main field in the plane, it produces like far magnetic sonic modes at the shock, and there is an imbalance. So we found a very good correlation with this kind of a simulation here we are using a Monte Carlo of simulations to understand what happens in the supernova remnants, adding turbulence and a shock. We found that the, here the magnetic field it has a, it had a degree 30 degree with the horizontal line, which is in agreement with the observations. They measure this magnetic field from other kind of things that I honestly don't know, but the agreement is very good. So we are finding that the magnetic field by a Monte Carlo of simulation until we have some sort of an anisotropy with them local mean field. So we vary the interaction of turbulence with us with a shock by using adding the mean field ingredient. So, since we know that where is the trick, the trick is that if I have a mean field I know that there is an isotropy. If I vary the mean field. I, I can see the fluctuation I can use this fluctuation to understand where is that where is pointing to. So that's the use of a simulation to extrapolate information from objects that are very far. So, oh, okay. There is a mean field. There is a mean field, but where does it come from. There are many models for dynamo. Each every model for dynamo assumes a seed of magnetic field you must have a small magnetic field, then you can add the turbulence shake your fluid, and this magnetic field will permit to larger scale. But the question was, where does the seed come from. Nobody knows everybody assuming nobody cares probably. I do care, do care. The seed can come from different. There are several explanations, I will argue that most of the plasma in the universe is collision less. So I have mostly a kinetic plasma. And so we did an experiment suppose I have a kinetic plasma collisions less, and I don't have any electric field and any magnetic field at the beginning zero. I have a plasma. Probably, I can steer it suppose I can steer it. I'm not now I'm cheating a little bit because I'm gonna have to stick. The Supreme. Okay, the Supreme will steer my plasma thermal convection convection over a star, or maybe the drug of neutrals with another winter the stellar medium suppose I still a plasma. Do you agree we are starting from up a step before m hd m hd knows that how to amplify it we want to create the first seed now. So we are stealing a plasma which is perfectly unmagnetized. And what we observe in time if I measure the energy, this is the total energy this is the bulk energy with this decreasing from zero. I have an exponential growth of the magnetic field from nothing. And this is the magnetic field spectrum. It starts from here at zero. And then it piles up piles up is the magnetic field is going to larger scales to smaller scale and is amplifying. It's sort of a local a dynamo starting from zero from nothing. But so where is where it's coming from here. This is a pattern. This is the typical flow, and this vast here is our local regions of magnetic field. So let's go back to laser plasma interactions. These are very familiar patterns for people that do laser beam interaction, for example, because it's a typical pattern with a sort of like serpentine. They are related to a very famous kinetic instability. Before I tell you the who is the killer. Now, let me show you some more data, you can measure from the arms load the various contribution of course you don't have magnetic fields. Okay, so we measured all the increase at a given time, the beginning we measure the contribution to all the homes low. We take all the possible contribution and we do a distribution. The same term is the divergence of the electric of the pressure, electron pressure is absolutely is incredibly large, the magnet, the electric field pressure, the electron pressure, sorry. The electrons became me produce a lot of divergence with the pressure via the following mechanism. So this is really too hard to explain it with a slide but let's try to follow me like this. So from lots of equations, I can take moments, and I can take the moments for density bulk flows and for the pressure tensor. Very easy exercise so far. There is a lot of but the exercise is very easy because I don't have magnetic field. Suppose you take the blast of moments without magnetic field, you will have this equation here for the pressure tensor. And when you do that, if there is the pressure is Maxwellian, the only term is going to dominate and you drive it without a divergence without shocks, for example, you will have a simple equation for the pressure times the stress tensor. So here is a is a is telling you that the pressure is varying according to the stress tensor. So if I put stress to my plasma, my pressure will become an isotropic. The electron president pressure becomes very an isotropic. It reaches our instability threshold, which is the way balance ability, the electron way bill is very fast. So I squeeze the electrons. They become too unstable. They produce the way ball instability that produce little magnetic fields. The process you get a current. Yes, yes. It's very, at the end is very basic. We measure the way bill growth rate and the way bill mode, I will call it. And so the picture is just as following. So there is someone which is staring my, you know, here it is a nice Michelangelo paint. I'm staring my, my, my, my, my plasma. The plasma produces the turbulence as local shears shears are too intense. They produce an isotropic and I drop isotropic gross is became a way ball unstable produces magnetic field magnetic field interacts with the, the, the turbulent it gets amplified in an hd way. And this is like a carno cycle that produces con keeps going on, but we created the seed of the magnetic field is not a novel thing to use the way bill to create small magnetic field but here is a more in a turbulent like fashion okay nothing new on this process way bill is very up is very efficient way to produce things. Now, I have like six minutes. Okay, six minutes. So now we are going far away. We saw the supernova remnants supernova elements of mean magnetic field we are trying to build the model to understand the mean fields far away from stars. Now we want to, I want to concentrate a little bit on this picture since in the past years I'm working on general relativity and numerical relativity. So, this is where it was very exciting when we was observed everything is concentrating on this black hole so tell me what you see here, you see a black hole, right. There's something black, but really what you are seeing is the orange shiny part around it. And now I'm going to speak about that part which is a plasma. So, we will do plasma physics in this region here, there is a lot of plasma physics around black holes. And I will say that this is like where research stands out now for astrophysics, for astrophysics trying to merge the communities working with different coming from solar wind probably, like probably me or other people and people that work on compact objects to do so, you have to really start from scratch, going back to Albert Einstein. So, you have to know, you have to start a little bit from the theory, which is beautiful and take some, some time to be digested digested. I don't know if some of you are familiar with the general relativity. Essentially, it's a, it's a model where matter tells space how to move and space tells matter how to move. Okay. This is a rolling wheeler definition. So, when I have really, really high densities, I can change my space time to orbit. Okay, that's the main idea. So close to compact objects, I can produce really singularities where the deflection of the space is absurd. Where is my Einstein theory is incredibly close to Navier stocks, indeed, Navier stocks is somehow embedded into here also not only Navier stocks and HD everything is it is a question at least except of quantum effects. Okay, so, at this level in the framework of general relativity everything about my energy my bulk flows is in the distance or team you know, so obviously, we want to do some crazy things. We want to simulate the Einstein equation on a supercomputer. First in vacuum. So suppose I have just in vacuum it means that the energy, there is a distribution of matter which is like point wise I want to see what happens around this point wise distribution of matter. So this is the curvature distance or here tells me all the space is not flat. The question is going to be equal to zero. So, to solve this equation. The problem is that nice, the Einstein equation is, is nicely compact, but it's horrible. It's horrible to solve it numerically is not really good to be solved numerically, because the derivative in space and time they convolute. We don't like it. Right. We don't like it we want derivative in time of something equal operators in space, then we like, as usual, everybody likes this. So we projected the questions on ISO surfaces at a given time, the technique is by bound goddess Shibata Shapiro Nakamura, and the equation a single equation which is very compact and elegant, then proliferate in a number of like almost 20 questions that are fully linear, the Einstein field equation is a nonlinear equation of order five. So imagine Navier stocks as nonlinearity order two, and we're already going crazy about Navier stocks, imagine something that has no linearity order five. So you need to treat this equation with a very carefully we have some numerical techniques, and we built a code which is spectral and solve as the Einstein field equation. The next finger is like spectra filtered numerical gravity code, which can handle block black hole black holes. So we did a lot of you need to test it for some years, you need to convince people that is working produce the right gravitational waves. The idea is not that to study what happens in near nearby compact objects, and we first want to test our model with the typical well known things like spiral in spiraling of black holes. But we did also went a little bit beyond that, and we did the three body problems with three black holes, because probably at some point they will measure radiation coming from three black holes, because there are galaxies that are dancing at each other, and there is more than one black hole, and they may produce a very small amplitude, radiation, gravitational radiation, so we'll be very small. So, next detectors like Lisa may will be able to capture what I'm going to show you here is like the typical signal to black holes that they dance and then they collapse. And when they collapse in the Einstein field of relativity, they lose momentum and mass, which that is a emitted away as a form as a gravitational waves. So in a, in Newton's law of motion, they will continue to spiral, but in the Einstein field equations, they will lose energy through radiation of gravitational waves. And it's a perfect emitter. This, okay, like a quadrupole limit. So we numerically we simulate all of this long simulations I will guarantee it's 1000 cube, and we put some detectors afar and we measure this gravitational radiation, but we also study the three body problem. This is the typical trajectory of three black holes. And when they spiral every time that they interact, they are spitting some mass emitting waves. So in a nonlinear way, because I'm suspicious that this is when there is a blast, the final blast when the three black holes come together and make only one black hole, and they really emit a large wave. When we in all of these experiments, we measure the power spectrum of the, of the wave. And of course, if it's a nice way, it will be only one mode. But if you have more nonlinear effects, you will probe the reminiscence of this wind is going to be like the solar wind. And maybe we'll export far away some information of this nonlinear interaction and probably there is a cascade in the of the metric. I mean the face, the, the, the metric, the metric tensor is creating small scale features like just like in turbulence. Okay, that will be that's a novel concept. I think the spectra here this is under review now, and there is there are some slopes there. I don't know this is novel thing. Yeah. I don't know if it's an energy cascade because really, this is a measure of a tensor, which is deflecting by this producing some nonlinear features. So it's something novel. We don't know. In our meeting that it was the Nobel Prize keep torn the ones we were asking something like this he said, hmm, I don't know we all, he said, we will see if we all about this. And, and it's a novel thing, we don't know, but no comment. Okay. Okay. So this is like an animation we are doing now we are coupling the gravitation Einstein field. I want to give me one minute and then I leave the then the gravitational wave to navier stocks in a self consistent way. This is like solving like the g mu nu equal T mu nu with only navier stocks. This is a typical example of turbulence produced by this kind of steering and not gonna speak too much about this. But what we did is then we went beyond and started to study the plasma in the neighbor of one of single objects and do it simulations. But I will leave you with this very recently working with the group from the image of the century there. We are doing simulations of plasmas around compact objects by using our pick codes by using a code which must be very specific, because otherwise they are very close to the company but it's special relativistic. So we have the big black hole, and we take little cubes around the black holes so locally you can apply just special relativity for Maxwell equations. So we are solving the Maxwell equation for particles in the vicinity of a black hole. And again you find the nice vortices, they are a little bit different. This is very high here of the thermal motion we are studying these effects. And what we observe is that really turbulence near black holes with these are typical trajectories of electrons and ions electrons gets very accelerated, like 0.99999 C. It's becoming very high accelerated and produces power law. Why these power laws are important because from this distribution of energies, we build the model to understand the shadow of the black hole. The emission of the black hole must be modeled through the this power laws on the spectrum that's where now is the challenge. We want to study local effects with genetic to extrapolate information and reproduce the real shadow of a black hole with higher resolution. So that's what we are doing now. And okay, this is another animation of this super ultra relativistic turbulence with particles, and is our particles are accelerated and they do the nice journey and get kicks into vortices and gets accelerated. But instead of the conclusion of my course, which are always boring and you want to hit. I would like to acknowledge all the collaborators in my in the in the recent years, because I always like to keep science multidisciplinary. I try to steal something from Navi stocks mostly and bring it to mhd from mhd to Navi stocks from from both of them to particle physics. So I do collaborate a lot with space missions. I even married one of them. This is my wife actually. Yes, and then so we do with people that really know how to manage data, but then you need to work with theorists like problem in in Matthews that are plasma physicists. They're working with me in the past years, but a lot with the hydraulic engineers, a lot to because I work on the river flows where we measure like third order laws in an isotropic systems. So, and I am just a physical and finally with general activity. Here there's also a student mind which is very good and very shy when I took it, I take a picture of you hit turn it right. I had a picture of him but he's very shy. So, thank you for your attention in this three courses and sorry if I was late. But it's a very uneven distribution mostly there. So we want to have to women in the process. And after who are the senior most people. So, but you're not a moment. And you said you're a professor and you're a professor. So, so I think that's what you like this. You want to see the most amongst you. How long have you been a professor long time. You. You. All right, so we probably don't have. Please come and join us. It's, it's not a cry. Thank you very much.