 Good morning. I am very thankful to organizers for possibility to make this presentation here. And I will speak about localization of convective currents. Does it work? Localization of convective currents under frozen parametric disorder and eddy diffusivity transport of a passive scour in the presence of these localized patterns. I would like to start with some brief introduction to what is under some localization for which is known from quantum mechanics and some non-quantum linear systems, conservative systems, because originally our interest to this work was related to similarity between something there and what we were doing. However, well, this similarity turned out to be nearly virtual. Nonetheless, I will keep this brief introduction and then I will discuss what we have in thermal conversion and pattern formation in this dissipative systems in the presence of this frozen parametric disorder. And you show excitation of localized patterns and discuss their localization properties and consider what they mean for transport of passive scour and consider some additional effects on this localization properties of these patterns and consequently on the properties of the system and what happens in the system. So what is under some localization? It is known for distributed systems with frozen parametric disorder. Let me start with very simple, let me explain this with very simple example. We have one dimensional particle, one dimensional problem of motion of particle in potential, which is constant in time, but in homogenous. If this potential is periodic, you know that you have running waves, you can have solutions which neither decay nor go. It's interesting that in the presence, when we have disorder in this U, when we lose periodicity, we will never have running waves. In one dimensional case, we will always have only exponential decay or growing solutions and the more we observe in the system are only these sort of modes. And they are exponentially localized and gamma is a localization constant for these things. Of course, when we go to higher dimension, two and three dimension, this interesting property will be some was, will be not so much absolute. So here for infinitely high energy, we will still have localization. But in higher dimensions for high energy, we can have propagating solution. Nonetheless, this effect remains very significant. Well, it first was discovered by Anderson and he's presented this work for the diffusion of spin states and some random lattices. And just to show with very simple model with Schrodinger equation, where potential U has zero overage value and it is Gaussian delta correlated noise frozen. And for this case, you have such a dependence of this localization gamma on energy U. So for negative energy, it's obvious that you have localized patterns even without their localized. But for positive values of energy, you still have localization and you have this localization for infinitely high energy. Well, and the same picture you can observe for other linear systems where you can have propagating wave propagation. For instance, for propagation of light in the wave optics, not in the quantum sense. Well, it's still quantum sense, okay. And for the sound waves. For instance, if you compare these two equations, Schrodinger equation and equation for sound propagation in homogeneous media, you can see the similarity between them. So the systems are essentially identical from this point of view. And just some demonstration. Unfortunately, I cannot show animation because it's PDF PowerPoint presentation was not reflected very well. Was not shown very well. So if you have string with point masses which and disorder in position of these masses by 10% here. And also masses are not identical, they're random. In this case, if you put such an excitation in your system, you will have this excitation surviving infinitely long. So the patterns after quite long time of waiting will be like that. So this is not only for quantum systems but also for microscopic systems. This phenomenon is also relevant. So what we have in thermal convection, let us consider the problem of thermal convection in a thin layer. We assume horizontal boundaries to be impermeable and nearly firmly insulating. And we have in homogeneous in space heat flux which is like that. Q, this is critical value of heat flux which corresponds to instability for a short time. We have such a modulation constant in time but in homogeneous in space. And epsilon square is just a characteristic amplitude of this variation of heat flux. And for this case, we can derive equations in the long wavelength approximation. We can derive such an equation. And here it is already specifically for the case of where Q contains Gaussian delta correlated noise. Think like that, we have such kind of equation. If you wish to interpret this equation also in terms of flows, specifically for the case of our specific problem, we consider the derivative of theta which is temperature along this layer. So it's nearly constant along vertical coordinate. It gives us, its derivative gives us amplitude of stream functions. So if we have for theta patterns like that, we observe patterns like that for given realizations of this Q, the flow structure will be this one. Well, just for illustration. Well, actually this equation we have, it is, well, it is a kind of modified Kuramoto-Sivoshinsky equation in contrast to classical Kuramoto-Sivoshinsky equation. We have here cubic term but not a quadratic one. It's actually quite natural for many systems to have, where you have pattern selection to have such kind of term. Because if your system has symmetry with respect to sign change, obviously you will have not a quadratic term but cubic. And also if you consider special modulation of oscillatory pattern. For oscillatory pattern, you will as well have inversion to sign symmetry because it is just a template. And as well you will have cubic term and this is just some list of different problems where we arrive to such kind of equations for one-dimensional case. Well, in this problem interpretation, well, the flow shape will be slightly different corresponding to this temperature field. But the equation for the temperature field, you remain the same. So in terms of these equations, our consideration will be much more general than this physical problem I initially introduced just for illustration where it could occur. Well, let us compare this thing with Schrodinger's study state equation just to understand why we were expecting something interesting here. So this is our equation with cubic nonlinearity and non-large values of q. If they are negative, either negative or near zero, the dynamics of this equation is a relaxation to some time-independent attractors. And for no advection term, we, well, our time-independent patterns are described by this equation. And if we have in this system localized solution in the area of tails where solution is small, we can neglect nonlinear pattern, we have such kind of equation. And I want to also recall that the interpretation for this theta, that it is amplitude of stream function. So if you compare now this equation to the Schrodinger equation for study states, you can see that these equations are simply identical. So in this term, in this sense, the problem of localization of patterns in our convective problem, it becomes identical to the Anderson localization in the Schrodinger equation. However, there is very significant difference in the interpretation of formal solutions between dissipative systems we have here and Schrodinger equation. So for our case, our modes are going or decaying, not oscillating for Schrodinger equations, our modes are actually oscillating and their amplitude doesn't change over time. It remains, well, averagely constant. In our system, nonlinear interaction between modes is essential, it is, it cannot be removed, it's just, it's always present. Localize mode in the linear problem, in the linear Schrodinger equation do not interact. So as long as we consider single particle, not particle interaction, not interaction of several particles, our equation is linear and these modes, these localized modes will not interact. So therefore, for Schrodinger equation, localization plays a crucial role in dynamics always and actually it's the main thing which occurs, everything comes just to be about this localization. In our case, for convective system, localized patterns can be of an interest only when they're very specially rare. Otherwise, we will not be able to see their tails and we will not be able to see their localization properties. So only when they are specially rare, so which means that we are deeply below the instability threshold, only in this case, we can have some significant effect from them and indeed one can calculate the dependence of special density of these localized modes of centers of excitation of these localized modes as a function of this mean criticality q sub zero. When q sub zero equals zero, it just means that we are at the instability threshold. So at q, at minus two and lower, we have quite small special density and indeed we can consider this localized pattern and here it's just illustration for different values of q. What we observe for some specific realization of noise of our disorder for q minus two and a half we have in this domain already single localized solution. So it's logarithmic scale, this exponential decay and here just noise of calculations and such kind of patterns we observe. And it's funny, but for this special density, these approximations works very well. It's drawn with solid line, this thing. So could it be something important behind these localized patterns when they are so sparse in space, when they are so sparse? And yes, there actually is something and let us consider the transport of passive scour and effective diffusivity. If we have some passive admixture, if we have fixed flux of this admixture, the concentration profile could be calculated and it is related like that with this flow structure and we can introduce actually effective diffusivity constant, this one sigma and it is related to such kind average value and if values of D is small, so the diffusion itself is small, molecular diffusion, we have nearly, we have very, well, in this case, in the area of flow we have nearly horizontal, nearly, well, infinite conductivity and some kind of resistance to these flux. It becomes purely diffusive in this area where we have no flows. And, well, one, we'll have the following picture for dependence of this effective diffusivity coefficient as a function of q sub zero for different value of molecular diffusivity. Actually, this D in our dimensionless units, it is a ratio of diffusion coefficient to heat conductivity coefficient to thermal diffusion. Well, small value, well, values which are close to one, they are more or less corresponds together, no kind of interesting effects here, but this value which is typical for liquids, well, actually for liquids more typical is 10 to minus three, but nonetheless it is close. We observe such a picture. We have very significant degrees of this effective conductivity. This level corresponds to the case of no currents. Nonetheless, we have here below the instability threshold, we have here quite a significant increase due to these currents and you can see that this increase can be by several orders of magnitude of this effective diffusivity coefficient. And actually one can make analytical estimation of what must be this diffusivity coefficient. And if we include in our estimation of this coefficient only two ingredients, localization properties of our patterns and their special density with this very simple thing, we have such a black curve which very well corresponds to our dependence. Well, of course, for large special density of this excitation center, it will be not very accurate close to this threshold, but here it is quite accurate. And well, why it is interesting to show these things here that because in this case where our dependence correspond to this analytical summation, it means that indeed everything which determines properties of our system is this localization property and special density of these localized patterns. So we have a kind of pure manifestation of only this specific effect, the influence of these localized patterns and their presence in our system. Well, one can also consider the effects of some, well, within our specific physical system, we had originally slightly more general equation. We had here this advection coefficient u, additional advection coefficient u. Actually, this advection is very small. When we have advection u appeared in equation like that, in equations like that, it actually means that this u, to appear like that, this u should be so small that it influence the pattern formation in our system, the dynamic of this theta, but it still is not very significant for transport of passive scour. So when we write equations like that for transport of passive scour, we still use the same equation for evaluation of the diffusivity coefficient. So this u will make negligible contribution to the diffusivity. And the only thing we consider, how this thing influence on the localization properties in the system. And if we write now equation for linearized form of this equation, just for our tails, to determine localization properties, the order of our system increases. Now it is a system of the third order and we have actually three localization points. Two of them will be relevant up. Well, against the direction of advection, one will be downhill the advection direction. So without this advection, we have pattern like that with symmetric localization properties. And here remarkable thing that for small, for finite by small, but small values of this advection, we have very strong delocalization upwards this advection flux. So in this direction, you can see that the patterns become significantly delocalized. And it is important that u doesn't tend to zero because if u tends to zero, this mode which is weakly localized, it seems to actually is proportional to u, it will go to zero. So it somehow should be small to have low localization of this mode, but not too small because this new mode is just proportional to this u. And just what we have for effective diffusivity when we have this delocalization effect. So what we have without advection, red lines, red curves, and blue curves what we have in the presence of advection and you can see that in the area where even our analytical theory is valid, so where we have our effects in pure form, even there we can have for quite moderate values of u, we can have increase of this effective diffusivity by one order of magnitude. And this is just, and this will relate it only to our localization properties and their alterations, the change of these properties under the influence of this advection. So let me go to conclusion. So we consider the excitation of localized pattern patterns under parametric disorder in dissipative systems. Well, we also discussed advective delocalization for these patterns, evaluated the change of localization properties and discussed the importance of these effects for the transport of passive square by convective currents. Actually, at the end this system, this problem became much less connected to the original, well, to that original motivation related to Anderson localization because there are too much differences in interpretation. So some part of very basic mathematics is similar, but a lot of everything else is different. So our results here, they're much, much, much smaller than Anderson localization or account of Anderson localization address for dissipative system because otherwise it would be something very, very big. But nonetheless, there are still some important implications of these, some important effects emerging from this specifically for transport of passive square. And that's it, thank you very much for your attention. Well, there are works which consider this system and actually there is even works by Zimmer and I don't know, maybe his PhD student where, sorry, Zimmerman, where they consider the same, the same equation. However, what they look at, they look at how the instability threshold is shifted downwards for finite size domains and discuss statistical properties of this. So nobody looks at localization properties of, well, at localized patterns in the system and their localization properties. Our focus here was exactly this localization properties of patterns due to parametric disorder, frozen parametric disorder. Thank you very much for your comment.