 talking about the nonlinear regime of Rayleigh Taylor instabilities. This is this one here. Thank you. About the nonlinear regime of Rayleigh Taylor and Richmeyer-Meshkov instabilities. And OK, so the starting point for the Rayleigh Taylor instability is correspond to, if you imagine, a droplet of dark blue fluid, the heavy fluid moving into the light fluid. Then because of the acceleration force, which is larger than the buoyancy force, then the heavy droplet will continue to fall down. And while a droplet of a bubble of light fluid will jump into the light fluid, and therefore this configuration is unstable. And the instability leads to the formation of a mixing zone that you can see in this glass. And you have spikes and bubbles developing, growing in the mixing zone. And the Richmeyer-Meshkov instability rises when a shock wave propagates through the interface of two fluids, and a similar process happens with also a mixing zone. And the Richmeyer-Meshkov and Rayleigh Taylor instabilities are known to happen to occur in ICF, for instance. This was shown yesterday morning by Bruce Remington. And also in astrophysics, it was shown by Isaac Park. And so the first observation of the Rayleigh Taylor and Richmeyer-Meshkov instability in supernovae has been observed in 1987 with the observation of this supernova. It is a core collapse supernova or gravitational supernova. And the early observation of heavy material was a signature of mixing deep inside the supernova. In a supernova remnant, a long time after the supernova explosion, then the ejecta move expands into the interstellar medium. And the interface between the ejecta, the plasma coming from the supernova, and the interstellar medium, which is here the light fluid, is unstable. And you can see that some perturbation, some corrugation develop at the base interface. OK, so for astrophysics or ICF, the base instabilities play a fundamental role. And it is, therefore, necessary to understand in which way the mixing happens and happens. And we try to model the time dependence of the length of the extension of the mixing zone. And OK, so the nonlinear stage occurs as soon as the initial perturbation is not any more symmetric with respect to the initial interface. And after the linear stage, then you have spikes. You can see the spikes here, there is a spike. And together with the spike, here you can see half a bubble. And on the edge of the spike and on the outer edge of the bubble, you invest a numerical simulation coming from Los Alamos. You can see the vortices are due to the Kelvin-Helmholtz instability. But here, this is not my point. And if you omit the Kelvin-Helmholtz instability, the interface develops such a structure, the rising bubble and the falling spike. And these objects, they define the extension of the time extension of the mixing zone. In 3D geometry, the instabilities create such a squid with the bubble here and the spikes. And in 2D geometries, which is the object of a stalk, then you have parallel curtains. And the way we model the nonlinear regime of Rayleigh-Teller and Riesch-Mellier-Meshkoff instabilities is following the laser approximation. And in this approximation, the interface close to the top of a bubble or the tip of a jet is given by this expansion. The eta 0 corresponds to the elevation of the bubble or the length of a spike. And eta 2 is the curvature of either a bubble or the spike. And we try to find, to derive, ordinary differential equation for that govern the elevation of the length and the curvature. The leading equation are the incompressibility of the fluid. The velocity is given by the derivatives of the potential. We have the pressure continuity at the interface plus continuity of the vertical velocity of the vertical velocity vz plus the property that the interface is the motion of the interface corresponds to the motion is equal to the velocity of either the light or the heavy fluid. And from these PDEs, partial differential equation here, that are developed around the top of the tip provides the ordinary differential equation for the various quantities of the model. And laser derived this approach the first time. It was 60 years ago. Then 20 years ago, the procedure was applied to the bubble velocity and curvature. A few years later, then it was applied to bubbles and spikes. And in 2002, Goncharov has been able to develop this kind of model for two fluids with light and heavy fluids. And therefore, for the first time, he had arbitrary value of the Atwood number. Before the work by Goncharov, the Atwood number is equal to 1 because the light fluid is vacuum. So this model was the first one, including two fluids. However, although it seems correct for bubbles, the spike evolution is not satisfactory. And I will explain it a little bit later. After this work by Goncharov, several models have been developed. And in 2013, Banerjee and collaborators decided to use this kind of ansatz for the potential in their heavy and light fluids. The first equation corresponds to the former equation we have written with the elevation or length of the object of the mushrooms and the curvature here. And in this ansatz, the authors have introduced two modes in the potential governing the velocity of the light fluid. In the work by Goncharov, there was only one mode, k, and this. But the one mode model is not really correct because Abarzi and collaborators have shown it in 2003. The equation of conservations are not satisfied. And therefore, this ansatz is, in my opinion, really an improvement for this laser approximation. And then you plug this form of the potential plus interface into the hydro equation, and you get the three ODEs, the three ordinary differential equations. And you can get rid of phi sub 1 and phi sub 2 because you can show that there is a linear relationship between the light potential in terms of the heavy potential. And if you call n the dimensionless curvature, then you can show that the three ODEs for eta 0, the curvature, and the single unknown for the potential phi 1 in the heavy fluid. And from that, you try to find the solution from the integration of the time-dependent ODEs and the energy and collaborators did it by numerical integration. However, what we did is that we noticed recently that these three nonlinear ODEs can be solved analytically. And we did it by changing the time, the new independent, introducing a new independent variable, which is n instead of time. And this change allows to find the curvature and the potential as a function of n, the curvature, and then the time is a function of n. And as a consequence, n becomes here the parameter, and you can obtain the full solution in terms of t. This is a solution for the Rayleigh Taylor instability. You have the elevation or length of the mushrooms, the potential, time in terms of curvature. And OK, this is not really interesting, but this is just to show you that for each time, for any time we have got the solution. And from this law, we can derive the asymptotic formula for the curvature and the velocity for both the bubbles and the spikes. And for spikes, the velocity increases linearly with time. We have a free-fall regime, which is different from the regime by Goncharov, because in his paper, he has obtained a constant velocity, asymptotic constant velocity for the spikes. And it is interesting to notice that this value do not depend upon the Atwood number. In contrast for the bubble, then we obtain this formula for the asymptotic velocity. The asymptotic curvature corresponds to the value obtained by Goncharov or Mikhailian. The velocity is different, but if you set 80 equal 1, then this formula corresponds to this one, which is rather nice, interesting. For Rich-Mayer-Meshkov instability, you just drop the terms that contain the acceleration, so it becomes easier. And the asymptotic formula for the spikes are given here. We have a constant asymptotic velocity. And you can see that it depends on the Atwood number. R is a ratio between the heavy density and the light density. And also it depends upon the initial condition. For the bubble, we get a simple relation formula. And if we compare the formula to the work by Goncharov and Mikhailian, we get something different. But for A equal 1, we recover the same low in the three models. And just a last remark, I mean, for this work, I have simplified the formulation. Because if you expand the potential close to the top of the tip of the mushrooms, you can write this expansion. But because of symmetry, phi sub 1 should be 0 here. And then from the Laplace equation, phi 2 is given by this equation. And therefore, you have a single function, phi 0. And this makes the derivation much easier. And in this formulation, the equations are made transparent with respect to the x dependence. And my conclusion is here. Thank you very much for your attention. Yeah, at the interface. No more, sorry, normal component, yeah, yeah, yeah. Right, right. Yeah, yeah, yeah, yeah, yeah. Yeah, sure, yeah, yeah, pleasure, yeah.