 And there is no signal here. Good. So I'm still a physicist and applied mathematician. So I would like to discuss today some, present some ideas regarding the fundamental properties of Rayleigh-Taylor mixing driven by variable acceleration. In principle, this work actually is an extension of the previous studies to some extent. So it uses some of the previous ideas, but it also explores the new parameter regime. And it started probably from the 2005 paper published in Physics of Fluids in a collaboration with Dr. Srinivasan and his student. What we know about the Rayleigh-Taylor instability, it develops whenever two fluids of different densities are accelerated against the fluid interface. We have a heavy fluid with the density of heavy here, light fluid with the density of light here. There is an important factor here, which is called atmospheric pressure. Because if there would not be an atmospheric pressure, we will have a completely different picture as we may see from the estimate for this hydrostatic atmospheric pressure neglecting the density of air compared to the density of water. Atmospheric pressure, which is 10 of the fifth power Pascal's, should be sufficient to support the water column 10 meters in height. However, even a small amount of fluid flows out from an overtone cup and the motion becomes pretty complicated. Even though, as we have just seen from the talk of Professor Meshkov, it actually, there are a lot of interesting features that make the dynamics simpler than we expect. And in fact, I would like to, maybe to express also my own opinion. I'm studying these problems, like, for some number of years. My PhD advisor, when I started to work my graduate school, he proposed me three problems. One of them, it was about like, he said, Snezhana, I have three problems for you. One is about how stones crack. The other, how the water flows from his healing and the sort I don't remember, but I do remember my feeling that stone cracking is not something suitable for a girl. So, the structure that developed in the Rayleigh-Taylor flow is pretty complicated because we not only have a lot of light fluid coming up, heavy fluid coming down, but there is also extensive shear and actually the shear is being accelerated and initially, at least, the shear results associated with instability growth. This is why Rayleigh-Taylor instability is sometimes called Baroclinic instability. Rayleigh-Taylor instability, they play very important role in a broad range of phenomena from astrophysical scales up to the atomistic scales in high energy density regimes, low energy density regimes, they essentially met everywhere. It's a fundamental problem also in the science of mathematics because when we are talking about the dynamics of Rayleigh-Taylor instability, it's essentially when we are solving our very famous equation for the stable oscillator, we just need to replace the sign from the plus to the sign minus and this would be the Rayleigh-Taylor instability. So it's indeed a very fundamental problem to consider. In fact, in majority of this phenomenon, the acceleration that is being imposed, which actually drives the Rayleigh-Taylor instability is usually non-uniform, whereas the bulk of existing studies is focused on constant acceleration or impulsive acceleration. And this work actually, I would like to refer to the previous studies with a constant acceleration and also to expand to the case of variable acceleration choosing the acceleration that can be a power law function on space or on time. And the power law function is selected because power law function are scale invariant, therefore they don't impose any particular scale in the flow and allows us to study as they interplay between the self similarities of various sorts. In fact, the main result is such that we definitely know that because Rayleigh-Taylor mixing is non-local in homogenous and isotropic and statistically unsteady, its property should depart from the properties of local homogenous, isotropic, and statistically static homogorgon of turbulence that has been studied by Dr. Shenevasan and Kalex to significant extent. For variable acceleration, this would be actually the finding of this work. The properties of Rayleigh-Taylor mixing makes it substantially depart from those that are prescribed by self-similar Sidov-Taylor and Guder-Ley-Stanikovich blast waves. So essentially self-similarity is the second and first kind. Primarily, because when we have acceleration, exponents sitting in this parameter regime, we don't have an acceleration-driven flow. We have a flow, actually, which is really depend on some critical exponent. It's a standard Rayleigh-Taylor problem. Taylor experiments performed in a flow in a cylindrical tube from 10 centimeters in diameter and about two meters in height. One of the first, I would say, classified applications of Rayleigh-Taylor and stabilities. This is a reticle problem. It is usually formulated in a pretty complicated way because it's a conservation loss in the bulk, which is a three-dimensional non-stationary equations quite often involving compressible fluids. So all these functions are functions on space and time. If it would be just like that, if it would be actually substantially smaller than that, then the problem would be already a subject to the million-dollar reward by clay institute. But it's more complicated, so it's a three-dimensional might be compressible. But on the top of it, there are also the boundary conditions at the interface. For instance, when the fluids are missable, we have continuity of normal component and pressure at the interface. And besides, there are other conditions at the outer boundaries of the domain, which might imply periodicity in the plane, normal to gravity, as well as the absence of external sources at the outside boundaries of the domain. And at the top of it, there is also the initial conditions that may impose symmetry as well as characteristic length scale and time scale in these flows. And as I mentioned, if we will have just a small portion of it, it's already a very complicated problem. It's called canonical classical, actually isotropic homogeneous turbulence. But with this set of complexities, we have really actually complicated problem, much more complicated. Because each time, even if we have missable fluids, we actually should solve boundary value problem and boundary value problems on contrast to initial value problems may or may not be solved and if solution exists, they may not be unique. So I will probably really go very quickly through the applications because Rayleigh-Taylor makes an important role in both of stars, such as the dynamics of auto-operated molecular clouds or in a death of star, such as a supernova type 1A or type 2, where it may provide special conditions for generation of heavy mass and iron peak elements in the plasma fusion, which might be both magnetic fusion as well as a natural compartment fusion. And I kindly point your attention to these pictures, which are actually 20 years distant from one another, but our understanding of this problem persists. So actually the computational tools are becoming stronger, simulations are becoming more powerful, but actually the ideas that have been developed some time ago, they're still valid. It's definitely related to the problem such as non-business convection, convection in sterile and planetary interior, bed and oil industry. In fact, actually it's pretty closely associated with the problem of the transportation of the quantified natural gas, which actually might be another as important as maybe fusion for the economy of many countries. Technology and communications, as well as impact dynamics of liquids and solace, which we have been discussed here as a problem. Usually the dynamics of Rayleigh-Taylor stability divided into three stages, such as linear regime, non-linear regime, as well as some similar mixing when the acceleration is constant. So there is an exponential growth initially, power, low dynamics later on, and then GT squared low, which does not depend, which is presumed not to depend on the any characteristic initial length scale. And in fact, actually this problem already tell us this formulation, what is the problem is about, because if you would have no dissipation, no losses, everything would go in a proper way, then the coefficient in this formula would be exactly equal to one-half, modified with some output number, dependence. However, it's not one-half, it's substantially smaller than one-half, essentially 5% of one-half or so, meaning that there is substantial amount of losses in this system. It's a picture of interface evolution in experiments like in Rayleigh-Taylor mixing, one of hero experiments, which we can see here, as well as molecular dynamics simulations that actually shows us that indeed formation of phase is an important factor of RMI evolution, because if you will take this little small parcel of fluid here, zoom it in, zoom it in, and zoom it in, you will see that at molecular scale, there is some mixing, there is molecular diffusion, whereas the mixing that we are looking for is essentially an interfacial mixing, so a phase should appear for some magic reason. Usually, because instabilities are considered as a mechanism of transfer to turbulence, for many years it has been believed that Rayleigh-Taylor mixing is actually kind of a super-turbulence, and this belief was based on the fact that when we have our velocity, which is growing this time, integral scale, which is growing this time, Reynolds number, which is growing this time, rate of energy dissipation, which is growing this time, viscous scale, which is decaying this time, and span of scale, which grows this time quicker than quadratic, then indeed one might expect that there might be something quite important as a super-turbulent regime, and these are the models that were developed probably starting from the Fermi and the Neumann models, to more recent models for many years, and the common wisdom was indeed that Rayleigh-Taylor mixing should be similar to turbulence. However, some experiments were conducted essentially back in 2000 in high-power laser facilities, where the characteristic accelerations were greater than 10 to the power 13 of the Earth's gravity, and characteristic Reynolds number was 10 to the power of six. A super-turbulence was expected to occur, however plasma flow exhibited a significant degree of order, and now a theoretical analysis explains why it can be like this. However, these observations, which is essentially experimental like that, they are not only the observations which report an ordered character of Rayleigh-Taylor mixing, this is the experiment of Nishkov, who presented them just right now, as well as existence of a very short dynamic range in a so-called quasi-turbulent regime, which is, for instance, observed at a very small Reynolds number here, and as well as association with the previous study, such as the turbulent, accelerated turbulent boundary layer in the regime Hensheny-Vasem, and Taylor, who considered the great criteria of development of turbulence for flows in a curved pipe. So, in our case, we would be presuming that Rayleigh-Taylor mixing might be driven by sustained or variable acceleration, and we develop a theoretical approach that is developed on the side of, I would say, an analogical modeling on the one-hand side and on the other side of the solving of a boundary value problem. So, boundary value problem, the solution we presented in the poster session yesterday, and today we'll be discussing the analogical modeling. So, if you're talking about Kalmogorov turbulence versus Rayleigh-Taylor, and I should probably admit the word turbulent mixing from here, we actually have quite distant processes, because Kalmogorov turbulence appears as somewhat more similar process than Rayleigh-Taylor mixing, because Kalmogorov turbulence is isotropic, homogeneous, statistically steady, and because it's isotropic, it means that there is no any force, no transport of momentum. Homogeneous, it means that there is no any transport of mass. Statistically steady, it means that how much energy we pump into the system exactly amount is dissipated. So, essentially the system is characterized by the transport of kinetic energy. In Rayleigh-Taylor mixing, it's somewhat different because it's an isotropic, we do have a direction of acceleration, it's in homogeneous, the fluid should have different densities. It's statistically steady, so we don't have actually any kind of diffusion mechanisms. So, the transport of momentum mass, potential and kinetic energy are actually this part to drive the flow. And again, we will be discussing Kalmogorov turbulence, which is a picture of maybe Shenyuan in 1999. We essentially have a simplified set of equations that have certain symmetry properties, including infinite number of Galilean transformations, temporal translations, spatial translations, rotations, et cetera. They also have a scaling invariance, and the exponent of the scaling transformation in the limit of Venetian viscosity that is Reynolds number approaches infinity, but is not equal to infinity, equals to one sort. In Rayleigh-Taylor mixing, as I mentioned, we have completely different set of equations, so it's a more complicated system, but most importantly, we have boundary conditions at the interface and also initial conditions. And inertial properties of Rayleigh-Taylor mixing, actually invariant properties are also different because dynamics is non-inertial. It has only transformations in the plane, not in 3D, and it has a different scaling invariance, such that when, because of this letter G, which is present in this equation, which is acceleration, which is essentially can be quantified as a rate of change of specific momentum per unit mass. So, and like, you know, as a simple picture, if you put a parcel of fluid in the flow, it would be influenced by the buoyant force and as well as the friction force, and keeping like, you know, some balances per unit mass of the rate of momentum gain and rate of momentum loss. We may write a sort of equations, and I should mention that this type of equations probably, I don't know, maybe was 1975 was one of the first who tried to present this type of model. What is important for us is that all the balances are counted per unit mass, not per unit volume. And second thing, these models equations actually have the same set of scaling symmetries as these complicated governing equations, which makes them convenient to study. So, when we have, we'd like to look at asymptotic solutions, then when the characteristic length scale for energy dissipation is horizontal, essentially constant, we have a steady solution. When energy, length scale for energy dissipation is amplitude, actually this scale rather than that scale, we have an accelerated solution. What is interesting is that there is a slight imbalance in this accelerated regime between momentum loss, which is a new and energy loss, which is dissipation, which is epsilon, and between the gain, which is momentum gain, new tilde, and energy gain, which is epsilon tilde. However, according to observations, this balance is very small, and this is actually our very famous coefficient alpha that various groups tried to calculate. If we will be looking for a time where an acceleration, for instance, when our acceleration is a function of time, the dynamics would be somewhat more complicated, because we will see that depending on acceleration exponent, we might have essentially two regimes. One of them, it would be an acceleration driven mixing, I would say Rayleigh-Taylor type of mixing, and in this mixing the exponent of the actual growth rate is exactly set by the exponent of the acceleration. So if there is an A here, there is an A plus two here. However, there is a critical parameter, A critical, when another type of mixing starts to dominate, and it's a dissipation driven mixing, and in this case, the exponent of the growth rate is set just by the drug coefficient, by the characteristic drug. So essentially, we have an acceleration driven mixing because it's exponent that's set by the acceleration exponent, and we have a dissipation driven mixing because it's exponent is set by the characteristic drug. This one, this mixing, it can be actually accelerated, steady, or decelerated. I mean, depending on what is the various parameters in the dynamics. However, it actually does not keep much memory on the initial conditions, where is the dissipation driven mixing. It's exponent, regardless what is the acceleration exponent is the universe also essentially the same for the same type of flow, and it's pre-factor is set by the initial conditions. If we will be looking at the critical value of this acceleration exponent, which is in this regime then, presuming that our coefficient C may vary from zero to infinity, which is a drug coefficient, we are getting A critical belonging to the interval from negative one to negative two, and for some magic reasons, this is exactly the interval to which all the blast waves, including first cell similarity or second cell similarity belong. If our acceleration will be a space varying function of time, which is, for instance, gravity is one of the examples, or we may try to organize such a special channel flows, if it would be done an experiment. We have different dependencies, of course, but actually the principle situation remains the same. There is a critical value of the exponent of the acceleration, which defines by the dissipation in the system, and there are two regime, acceleration driven regime, and the dissipation driven regime, which is here. So acceleration driven regime is set by the acceleration and is independent of the initial conditions, where the dissipation driven regime is set by the characteristic drug, and its pre-factor is actually determined by the initial conditions. Now, just repeating, reiterating the same, so we have that in both cases, there are two sub regimes, acceleration driven mixing, when the exponent is greater than critical, and its exponent is set by acceleration pre-factor, depends on the acceleration strength and exponent of the drug, strength and exponent and on the drug, and there is a dissipation driven mixing that is smaller than, when acceleration is smaller than critical, and its exponent is set by the drug and pre-factor is set by the initial conditions. And I would like to point your attention about this particular regime. In fact, in this case, if you will be looking at the solution more cautiously, acceleration is needed to start the flow to accelerate, to push the flow, however, later on, the flow moves quicker than the acceleration prescribes. This may sound counterintuitive, but still, it moves quicker than the acceleration prescribes because acceleration is so, so small that whatever initial V zero we would have at the very beginning, initial conditions, it will actually dominate the flow. So, yeah, and this actually sends our critical values, they're sitting in the blast wave interval. And if you will be comparing our various asymptotic solutions, we see that definitely there are the balance dynamics and all the losses are balanced, this gains are balanced, and there is an imbalance dynamics, which is actually we have characterized by the amplitude being the characteristic scale. A transition from here to here might occur if the horizontal scale may grow, would grow this time, and if the verticals grow, the scale would be the scale for energy dissipation. What essentially means that the merger of the structures which is quite often observed in Rayleigh-Taylor clothes and which is considered as a primary mechanism of transition turbulence is possible, but it's not a necessary mechanism for Rayleigh-Taylor mixing to accelerate. Microscopic properties of Rayleigh-Taylor mixing and Kalmogorov turbulence are different because in one case we, for instance, need an energy source, we have actually just a transport of kinetic energy, and we have a mean transition of the center of mass being independent at a relatively small drug, whereas in the other case, we don't have any energy source rather than gravity, momentum and energy both gained and lost, and the typical drug coefficient here is pretty large, which actually an indication that the flow may be more laminar rather than turbulent. Symmetry properties, I think probably, like, you know, maybe just need to formulate this light and then from this light I will jump to very conclusive light, which is very, very dense. So in this light, what we have, we essentially, for each of our typical cases, we have different set of invariance values. If it would be Kalmogorov turbulence, our basic invariant is a change of energy dissipation, essentially, rate of energy dissipation with the power of external motor, how much energy is being supplied to the unit mass, and characteristic scaling exponent is this one. For Rayleigh-Taylor with constant acceleration, it's a rate of change of momentum loss and characteristic exponent is this one. For a momentum, for an acceleration driven Rayleigh-Taylor mixing with variable acceleration, the measure of the scaling invariance is a new variable and I would call it, we would call it rate of modified momentum, which is essentially this value, I mean, divided by t to the power a, essentially we need to look at the h-order derivative of this value, so it's essentially h-order derivative and there is a characteristic scaling exponent here, as well as for the dissipation driven mixing, it is here. So I probably will not be discussing too much the correlation, fluctuation, and spectra of Rayleigh-Taylor with sustained acceleration, I just will tell that, you know, it has a different invariant properties compared to Kalmangorov's series, different scaling and correlation properties, different Rayleigh's number dependence, viscous and dissipative scales, as well as a different fluctuation spectra, just because in Kalmangorov turbulence, we have one parameter, key parameters drive the flow and in Rayleigh-Taylor mixing it's a different parameter. It's quite important to see because usually turbulence is considered as a state of a strongly fluctuating system where fluctuations are set just by the system and not dependent on the initial conditions or forcing or some external conditions. And actually looking at this, just trying to compare different processes such as the standard diffusion Kalmangorov turbulence as well as Rayleigh-Taylor mixing with a constant G, we see that in a standard diffusion process, our land scale is scale usually with the time of the power, t to the power of one half, as was actually discussed today by Sergei Fedotov. Whereas in Kalmangorov turbulence, we have a different scaling, yet the velocity scales this time as t to the power of one half, meaning that we do have a Gaussian distribution, standard diffusion of velocities, similarly to what we have in a standard diffusion process, whereas in Rayleigh-Taylor mixing with constant G, the motion is essentially ballistics and the fluctuation, if they might occur, they should be due to the initial conditions. And indeed, if you will be looking at the fluctuation properties, we will see that in Kalmangorov turbulence, it really does not matter what the initial conditions were because turbulence produce much more fluctuation than any initial conditions could occur, whereas in Rayleigh-Taylor mixing the fluctuations might be really well-frozen to the initial conditions. And now I would like to show a really clean table. It's quite detailed, but actually what does it tell us? It tells us that if you will be comparing say, Kalmangorov turbulence, Rayleigh-Taylor mixing, Rayleigh-Taylor mixing of RT type, when acceleration is greater than acceleration exponent on Rayleigh-Taylor mixing of RM type, I would say essentially it's a dissipation driven mixing acceleration is smaller than acceleration exponent, we will have different dynamics because the flow would be statistically steady here and statistically steady in all these cases. We will have different set of invariant values because which would be actually time and scale invariant here and we will have a much different invariance here. We will have different scaling exponents and because we will have different scaling exponents, we will have different scaling for the velocity scaling, structure function, Rayleigh-Taylor's number, viscous dissipation scale, as well as for the energy fluctuations and this is just like the invariance and the scaling exponents are listed here. Now, of course, because we know how the solution behaves, we might try to study what would be the behavior of different parameters such as Rayleigh-Taylor's number, energy dissipation rate, rate of momentum loss, how the typical Kalmangorov scale would behave this time, how this kind of scales would behave this time and there is the entire loop of these dependencies because they do depend on parameters quite extensively. However, if I would like to consider some important, like really for the applications and also to actually conclude the talk pretty soon, we would essentially need to say probably the following. We have two types of mixing when the acceleration is variable and power law. One of them is a Rayleigh-Taylor type or acceleration driven mixing when the acceleration is set by the acceleration exponent and it has this invariant value. There is in, when we have an acceleration smaller, the acceleration exponents more than the critical value, our actually exponent of the dynamics is set by the acceleration critical value of the exponent and we would call it Riecht-Meyer-Meschtl mixing. And if we will compare and say diffusion, turbulence, Rayleigh-Taylor mixing with constant G, Rayleigh-Taylor type of mixing and Riecht-Meyer-Meschtl type of mixing, roughly speaking, what we may say from the scaling considerations that Rayleigh-Taylor type of mixing, if we have a strong acceleration, then strictly speaking, landscape and velocity scales change with time quicker and the relations are stronger than those in the processes of diffusion and or canonical turbulence. And our weak acceleration case is actually leading us out of the turbulent regime because in this case, for instance, Rayleigh-Taylor number may start to decay or energy dissipation rate may start to decay so essentially we don't even think about possibility of turbulence in this dynamics. If we will be considering, for instance, dynamics which is set by the blast waves, as I mentioned, because our critical exponent does belong to these blast wave-driven regimes when we look at the classical solutions of the Dov-Taylor as well as Guder-Lien-Stejnukovich, then in this regime, as a rule, Rayleigh-Taylor mixing would be behaving as a dissipation-driven mixing if my Riecht-Meyer-Meschtl type and this Riecht-Meyer-Meschtl mixing would be definitely slower than canonical. Turbulence and other energy sources should be required to maintain kind of turbulent dynamics in this Riecht-Meyer-Meschtl mixing. I probably will not be discussing the stochastic modeling of this sense. I just would like to mention that if we will try to compare with experiments, we are capable to achieve a good agreement. And now, because I'm trying to learn to be a good politician, let me use some fancy words. For instance, we can have some outcome of experiment. What was the power of traditions in this community? Traditional scenarios predates the picture to the Rayleigh-Taylor mixing with constant acceleration in the following way. So we have in the linear regime small scale structures with certain periods that is growing quickly. Then in linear regime, the dynamics still remain the single scale that at this scale dominates and amplitude is essentially behaving the same way as period and accelerated mixing may develop solely because of the period growth. As a result, the flow becomes disordered, Rayleigh's number increases and the flow is transitioning to a regime of isotropic turbulence. We don't see this actually quite because we do know that we do have a small scale, the growth of small scale structures. However, in the linear regime, dynamics is a multi-scale. The accelerated mixing might occur due to momentum imbalance. The raw growth of the period might not be quite and the Rayleigh's number while growing and it was found in the seminal work of Taylor in 1929 as well as in the regime of Srinivas in 1973. You know, accelerated flows may do the laminarize as the flow may keep significant degree of odors and fluctuations might be frozen to the initial conditions. If you will be talking an application to the experiment, then this is the idea that I'm trying to push somewhat actively with all my remarkable energy for certain number of years. Usually, a lot of actually energy and dedication is, for instance, for the national ignition facility being placed to polish the capsule, to polish the target, to eliminate all possible perturbations. However, what is being lost in this consideration is that Rayleigh's ability is developed one way or another but they will be out of any control. So instead of being polishing it, it might be better to scratch it essentially to force the system to behave in the regime that we would like to see. And the other thing is that it's a control of acceleration because what we might see from this picture is that there are essentially two parameters which is a combination of velocity, position, and acceleration type. And these parameters, they are important because in fact, actually in a proper regime, for instance, parameter pi is a value on the order of one and in the improper regime it starts to diverge. There is a parameter pi tilde, the parameter that characterizes the dissipation driven regime and in the improper regime it starts to behave improperly. In fact, when we are talking about power or accelerations, this is what I've learned from interactions with turbulence people and with Dr. Srinivasan, particularly the dynamic range is a really key scene. And it's really hard to implement the flow with a really dynamic range in this type of circumstances. So we might have actually a parameter which would be telling us if we are sitting here or if we are sitting here. And in this particular case, the critical exponent was minus 172. So our theoretical analysis applied group theory for study unstable interfacial dynamics and identified fundamental properties of Rayleigh-Tiller instability and that's actually found under Rayleigh-Tiller mixing. And this is the conclusions. Thank you so much for your attention, for your patience. And I hope that we will survive.