 by our group at the Humboldt University of Berlin, and Alex Niponnich from Technion in Israel, this project was supported by German-Israeli Foundation. All of us know that large-scale turbulent flows feature complicated dispersion for particles. Passive and active, well, on large scales, on moderate or large spatial and temporal scales, normal diffusion dominates. But if you have coherent structures in your flow, you may easily obtain anomalous transport properties. And here, we would like to show that such anomalous transport properties can be observed without turbulence. So I will show a few examples. So first, without any noise, deterministically, the flows are animal laminar. So these are just flows past vortices, soft obstacles, and past hard obstacles, solid obstacles. And then I will also, in the end, I hope you'll have some time for that introduced noise and show how the intermediate time scales are affected by anomalies. So let me start with, OK, this is an important remark here. So I have anomalous transport. There is neither molecular diffusion. There are no fluctuations. Nor even Lagrangian or Eulerian chaos there. Everything is quite regular. And nevertheless, you will see that the results are not somewhat surprising. So this is probably redundant in this audience. So just the basics of fluid motion. I just also would like to focus on the Eulerian description. So everything is fixed locally. Or you let your observation devices float with a tracer. And then you have the Lagrangian picture. And of course, for transport, the Lagrangian one is much more convenient. And that's probably the reason why our talk was shifted in the program from the Euler Hall to this particular place. Of course, there it would be bad in the place. So again, what do we call here normal transport? So we take an ensemble of particles. And if the variance in that ensemble grows linearly with time, we say it's normal diffusion. If it goes slower, we call it subdivision. If it goes faster, we say it's super diffusion. OK. And then to make the things quite simple, I go to the plane, two dimensional flows. And I discard time dependence. So for Eulerian observer, this would be boring and trivial. For Lagrangian observer, the velocity is constant in every point. And this means that your plane turns into a phase plane of a dynamical system. And a tracer moves along this plane, along the phase trajectory. So we have incompressible fluid, a two dimensional steady flow. This means phase volume there is conserved. And when you reduce a stream function, it plays a role of Hamiltonian. So fluid particles are carried along the isolines of Hamiltonian. And if you know this Hamiltonian or a stream function, well, it's an integrable Hamiltonian system. So in principle, you should not expect especially complicated things. And this is just two dimensional. Your phase space there is no chaos, neither Eulerian nor Lagrangian. Well, I will consider forced flows on a unit square with periodic boundary conditions. In five minutes, I will explain why I need this geometry. Now, you take just as it is. So a square with periodic boundary conditions is a two dimensional torus. So I impose periodic boundary conditions. And I would like to have mean drift. So there is some forcing there. I would like this ratio between the rates of drift to be irrational for the reasons which we will see later. And the forcing term is spatially periodic, is constant in time. And let it be very simple. Trigonometric sinusoidal in x and sinusoidal in y. So you can write down this forcing term in the Navier-Stokes equations. And you can write out an explicit solution of that equation. And you will see that when you increase the amplitude. So if there's no amplitude, you have straight streamlines. When you increase the amplitude, you have a topological transition. So from straight streamlines which cover the whole square or a torus, you see how the vertices are formed. And then after that, you have a mixed phase space. So you have still streamlines which go around the torus or along the square if you lift it onto a plane. And you have isolated vertices. And those vertices are encircled by separatrices of stagnation points. The bifurcation finds takes place here where you get a turning point. This turning point there dissolves into two. An elliptic point in the middle of the vertex and a hyperbolic stagnation point there. OK. So that's how it comes. And there you have two vertices. And OK, one of them was rotating clockwise. The other one was counterclockwise rotating. You may take a similar step with a little bit more complicated forcing pattern. And then you get co-rotating vertices. And from the transport point of view, there is a difference as we shall see in a couple of minutes. So OK, now let's compare. Let the tracer move, be carried along this velocity field. And then we see there is an interesting property there. If you have a look at the fully spectrum of the velocity carried by this flow, then you will see if there are no edges, this is a quite usual discrete fully spectrum. It's a quasi periodic process. So you just have those dense, countable set of delta functions. But if you have vertices on your flow, you have a different picture. You have this complicated pattern. And it can be shown this is neither discrete nor absolutely continuous with respect to, say, to the leg measure. It is singular continuous. It is supported by the fractal set of the frequency values. And there you also have an unusual behavior of autocorrelation. So you can measure the autocorrelation function for the passive tracer in that flow. And you see this autocorrelation decays. It decays although you have an explicit integral, the stream function. So a decay of correlations in an integrable system. OK, we know that in principle, as Ergodic theory tells us, transport is a spectral property. So you have those unusual things in the spectrum. You may expect something interesting in the transport. And indeed, first let's quantify the transport. So let's take the variance. So all particles are carried along the streamlines. So we take the elongation of the package of the ensemble. And now we introduce it. And just that's mean squared deviation, what we are going to look at. And we see how it goes. For counter-rotating vertices, in both cases, it seems to grow. For counter-rotating vertices, it grows reasonably slow. Here it goes faster. In both cases, it goes, well, the slope is lower, definitely lower as 0.5. So it is what you would call sub-diffusion. Don't pay much attention to those numbers. We will see later that, OK, they should be different. Well, OK. Now I take a different example. Again, a square, I take a lattice of cylinders. This is actually the applied problem. We would like to cool the rods of your reactor. So you let the flow go around it. Now you formulate it again in a similar way. Just take you discard nonlinear terms. You say it's a stokes flow. So it's a biharmonic equation. And you would like now to have no sleep boundary condition along the whole border. Now your function vanishes on the, now your velocity vanishes on the whole lines. And still you would like to have periodicity here. And mean flow along both axes. You parameterize it in the same way. And that's how your flow then looks like. And then, again, you see that the stream flow lines go around and again. So again, you can show that this, if you started the transfer properties of this flow, just you measure Fourier spectrum. You measure outer correlations. You would see that outer correlations displays power or decay. And the spectrum sits on the fractal set. And as for transport, now you see something which looks like a weak super diffusion, slightly larger than 0.6. So that's how the variance in the ensemble of particle grows. Why does it happen? Why do these flow behave that way? That's pretty easy. So here is the flow around the AD. Here is around one vortex. Here is the flow with one obstacle on a square with periodic boundary condition. I take here, let's say, 10,000 of tracers and let them drift. And as the time goes on, I see how they are stretched, how they are going around. And, well, if I wait sufficiently long here or there, I find my tracer particles everywhere. So there is mixing here in the system. It's a weak mixing here. It's a faster mixing there. But in any case, there is mixing. And what is the reason for the mixing? The reason is straightforward. The reason is the passage near the stagnation point. So if here, or OK, the reason is a slowdown somewhere. So there are particles which slow down. And there are particles which go elsewhere. And at this moment, do not experience a slowdown. So this stretches the distances. And for the effect to be strong enough, I need such zones in the flow. That's why I need stagnation points or obstacles where I have such continuous, so to say, of stagnation points. And I need repeated passages close to such zones. And for my repeated passages, I need, well, two-dimensional torus gives me a perfect geometry for that. With irrational rotation number, I come arbitrary close to any given point, including those things. So if I have an isolated passage time across the square, across the unit, it has a logarithmic singularity in the case of isolated stagnation point. And it has a power law singularity in the case, a stronger singularity in the case of the continuum of stagnation points in the case of the solid obstacle. And then you would like to have a look at the transfer. And then you realize this is a slow effect. And for it to be observed, you need something better than, indeed, the Stokes equation or the Navier-Stokes equations. Although you have your explicit solutions, you won't observe, you won't get reliable numbers from a thousand of iterations around the torus. Not neither from 10,000. You need something like millions and tens of millions. But solving Stokes equations or Navier-Stokes equations for that time is not very reliable. You get errors which accumulate. You need a model which works much better. And there is such model. OK, so it's indeed a very straightforward introduction of that model. So what you have there is a non-random walk over the lattice. So you see there is a streamline. For example, this red streamline, sometimes it goes relatively fast across the square. Sometimes it comes close to the border. And then it hovers for sometimes the mapping which you would like to introduce here from a border onto the next border. It is straightforward. It is just a circle mapping. So for y, for the vertical coordinate, it's just a shift. And here it's a circle mapping. And alpha to beta is a rotation number. But you need somehow to incorporate the time for an iteration of this mapping. So you need somehow in the dynamics to take into account the fact that this mapping here occurred pretty fast. And that one close to the obstacle took quite a lot of time. When you compute transfer properties or stectra or correlations, your average over trajectories. And therefore, such passages, such close passages, they have bigger contributions. They have bigger weight in those things. And you need to take that effectively into account. And there is such construction in ergodic theory which is known as a special flow. It was introduced, well, OK. It was introduced by von Neumann. So let me say, that's how it is. It's just piecewise continuous dynamical system. So you start with some, so y now plays the role of time. You start in some point of x. And you move here with constant x. And constant velocity until you hit somewhere, the border curve. You hit it, and then instantaneously you make a jump. You iterate your circle mapping. And then you start your motion again until you hit the curve. And then you make a jump again. So the mapping here can be a circle mapping. And the border curve is your passage time past the obstacle. And you see this passage time can be bounded in the case where you do not have singularities. Or it can be unbounded if you have stagnation points or stagnation zones. This construction was, as I say, introduced over eight years ago by von Neumann. And now it is straightforward to see what kind of singularities we need there. It's a very simple construction which allows to compute 10 to the power of 12, 10 to the power of 14 iterations in a reasonable time. So for general saddles, these are logarithmic singularities or generic saddles. For non-hyperbolic saddle points, you have singularities of the power. That's how they look like. So that's the normal form for your bifurcation, depending on the degree of the degeneracy, n. And n is here. So you see all those couples are smaller than 1 over 2. So these are powerless singularities and they are weak powerless singularities. And finally, for the flows like that, where you have a continuum stagnation point, you can take the explicit solution of the Stokes equation high-order terms you need to get from this isolated square lattice. But from this stuff, you can already estimate the passage time. So if x0 is the entrance of the separatrix, which hits exactly the cylinder here, then your passage time diverges like an inverse square root law. And then you can play, for example, with logarithmic singularities with a special flow. And this is the result which you obtain there. So the growth is non-monotonic. And this is remarkable. This is not diffusion. So there are the ensemble stretches and short ends again. And stretches and short ends again. And if you take the golden mean rotation number, the minima correspond, build the Fibonacci progression. And they correspond to the best rational approximations to your rotation number. If you take the power laws, well, that's what you get. So you get the power law. And numerically, the exponent which you get there is just the double kappa, the double exponent of the singularity. This is still sub-diffusion. So you can. So we made a theory which estimates that exponent. And we show that it indeed goes for a special flow. It's such a simple dynamical system that there you can obtain the closed expressions for that. And there you see that after a certain time, you indeed observe a sub-diffusive process. OK, now this doesn't work for stronger power law singularities. And there we made a different approximation. So let's say a point. It's indeed a special flow, but the mapping which sits down there is not a simple certain mapping. It's a random certain mapping. So altogether, you have the Rw. So the object which was discussed today in a lecture of Sergei Fidotov. And this is continuous time random work. And there's a well-established mathematical apparatus how to deal with such things. And you see the second moment of the distribution goes like 3 minus. So after you make all the necessary Laplace transformations, you get the number 3 minus 1 over kappa. And kappa is larger. If kappa is larger than 1 over 2, this exceeds 1. This is super diffusion. So altogether, these two lines sub-diffusion for kappa below 1 over 1 half and super diffusion up there, they match in this point. So you have here sub-diffusive, slow diffusion. Here, there you have super-diffusive processes. Exactly in that point, the preceptor in the integral divergence. So there is a difference. Again, you can make a limit transformation. And then you observe there. You see the second moment grows like t logarithm of t. And indeed, it is well supported by our numerics. By the way, this transport is also multi-fractal. So if you have a look at higher moments, so the continuous time random work gives you the scaling. It goes like that. There is a term which does not depend on n. And this term provides multi-fractality. Indeed, this can be also confirmed by numerical things. One thing maybe here in the end, you go back to the logarithmic thing, the logarithmic transport. You remember, there were those returns. If you have a closer look, you see they happen here at the moments which correspond to Fibonacci returns. And so in the logarithmic time scale, it looks like a lattice of such returns. And again, it seems that this interval here reminds the larger one. This one has more inside it. The next has still more inside it. But altogether, there is some degree of similarity. So let's try to map all of them onto each other. And you start with one interval. Then you take a curve from the next interval, from the next interval, from the next. And you see how you are getting a more and more complicated object with a minimum at the same places. So it looks like a generation of a fractal. It's generation of a fractal in real time. Variance behaves like a fractal. So I do not have time now to talk about noise. OK, but you can do it here, of course. If you introduce noise in a usual land-driven way, this white noise, for example, here, and you add it to the cellular flow there, then a typical trajectory looks like that. So it's, again, it's CTRW, but it's an additional complication. Due to the cellular structure inside, you know there is a circular motion there. So the traps there, they have internal dynamics. And this can be done. This can be computed. So we also observe, of course, noise here ensures that in the end, at big times, that's efficiently big times, you get normal diffusion. But there is a lot of different intermediate regimes, which can be sub-diffusing, which can be super-diffusing somewhere. It depends now at intermediate times case where you start in the middle of the eddy or close to the separate rigs. If you start near the separate rigs, the initial motion is ballistic. If you start near the center, the initial transport is diffusive, and then you have a super ballistic transient. And this is, OK, it depends on initial condition. This means you have a property of aging. And this aging is non-monotonic here. So we also provided the theoretical estimates for that, which was reasonably good matched by our numerics. We also made some computations for the noisy cat's eyes flow. Cat's eyes, this means that you have a parameter here, where you proceed, which helps you to proceed from a cellular flow to jets. And the jets correspond to the transport mode of a levy walk. And the confined motion in an eddy is a trapping event. So altogether, if you match all that, then you obtain different transport to jets at intermediate times scales. Well, you also observe interesting aging properties. All that can be found in our recent paper in Physical Review E and for the cat's eyes flow in our arc sieve. And summarizing, I would like simply to say, even the flows with a very simple temporal structure may have highly non-reveal transport properties. And they can be captured by a reasonably simple mathematical model, a special flow, which gives you some exact values in the end. So thank you for your attention and for your patience.