 And so we will talk about the evolution of current important city sheet in a collision less plasma turbulence. Well, I'd like to acknowledge the person I've been working with for this work, which are Dario Borgogno and a colleague of mine at CNR, Beatriz Ciacchilli, his previous master students, and Massimiliano Rome from the University of Milan, and Luca Comiso from Columbia University. No, this way, right? OK. So an introduction. Current and important city sheet, I hope I convinced you yesterday that can be thought of as a fingerprint of magnetic reconnection and turbulence, respectively. So here are two plots I showed yesterday. This is the formation of a plasmoid. And this was a current sheet elongated and thin, and who gave birth to a plasmoid. And this was a Borticity sheet, which was developing turbulent structures in a simulation of magnetic reconnection. So magnetic reconnection is ubiquitous in several astrophysical environments, where due to high Reynolds number, the transition to turbulence is unavoidable. So in the presence of a strong turbulence, a magnetic field lines reconnection is continuously met along the flow, because you are forcing the magnetic field lines to get closer and closer and to reconnect. Both the processes, magnetic reconnection and turbulence, involve energy cascades towards the small scale. So in some sense, we can say that the magnetic reconnection is an intrinsic element of turbulence and of the turbulent cascade and vice versa. I'm sorry that it's not possible to see very well this plot. But let me say that many works have analyzed the turbulent cascade in resistive MHD, defining under which condition the plasmoid instability influences the energy cascade. And here is a plot, which I'm sorry, it's not very clear to see, but it's a plot taken from a Pierre Eilk paper by Dong and the co-authors in 2018. And so they made a turbulence simulation of plasma turbulence or resistive plasma turbulence. And by these following and subsequent zoom, they were able to show that the structures that emerge from the turbulence cascade are generating plasmoid. So here you see there is a smaller magnetic island, which is a plasmoid. So in such a way inspired by this work, we decided to look at the same problem, but in a collisionless plasmas. Because the collisionless plasmas are most of the time a space and astrophysical plasmas are collisionless. So we analyzed the development of plasmoid in a turbulent collisionless plasma, where the reconnection mechanism is provided by the electron inertia and not by the resistivity. As I tried to explain yesterday, these two mechanisms can give birth to magnetic islands. And we see that actually we found a much more complex situation with respect to the resistive case. Because due to the presence of the strong velocity shear, the typical plasmoid formation has to coexist and compete with the Kelvin-Henmos instability, which I introduced also yesterday. So the outline, well, I will briefly go to the collisionless model I introduced yesterday. And I will review with linear digression the shear flow effects on the reconnecting instability. Well, the numerical setup will be introduced. And then we will try to elucidate when the current sheet disruption goes through the plasmoid instability or through the Kelvin-Henmos one. We'll see how the energy cascade go in the plasmoid magnetic turbulence regime. And we compare this theoretical prediction with the energy spectra. To conclude, a comparison with a resistive case will be carried out. And basically, this is the outline taken by an astrophysical plasma journal paper we published a few months ago on this topic. OK, so the collisionless model, I will go very quickly because it's the same I introduced yesterday, the reconnection model. So this is the question which comes from the generalized where just for the comparison I will make in the end, I kept the inertia term here and here and the resistive term. This is the plasma motion equation. So the vorticity is related to the stream function, which is defined here, and which goes at the across bediff velocity. And here is the current density. And the magnetic field is assumed to have this form. The equation are integrated by a numerical solver, scope 3D, which is a solver for collisionless plasma equation. And it is a three-dimensional, even though we just run here in the two-dimensional case. It's pseudo-spectral explicitly in time and MPI parallelized code. So let's recover what I told you yesterday about the big legit stability. And we want to see the growth rate of the reconnection instability and the Kelvin-Helmholtz instability for a current sheet-like equilibrium. These things can be found in the Biscump book for the resistive case. I will show you how it goes in the collisionless case. So we have a magnetic field, which is the tangent, the blue line. And we have the big legit, which is basically invisible, but is given by the yellow line, which goes like this, this, and then it goes like this. I'm sorry. It's not showing. The linear solution of this model equation, assuming the equilibrium I have specified here, can give even function, the so-called Kelvin-Helmholtz instability, where phi is even and psi is odd. And the pinch mode, where phi is odd and the psi is even. OK? In the case of an ideal plasma, the numerical solution of the linearized equation I showed you before give a stabilizing effect of the magnetic field on both the classes of solutions. But what happens when we have also the reconnection instability, the magnetic reconnection phenomena? So in presence of non-ideal contribution in the plasma home slope, the reconnecting mode are strongly affected by the presence of the shear flow. Here we have taken the parameter for the resistivity and the electron skin depth in such a way to have the same linear growth rate, which is shown here in yellow and blue. So the yellow is the collisionless case and the blue one is the resistive case. And we linearized the system of equation before and we solved it. And we carried out the growth rate for several modes. What happens with the effect of the shear flow? OK, the combined action of the shear flow together with the shear magnetic field in both of the regimes causes a broadening of the spectrum of the unstable mode. You see the red curve is the resistive case and the green one is the collisionless case. So we have more and more unstable. There is an increase of the growth rate. And there is a shift of the peak growth rate towards higher k or higher mode numbers. But there are some differences between the two cases. And first of all, the spectrum of the unstable mode for the collisionless case, broad and less respect to the resistive case. The growth rates on the other side are much higher. So these results already suggest that in a turbulent collisionless environment, the reconnection is the busiest. We receive a booster respect to the resistive regime. So I ask you to keep in mind and back your mind this picture, because I will rely on this later in trying to interpret the results of our simulation. So the numerical setup is a freely decaying turbulence problem, as in the paper. So we are solving the question in this domain. And for the collisionless case, we took the resistivity. We take the resistivity is equal to 0. We had an artificial viscosity in the sense that it is there for numerical reasons. And then this is the value of the electron skin depth. We start the simulation with 10 times 10 uncorrelated equipartition of psi and phi fluctuations in Fourier harmonics. And this is the composition. And the energy is initialized in the range of the wave number mn, which goes from 0 to 10. So we start with 10 at this. Your box is filled with 10 at this turbulent at this. And the energy phi 0 and psi 0 are defined in such a way that the energy is equal to 1. This condition, why we choose this parameter? There is a reason. And this condition is convenient both to achieve early and adequate state of turbulence and also to have thin and elongated current sheet with a high aspect ratio of order 200. So LCS is the length of the current sheet and DE is the electron skin depth, which is basically the width of our current sheet. Because DE is the smallest scale we have in our problem. And this condition has been shown to be necessary for the onset of the plasmoid instability by Comiscio Cironi in 2008 and 2009, sorry, 18 and 19. So here is just an example of how the evolution of our system goes through time. And we have performed the numerical simulation with several resolution to make a convergent test, and so in the following, I will show you only the highest resolution run that we have. And I anticipate that the plasmoid activity starts for the collisionless case well before respect the reaching, well in advance, respect the reaching of the maximum turbulent activity measured by the maximum of the root mean square value of the current density. So I am sorry for this. Maybe I can try to close this light because it's impossible to see anything which is this. Well, not much better. I say this, I like this, but that's OK. Yes, if you can, that's a good idea. Well, you can look here. I might be a bit changing it, I don't know. But the next picture should be more visible, I hope. OK, thank you very much. Anyway, I'll try. Yeah, but not now. OK, so here, well, the initial light is merged, giving rise to the formation of thing and elongated current sheet. Here is all the integration domain. We have to divide it in four subplots because otherwise we couldn't handle such a big number, such a big data. And in, well, in principle, in each subdomain it's possible to see current sheet evolving in plasmoid chain as well in cabin animals instability. And I hope in the next slide this will be more visible. OK. So to highlight the formation of the plasmoid, we overplotted the magnetic flux surfaces. OK, so we are in 2D, so the magnetic plasmoid surfaces are well-defined and they actually can individuate the magnetic islands. And so here is a zoom for each of the four subplots I was showing before. And then you see that it's possible to see plasmoid formation here, OK, here in this other plot here. You see there is a plasmoid chain. Here there is another one and here as well. But at the same time, you see that there are a region where the magnetic field is much weaker. So there is no structure of the magnetic field. And you see that there are anyway laminar structures which are evolving through the Kelvin-Ellemann instability. So at the same time, you have both the evolution of the current sheet. So let's try to see if we can understand when the current sheet is going through to be disrupted. Yeah, I don't know. I think contrast. Contrast, because it's possible to see on the screen of the computer. OK, I don't think I have the remote hand. OK, I go on if you find it and you can try to manage. OK, so let's consider this particular current sheet. OK, you see this is evolving through plasmoid. So this is a snapshot at the time before. And the current sheet was still in its laminar formation. OK, so we have evaluated the velocity in red and the magnetic field in blue across the current sheet. OK, and you see that the magnetic field is much higher or it's higher than the velocity field. A condition which I remember is stabilizing for the Kelvin-Ellemann instability. OK, so the reconnecting instability is favored by the fact that the magnetic field is higher than the velocity field. What happens to these current sheets going on with time? Because, of course, inside the plasmoid, turbulence is anyway generating. OK, so in the case of the plasmoid chain formation, the turbulence is bounded inside the magnetic structures. Where the vorticity and the current density exhibits small-scale vortexes. OK, when the plasmoid becomes larger, the mutual interaction leads to the coalescence between the magnetic structure that belong to the same plasmoid chain. And these results in large turbulent regions where the magnetic structure eventually disappear. What I did, I don't know, it may close like this, right? No, it's not a graph. Thank you. The magnetic connection between nearby turbulent region finally drive the appearance of macroscopic turbulent domains, OK? So it is important to note that the Kelvin-Eilmonts dominated current sheets never show the plasmoid formation, OK? But just to tell you that in this case, I was showing the vorticity sheets because vorticity and the current density for this collisionless model behaves exactly in the same way, OK? So now let's go if we can understand which is the situation under which the current sheet undergoes the Kelvin-Eilmonts instability. So in the same subplot I was examining before, I am now considering this structure, OK? This current sheet. And again, this is the time before the current sheet, OK? At the time before he got disrupted. And here are, again, the velocity in red and the magnetic field in blue across the layer. And in this case, we see that the velocity field is much higher than the magnetic field, OK? So in this case, the Kelvin-Eilmonts instability dominates. And the current sheet evolve directly into turbulent regime. And as I said before, this kind of current sheet never exhibit plasmoid formation. So there are two distinct paths along which they can go, they can develop, OK? So here is, again, the vorticity at a later time. So the vorticity layers start to develop the Kelvin-Eilmonts instability leads to its complete disruption in many small vortices. So the layer I have examined, of course, are just representative of the many ones that form and disrupt according to either the plasmoid or the Kelvin-Eilmonts instability, depending from the relative magnitude of the magnetic and the velocity field. So although distinguishing between these two evolution paths may be very difficult, well, we believe that they could exist and is crucially in explaining the energy cascade, OK? And I will try to convince you now about this. So here are the energy spectra for the energy cascade for the resistive case, OK? So in the resistive regime, the onset of the plasmoid instability produces stippening, OK? Then this, I'm not very good with the point, with this point. Any, OK, here, there is any whose slope is given by k to the minus 2.2, OK? This is the result of Comiso et al physics of plasma 2016. So k star is the k at which the plasmoid instability starts. In the collisionless case, thanks to the collaboration with Luca Comiso, we made exactly the same calculations. And we found that the stippening, the slope of the change when the plasmoid magnetic regime is developed has to go like k to the minus 3, OK, to the zero order. The plasmoid-dominated regime in this case ends at the available scale length, which for us is given by the inertial scale, the collisionless scale, skin depth, please. So one scale, but here we are seeing multiple scaling. So does it mean there are multiple scale at least the energy is injected? Does it mean that? Well, multiple scale at which the energy is transferred, because you inject the energy at the beginning at a large scale, and then you go through the turbulent cascade. Well, here because there is a new regime in which there are other instability which take over, which is the plasmoid instability. And the plasmoid instability affects also the turbulent cascade, OK? That's the reason. And OK. So let's see if we can recover this k to the minus 3 spectra in our simulations. Oh, no, no, that's OK. Let's try on this picture. So that way. Oh, so now you can see. You can redo the top. It will be uploaded, so OK. So now on the other screen. OK, so let's wait. This one is not working. Here is not. You can appreciate here. Yes, it is working. OK, thank you very much. OK, OK. I'll just leave it here. So here are the magnetic and the kinetic spectra. OK. For reference, here is the minus 3 slope. And you see that we find a smoother slope than the theoretical prediction, OK? The magnetic energy spectrum tends to flatten consistently with the observation that the fully plasmoid mediative regime is somehow inhibited by the presence of the turbulent dominated, of turbulence dominated by the Kelvin-Helmets instability. OK, so this is a difference respect to the resistive case. And I will show this later again. And the kinetic spectrum is completely the culprit from the magnetic one. And it's different. While in resistive MHD turbulence, they goes together thanks to the collisions. Here they have a different slopes, OK? And indeed, while here you can recognize the presence of a knee, which is due to the developing of the plasmoid mediative regime, here you cannot see this sign, OK? There is no need. And here is the comparison. So here we run for the 19,000 resolution. The two cases in which we choose this value of the resistivity. And we have no electron skin depth. And here is the reverse, OK? So no resistivity and the electron skin depth. The structure are similar, you see? We have basically the same structure. But resistive current sheets are less sensitive to disruption and evolve following the laminar behavior, OK? While at the same time, you see that here you have already plasmoid and you have already evolved in a turbulent, in a completed turbulent regime. And this is a big difference because the number of plasmoids in the resistive regime significantly increases as we follow the turbulence evolution. And we get closer to the activity, to the turbulent activity peak. While in the collisionless case, the plasmoid start to form well before the turbulent activity peak. So this is a difference in the revolution. So basically, we stop this simulation before we could get to the plasmoid mediating regime. So but from the linear analysis I made at the beginning about the effect of the Kelvin-Helmholtz instability on the magnetic reconnection instability on the plasmoid instability, we could expect that the impact of the velocity shear on the growth rate of the reconnecting mode to be greater in the collisionless regime. Because we saw before that the peak of the peak, it was basically something like this. And then there was for the collisionless and the resistive regime, they were DE and ETA, they were almost the same. But this was the DE case, while the resistive case was like this, something. And so we could expect, oh, thank you so much, we could expect that in the collisionless regime, due to this announcement of the reconnection instability, we could expect that the plasmoid started to form before. So it is reasonable what we see. And the second difference, the big difference is that in the resistive regime, the current and the vorticity sheet are not affected at all by the Kelvin-Helmholtz instability. So this is a difference that they stay in the laminar structure until they evolve through the Kelvin-Helmholtz plasmoid instability. Here is a comparison of the spectra. And you see the spectra, we are not so close to the minus 2.2 resistive spectra. But the reason is that we stop the simulation because we could get to the plasmoid mediating regime. I mean, the results in the literature are solid. And we do not have to check them, but just to help us in understanding the difference with what we were seeing. OK? So plasma collisions make, and you see that the kinetic and the magnetic spectrum are quite similar in the resistive case because there are collisions that allow the two spectrum to evolve in the same way. So this is what I wanted to say about this slide. OK. So conclusions. And then I will show future perspective, try to remember some of the things I told you yesterday. So this problem has been addressed. The problem of the turbulent cascade in collisionless regime has been addressed for sure with a different perspective from what is usually done in this field because we were focused on the magnetic, on the structures, and on the interaction between the magnetic and the fluid instabilities. OK? So there was a question that has no answer at the time we made this work. And the question was, do plasmoid forming collisionless plasmas? Yes, they do. And the plasmoid instability develops also in collisionless plasma under appropriate choice of parameters. So you have to make a choice of the parameter that allows you to get the thin and elongated current sheet that has a ratio of order LCS over DE of order 200. At least. OK. And if they do, this is the second question, if the plasmoid form, how do they affect the thermal and energy cascade? So the energy spectrum of the turbulence in the plasmoid mediated regime has a different slope. And most importantly, the magnetic and the kinetic spectra are decoupled. OK? So they have a different behavior. And how do the plasmoid interact with the Kelvin-Emmels instability? Well, the plasmoid instability is enhanced by the presence of shear flow. OK? OK. So future perspective. Well, yesterday we saw the generalized home slow with many terms. And one of these terms was related to the ion sound larmor radius, which is related to the pressure gradient in the generalized home slow. So trying to put more ingredient in this framework and trying to still relying on the fluid model, which are more tractable for many reasons, it is possible to take into account also the scale length, which is bigger than the electron skin depth. OK? And so it will be, in our opinion, our next step is to put this ingredient into these turbulent studies. And we already make the linear analysis. So I give you just the flavor of what's going on. And so this is just the plot I made. Well, I try to make it again on the blackboard. Collisionless resistive case and resistive a collisionless case with Kelvin-Emmels instability. And here in the dash of the black line, these are the growth rate for the collisionless reconnection rate in presence of ion sound larmor effect. And in the solid line is what happens when we put on top of these shear flow instabilities. And you see that the spectrum of the unstable model grows much more than before. OK? And then there is a suppression in this area. So well, it will be interesting to see how these strange behavior will affect the turbulent cascade. We have to address this problem. And the second problem we are interested in, of course, is to go to three-dimensional simulations and three-dimensional effect. Well, from the physical point of view, it will be really interesting to have turbulence and magnetic chaos. Yesterday I showed you that when you go to 3D, the magnetic field lines equation are no longer integrable and you develop chaos. So what's going on? I don't know. But we believe it will be a very interesting topic to see how the interaction of the turbulent motion with the turbulent magnetic field with where these will drive us. And of course, there is an issue because for doing simulation in three-dimensional, a technical issue, for doing simulation in three-dimensional setup, it is necessary to go from CPU to GPU because otherwise it's not handable from the point of view of the numerics. Or at least you can do very low-resolution runs, but you need very high-resolution runs here. And so I think I stop here. And I just want to thank you. I have a garden. It was faster. I'm sorry. But so if you have a question. Yeah. I have a question about your last comment. I found that deep in you. Yeah. In my experience, I mean, I think all fields doing turbulence are going through this transition. And in my view, it's an awkward transition. Yes, it's really awkward. Because the codes that you've optimized, written and optimized for CPUs, it's hard to kind of transition them to GPU. If you start with scratch with GPU, it's not such an ordeal. But transitioning old codes to GPU is hard. And it can take full time. Good time. Yes, sure. Absolutely. Indeed, we started and we are not succeeding in finding someone who can do that because we need a person dedicated. You cannot put a student to do that. Because, I mean, a PhD student and then what he has in the end. So it's now you need the computer science exactly. But it's difficult to find. And we are trying to. You're funding but not personnel. Exactly. And well, we are trying to collaborate with the bigger computer center in Italy, which is the Cineca. And well, actually, Dario Borgonio participated in a hackathon with them. And they started the collaboration. But he cannot do that the full time. And so we are slowly. It's a slow transition. Are you trying to rework old codes or kind of build up on scratch? No, we are trying to. Well, they made in this hackathon some optimization. But it's a hard work because yes, you have to look at for what I understand. Because I said that I don't want to be involved in the GPU transition from the technical point of view. But basically, they have to look at all the communication and to understand where the communication are low and why they are low. So it's a. And then there are many, many. You are taking old code. Yes, we are taking the old code. Yes, which is older. But it has been renovated so many times. So it's not so old. But it was written for CPU for sure. And then we received many, many invitation from the Chineca for quantum computing courses and experiences. So I'm wondering, do we really need to go through GPU? And then we will be asked to go to quantum computing. I don't think so. No, I don't know. But that's a joke. But it's a field which is going fast. So if you don't have full-time people to work, you lose the train. And you have to think already of the next generation. I don't know. Because of all the work that people have done for the message passing techniques for the running blaster has to be done with that. I kind of feel like the MPI GPU era was OK. You know, it could be a physics system learning that's part of your toolkit. Exactly. That's it. I stop it there. That's not the case anymore. You really have to be multidisciplinary. You need experts in this area working on this. Yeah, it's difficult. I agree. There was a question over there. The driving is the electron skin depth or the resistivity because the plasmoid is a magnetic instability. And so it is driven by or in this case, the resistivity or the electron skin depth scale lens. OK. That's the mechanism which breaks the frozen incondition. OK. That's it. The sequence is of order D of this electron skin depth. Well, in a tokamak, I would say it's two centimeters, probably something like that. And of course, if you go to astrophysical plasma, it's much greater. Don't make me to say numbers now because I could be wrong. OK, you're welcome. Well, this is something I have the time. I can put the other presentation. So I can show you all the magnetic fillings. Exactly. As if it were time, then they tend to divert it on an electron path. That's what chaos is. OK. It will be time. Yeah. That's OK. Well, here is an idea of, no, no, no, I was wrong. Sorry. Where is here? OK. So these are the equation of motion. So this system can be the equation of motion for the magnetic field line. OK. So when you have the third dimension, which can play the role of the time, so at fixed time, Z can play the role of time for the magnetic field lines. OK, so this equation, if there is a Z dependence, of course, are not integrable. The hae, me, which equations they are? Do you recognize them? Hae? Haemma. Haemma. That's why. So these are the Hamilton equation. OK. It takes away time. You look at the Z as a time. It's your time. And the psi is the Hamiltonian of your system. OK. And so these are the conserved quantity, the conserved energy of your system. You can think in that sense. Yes. P and Q, exactly. P and Q. And so you cannot integrate them. Or you see, since as far as you have, in this case, it was two magnetic islands at the beginning, OK? So as far as you can consider them as linear and so not interacting, you can visualize the two magnetic islands. But when they start to grow and to overlap and to become closer, so there is a critical criterion which says that if the magnetic islands overlap, they develop chaos, OK? And here you see there is a developing of chaos which started from the first, the most unstable magnetic surface, which is the separatrix of the magnetic island. And then chaos develops in all the domain. So back to the question I was interested in, as a further word, so what I've shown today is, OK, in such a kind of situation where we know that there are also hidden structure in this magnetic chaos. So what happens if we have also a turbulent fluid? This is our goal, final goal. Thank you very much.