 has been done in collaboration with Ernazar and David from Nazarbayev and Princeton University, OK? This is a brief outline. I will skip it because we have not too much time. But the thing is that why are we interested in studying how this shock is affected by nuclear dissociation right behind it when it encounters perturbations upstream? At the context, I think that I can skip this part because Eric did a pretty good job in that way better than I will do. So the thing is that when we have in that start that is running out of fuel, of course, there is no pressure to counteract the gravity effect that it collapses. It collapses supersonic. So from the freedom of expression of view, what happens is that to match that boundary conditions and in falling supersonic flow with a zero velocity at the center, then a shock wave is formed. That's just looking into the fluid dynamics point of view. This shock wave, however, dissociates heavy nuclei. And this is an endothermic reaction. So in this problem, I'm going just to look at this small piece of the problem. This is a multi-physics problem, as has been shown before. There are many physics involved, many scales. Just modeling this problem is quite complicated. Solving this multi-scaling in time, in temporal, and space scales is quite challenging numerically. So I think it's worth some time de-kerneling these problems into simplest scenarios trying to gain understanding. So for these reasons, I will look from the theoretical point of view at the interaction of these nuclear dissociating shocks when they encounter isotropic turbulence upstream. So this is like a brief sketch of the problem. We have a shock wave that right behind it, it's dissociating heavy nuclei. And this is an endothermic process that's important. There is perturbations upstream, and these perturbations are amplified after the shock front. So the shock reacts to these perturbations, as it reacts to the nuclear dissociation degree. I will look at this problem from an intermediate-scale problem. So the thing is that for us, the characteristic side of the perturbations upstream is much larger than the whole living shock and nuclear dissociation layer thickness. It means that for us, this dissociating wave is discontinuous with respect to that perturbations upstream. The other thing is that we are looking into the planar limit. It means that these perturbations are much smaller than the shock radius. It's sometimes, of course, at the beginning of the shock generation, it does not apply for sure. But the thing is that how these infalling perturbations are amplified by these nuclear dissociating shock waves. So this is again the assumption done from this fluid dynamics perspective of the problem. We have what I said that upstream we have isotropic perturbations. The first talk in this session, they show that for highly compressible turbulence, they tend to be non-isotropic. So these hypotheses may break down in some conditions. This is the restriction about the scales shown before. And the other thing is that we are decoupling the problem. The problem is fully coupled. I mean, we have a shock that reacts against perturbations, and these perturbations affect the flow properties downstream and affect the nuclear dissociation degree, affect the neutrino transport, and all of these in the previous talk has been very well explained. And in our case, as we are solving this from the theoretical point of view, we are decoupling that. Let's say that we are solving this in 1D problem, and we are using these parameters to compute the interaction with 2D and 3D perturbations. So let's go first to the shock front. This is the shock wave in the planar limit when there is a relative infalling velocity crossing the shock front. Of course, the conservation equations applies here, and here we have that, unlike regular shock fronts, the difference in the internal energy is affected by the nuclear dissociation degree. So if this term is zero, we have the black here, here, which corresponds to regular adiabatic shocks. However, when we have this energy dissociation right behind the shock, then, which can be scaled by the free fall velocity, then we have that the ranking curve is affected by that. So the final conditions is that the intersection of this rail line with the corresponding ranking hugoniot, but every different dissociation parameter. So for our case, the problem reduced to how this turbulent is affected when we are in a different Mach number and in different dissociation degree. So that's the problem to solve. Now, well, this is actually the corresponding jump condition that we have, the density compression ratio, pressure, and the corresponding strong shock limits. Let me say something that, in our case, this is not a self-sustaining reaction. I mean, the stronger the shock, the stronger will be the nuclear dissociation degree. So for this reason, the strong shock limit value at the end is different depending on this dissociation parameter. This is different to that what we find when we are studying detonations. In detonations, the shock wave trigger a reaction that this self-sustained and the energy released in detonation does not depend on the shock intensity. For this reason, when moving to the strong shock limit, in detonations, all of these curves collapse. This is not the case here due to the reason I just told. So now what happens when, instead of having a planar shock, we have perturbations coming from upstream to reduce the parametrical dependence of this problem, we have here, we are only considered when we have isotropic weak perturbations in the vorticity field. So how are we going to address this problem? We have vorticity perturbations. We study what happens when we have one mode and then, assuming, because we are within the linear theory realm, assuming isotropy, we can do the corresponding average and take into account the interaction with the whole spectrum. So let's focus first what happens when we have such shear waves impinging on the shock front. Of course, the real turbulence upstream will not be isotropic and will have density and very likely acoustic waves upstream. So for that, we should have a real model of the upstream turbulence to compute really what happens with such non-homogeneous flow interacts with the shock front. So the equations I showed before for the nuclear dissociating shock must be linearized. Mass momentum are not affected by that. And like energy, energy is clearly affected by the nuclear dissociation taking place right behind the shock. And these two terms are what differs these shocks to regular shock waves. For the case where there is no nuclear dissociation, this term goes to unity and there is no such term. So these two terms must be accounted to fully account for the effect of this effect. Now, the post shock flow right behind the shock is assumed to be adiabatic and invested in this time scale. I mean, the shock moving very fast, there is no real time for viscous dissipation and thermal dissipation. So in that sense, we are computing the values right behind the shock. And then this post shock flow can be used as the onset of the real turbulence evolution. So we have to solve this Euler sound wave equation, which is periodic symmetry in the travel direction, to link the perturbations downstream with the oscillating shock front. So to write a well-posed problem, we must provide the boundary conditions at the shock front and the initial conditions. This is the equation here. I can go further. The thing is that due to the nature of the characteristic function in the shock pressure, that one here, the natural expansion to solve this equation is by means of the vessel series. Gustavo gave a talk yesterday, and he explained that pretty well. So the thing is that we can compute the shock pressure as a function of the linear superposition of vessel functions. Then we can get the transient evolution from the first time that the shock encounters the perturbations up to the asymptotic field. The asymptotic is provided here in terms of harmonic functions. And they fit pretty well. Once we have the shock perturbations, as we are working here with linear perturbations, we can, with more or less straightforward manner, we can get the entropic perturbations downstream, the borticity generator right behind the shock. And after some straightforward algebra, we can get the rotational contributions too. So we have a shock wave that is moving through a non-homogeneous borticity field. And it generates entropic perturbations, borticity modifications. Actually, there are two contributions. One is a 2D or 3D contribution. And the other is just given by the mass compression ratio. I mean, we have a given borticity field upstream, and the borticity field is shrinked by the shock mass compression ratio. That's this effect. But there are many other, actually, that one. So this can be computed. I mean, the thing is that the borticity perturbations and the entropic perturbations and the amplitude of these perturbations can be computed easily by means of linear theory. But more interestingly is that the amplitude of the perturbations depend on the characteristic angle of the perturbation impinging on the shock front. What I mean is that if the bortices are highly stretched, the perturbations are given these values. Sorry, that one. When the bortices are highly shrinked, then we are obtaining these values. The dot lines includes the value of the acoustic perturbations. And it is seen that acoustic perturbations are only important here. But when looking into the transverse direction, acoustic perturbations are almost negligible. So the main kinetic energy, the main value of the velocity perturbations downstream are due to the rotational contribution of the velocity field. So we have how is the amplitude of the perturbations across the shock, the amplitude change, across the shock for any given angle. If we have a whole spectrum, we can compute how would be the kinetic energy, the average, of these perturbations by doing the corresponding superposition considering the probability density distribution. This is the case here. We can compute these averages. Assuming that we have the isotropic condition upstream, we can compute how is affected the turbulent kinetic energy when the shock is moving outwards. So that's the idea here. The idea is to compute the kinetic energy amplified by the shock front when it encounters that isotropic turbulent field upstream. And the idea too here, especially in this astrophysical context, is to see how this turbulent kinetic energy amplified by the shock is affected by the nuclear dissociation and parametrized this interaction. And there it is. When we split the contributions into longitudinal and transverse kinetic energy, and of course the total kinetic energy, we obtain the following pictures. Seeing from the big picture, the longitudinal kinetic energy would refer to the radial kinetic energy. And the transverse kinetic energy would refer to the interaction, to the computation of the kinetic energy transverse to the shock front, moving parallel to the shock surface. So here we see that the transverse kinetic energy is highly affected by the shock front. So the main contribution of the total kinetic energy is given by transverse perturbations. So we cannot neglect transverse perturbations. Transverse perturbations are found to be really important, as has been explained before, with 1D models, you cannot predict explosions. But in fact, 2D are important, for sure. And 3D, sorry, I'm referring to the non-radial perturbations. You have to be able to capture the perturbations in the non-radial component to properly set up the condition of possible eventual explosion. So that's why I put the stress in the transverse perturbations, in that case. What it's seen here is that the black curve corresponds to the adiabatic shock front, regular adiabatic shock front. And the different curves correspond to higher values of the nuclear dissociation degree. So we have higher dissociation for the red curve, nothing for the black curve. So the thing is that in our parametrical spectrum, we can compute how the kinetic energy is amplified as a function, the Mach number and as a function of this nuclear dissociation parameter. To put that into the space, we have SORF co-worker, Ernazar, do the computation by using a 1D code in order to parameterize this how the shock front, the shock intensity, the Mach number, and the value of the dissociation parameter as a function of time for three different scenarios, depending of a parameterized heating factor. It is found that for the heating factor sufficiently low, the rodents of the shock goes like that. We may have oscillations when the heating parameter is sufficiently high at very strong oscillations. But I'm sure that here the 2D and 3D effects become dominant, and this 1D perturbation is not real. Especially when you have a shock wave that is moving very close to the center, any perturbation, any non-radial perturbation becomes of the same order of the shock radius. So here, going very close to the center, the 1D code cannot reproduce really this evolution. But the thing is that for these two models, we can compute the dissociation parameters, a function of the shock radius, and the Mach number and the shock radius. Parametrizing that in a very simple, naive way, we get these heuristic formulas obtained by those simulations which can be used to compute the kinetic energy as a function of the shock front. And this is what is done here. The kinetic energy amplification is split into the lateral transverse longitudinal and total kinetic energy as a function of the shock. For when we are here in this limit, when the shock radius is very small, we have to bear in mind that it cannot be that small that curvature effects have been neglected. But in this limit, the strong shock limit applies. It's like having a blast wave moving outwards. In that regime, we have the saturated regime. There is no longer change in the degree of dissociation through the shock. So we can get like a first snapshot of the turbulence generated downstream. If we have information about the turbulence upstream, then we can use that to set up the turbulent condition downstream for any radial locus. Of course, there will be a diffusive effect and the turbulence will evolve. There is a transfer of energy between different length scales. And all of that has not been accounted. The thing is that the shock moves really fast before all of that happens. Then the turbulence structure will evolve accordingly. Finally, we just do another calculation which is how this perturbation downstream can affect the critical luminosity. So just linearizing this expression, it depends that we have the post shock Mach number is really affected by the perturbations. And they can be computed and they can be split into the entropic rotational contribution and the acoustic contribution. If we can compute that, we can therefore compute the average value of the post shock Mach number and then compute how the critical luminosity is affected by this effect. And doing so, we get these two figures where the main characteristic here is that first, the entropic rotational contribution is dominant with respect to the acoustic contribution. Acoustic disturbances are much smaller. I mean those generated right behind the shock than their entropic rotational. So this must be really, this can be neglected for sure. The second thing is that they increase, they increase with this nuclear dissociation parameter. So in any value of the parametrical space, the critical luminosity is reduced by the effect of the nuclear dissociating shock from. In other words, this effect, what finally produced, is an easier way to finally lead to explosion. So just to wrap up the talk, this is the main conclusions here is that the interplay between the shock wave and the turbulent flow, it's important to characterize the post shock flow that will later, eventually, lead to an explosion event, OK? The perturbation of the nuclear dissociation is found to be important. I mean, we are not, we are solving how the nuclear dissociation is affected to by these perturbations. And finally, just saying that critical neutrality, necessary for producing explosion, is found to be decreased by this effect, OK? Finally, of course, this is a very naive model. We don't want to reproduce the real event. We just want to decouple the effects and trying to study them in order to gain understanding. But this model can be, you know, will be benefit from including effects like including intermediate scales, including the converging of the flow, nonlinear effects, and, of course, improving the turbulent flow option. This is, of course, the kind of turbulent that we will find when the shock is expanding is it won't be a isotropic morticity field, OK? But I think that this is worth doing solid steps in order to understand a more complicated phenomena. So that's all. Thank you for your attention.