 I have time till 3.20, yes? Till 3.20, I have time, yes, thank you. My name is Ilyas Ipgatulin, I'm from Moscow State University. Also, I collaborate with the Institute of System Programming, UNS de Lyon in France, with Laboratoire de Physique, the École Normale Supérieure, also with Institute of Oceanology in Russia, and Lavrentiev Institute of Hydrodynamics in Novosibirsk with Evgeny Irmanyuk. I will show my courses later in presentation, so it will be about gravity waves, but not surface gravity waves. I will like, you can see obviously on the surface of the sea, but the waves which propagate inside the continuously stratified media. So if stratified, if you have a layer which is stratified, in case if density grows with height, you will obviously have convection, otherwise it will be asymptotically stable because this time it will in presence of viscosity and any perturbation will go close to zero, but this case is also very interesting because they may propagate the waves, they are called internal waves, and from the simple consideration you can see that the frequency of the proper frequency is equal to the square root of the equation of the oscillator, like it is written here, it is called the frequency of Bruntweissala, or buoyancy frequency. This is all well known things, I will just show you the example of internal waves, this is a picture of an experiment, and you have a cylinder which oscillates and you have internal waves which propagate obliquely, so they are inclined vertically, and from this picture you can see also that they can reflect from the surfaces. Here is how it is important in nature, but I will just skip it, and go directly to the first problem which we studied. We studied propagation of internal waves in special kind of geometries experimentally and numerically, my part is direct numerical simulation, so we solved full Navier-Stokes equations in Boussinesq approximation, and Boussinesq approximation works very well in these conditions. Here I just remind linear theory, and from linear theory you can see that frequency and wave vector are connected just by this equation, so frequency doesn't depend on the length of the wave vector, only on its direction. This is very, very peculiar to internal waves, and because of this we will see absolutely astonishing from my point of view results. The fact that after the reflection from the inclined walls, the internal waves conserve the angle with vertical direction is well known from ancient times already, Philips Gabus Veshnikov in the 70s, 80s wrote about that, but interestingly that only in 1995-96 by Leo Maas it was discovered that in some kind of geometries, he tried many kinds of geometries actually, but it happened out that very simple geometry, just trapezoidal, if you put inside a source of monochromatic wave, so for example a cylinder oscillating, the waves after some number of reflections actually will tend to be on some kind of trajectory. This trajectory was called, this path was called wave attractor, in this case internal wave attractor. So here the pioneering works of Leo Maas in 1995-1997, and so what we did now, the conventional linear theory is well developed and what is most interesting now is non-linear effects of this phenomenon. So actually what we solved numerically is Navier-Stokes equations. This is the experimental facility which is located in Ecole Normale Supérieure de Lyon in Laboratoire de Physique, so they put with the help of two bucket techniques, they create a stable stratified aquarium tank of salted water. The dimensions here is the height in this particular experiment was about 30 centimeters, also there is another experimental facility with height 1 meter and different widths, and here on the left side you can see the, on the left here you can see wave maker. So it consists of 50, actually 50, eccentric plates which are moving, so it's quite a complicated experimental stuff. I will not stop on this now. Here just several of the papers devoted to, devoted to in this study of wave attractors, but the problem with numerical simulation was that here we have a high Schmidt number. A high Schmidt number already was mentioned just before yesterday that high Schmidt number actually induced the characteristics of turbulence become much more complicated and here we will see it also. And from the point of numerical point of view, you have not to resolve Kolmogorov scale, but much smaller scale if you go to turbulent regime. So these are experimental results, it's amplitude of velocity field. So you understand that here there is no motion of a fluid from one part to another, it is wave motion. So every particle of a fluid just oscillating about its initial position somehow. So this is, if you will draw in each point, time dependence of some characteristic, for example velocity, you will have a picture. And this picture will differ in different points only by amplitude. So here we show with the help of Gilbert transform the amplitude of this field and you see clearly that most of the energy concentrates of the first branch of the attractor. It is called the first branch. So here is some details of numerical simulation because as I said it is quite challenging for modeling turbulence because of small scale structures which can appear in the boundary layers. And also we needed to make numerical simulation for long time and account for non-linear interactions. So we use a high order approach and in this case it's spectral element approach and we modified code by Neck 5000 by Paul Fisher and use it finally very efficiently. It is a structure of one finite element in a spectral element approach. So this is one of my first simulations. You can see, so this is already numerical simulation, the left wall begins to move and after some time you see that actually attractor is formed with the color corresponds to a component of velocity, a horizontal component of velocity, yes. And it allows just the direction of the velocity because it would not be reasonable to draw the length, otherwise you would see nothing. But you see that actually it's moving like this. So the wave propagates like this and the transversal wave, particle in the wave move like this. And so we studied lots of things about that but I will point out now only the major facts. So on the left side you can see distribution just of density, so salinity. And on the right hand side component of velocity or it may be also the same absolute picture for the density gradient, for example, d rho over dx. So you can see here that it is quite, it is not just optical effect. You can see quite clearly even on the left hand side that something is going on in the fluid and on the right hand side you can see that first attractor is formed and then it becomes unstable and we have larger input, larger external forcing. So actually you can see almost turbulent motion and the first thing which we have done is we analyzed this with the help of Hilbert transform. So we could separate the phase and amplitude to prove the fact that actually equations for triadic resonance are fulfilled here. So for this we constructed time-frequency diagram. So first you will see only external frequency, not just the frequency of the external forcing but later there appear dotted waves. So this instability is triadic resonance. So we have studied it in details in 3D simulations also and unstable, also unstable case. And for the first time we finally obtained very good comparison between experiment and numerical simulation because before the discrepancy between numerical simulation and in previous tries, for example in 18.8 in John Fluid Mechanics there was very good paper but they couldn't and they reproduced numerically in two-dimensional simulation attractor and by characteristics it differed almost twice. So here the effect of the wall happened to be quite important. It's like a stock slayer and we studied its effects in details. After that we had actually received this. We went to the Brasserie-Jorges in Lyon. So these are my courses. Thierry Duxois, director of the Le Bruttard physique in Lyon and Evgeny Irmanyuk. It's Roberto Kamasa just was there also. So we published this paper about this, John Fluid Mechanics and Europhysics Letters. And just now we submitted a paper with Felix Bakibanza concerning the effect of the lateral wall which also may be important for geophysical applications. So here you can see the appearance of the instability on the attractor internal wave attractors. Here it's how it looks like if you can see the time tie dependence of a variable on the branch of the attractor. And here's the time frequency diagram, one of which is experimental, one of which one of them is numerical and from my point of view it's even difficult to say which one is DNS, which one is numerical. So I will now go to the next representation of the internal wave attractors. We can study them with the so-called by spectra. So on both axes you have frequency and by color you can represent a quantity proportional to product of the amplitudes. In this case if you can clearly see here that here we have triadic resonance because if we will sum this value on horizontal axis plus value on vertical axis, it will lie on the anti-diagonal only if omega 1 plus omega 2 is equal to omega 0. And in such a way if the external forcing is very intensive, we will have cascade of instabilities. So here we can see that we have cascade of triadic resonances. So first the external forcing produce the two major dotted waves but these dotted waves also become unstable and produce also dotted waves and so on. So we have cascade and we have finally the spectrum which is spiky, which has many lines on the background of some continuous spectrum. So here we have investigated this and also what was important for us is to investigate the distribution of varticity because there is a criteria of miles forward. If varticity is higher than a certain value, varticity divided by Bruntweis frequency if it is higher than 2, then there is a probability of overturning means intensive mixing. So we studied experimentally and numerically actually here, for example, for some time we have probability density function of varticity in the whole domain and these branches, these left and right arms, they go beyond 2. So it means that there may be mixing, overturning and intensive mixing. So we have studied this. This is from experiments. So for different amplitudes we have different regimes and also I will not go into details of this picture just what it means. It means that we have cascade wave instabilities and of course we get longer waves and shorter waves. So shorter waves are more prone to overturn if they have a higher amplitude. So this is what actually it shows but I will not stop on this. I'm sorry for rushing here. But just to say that we studied this. Also now I will show you the importance of the account of high Schmitt number for the laboratory. Here we have large amplitude of the oscillation of the left wall and this time you see that in contrast to the previous simulation they appear very small structures because here the Schmitt number is 700. First I thought that it is some numerical stuff but actually what we have is small scale structures which propagate inside the area and we have such picture much more small scale. But anyway if you will filter it on external forcing you will get attractor. So it is moving but very very stochastically turbulently but anyway there is attractor structure in this simulation. So what next I would mention is that if you will oscillate not all the left wall but only a small part of it here just I didn't draw the inclined wall but as before it is a trapezoidal domain. You see here that after some time actually you have again almost two dimensional attractor. This is amazing to my point of view because from the local source if you will just drop a stone to the water of course you will have cylindrical waves but here you have a localized source just near zero but you don't have cylindrical waves around it or some kind of cylindrical symmetry but you have almost two dimensional attractor and two dimensionality we analyzed with the help of correlation functions. So this is what we are working now. And the next thing I will switch to another topic but related with internal waves not to waves attractors, not internal waves. So we will consider now experiment with rotating homogeneous fluid so now there is no stratification only rotating annulus. So it is like it is a vertical cross-section and it is actually symmetrical. It has a rotation and up a surface on up a surface we impose some forcing. So in this case I will now switch to another. So for this there is a construction of experimental facility also in Lyon. I will not comment it. I will only now show the results. And just a second. So this work we are doing now with Evgeny Romanovich. He doesn't know yet about all the details of this and for its graduate student in Moscow University. So we have such, if you will cut this for the depth z minus 15 you will get such a picture. So actually nothing especially interesting just some presence of internal waves. So this is not something special. Next we will do in the opposite direction. So first we moved the up surface like a precession in the same direction as the rotation. But if you will make it in opposite direction you can see clearly that there are regions of accumulation of energy. So here you have it very clearly. It's in a young picture from Chinese philosophy. And if you will cut it vertically you will finally have very nice attractors. So you can see that they are in counter face the left and right sides. But yes you got the attractor in rotating absolutely homogeneous fluid. Also we studied different cases and also for liberation. By liberation maybe it's not very correct word but I mean that if you just make a vibration of the upper surface it not moves like this as a effect of tidal effect but just vibrate in the frame of reference rotating frame of reference. You will also have the internal wave attractor. So these are completely new results and actually this is the first conference we presented and we are working now on publishing this. So this is completely new and for the first time it is presented on any conference. Thank you very much. Yes of course. There are actually a number of problems which are very interesting. Yes it's combining internal waves and inertial waves and also studying it in a much longer geometries and to compare it to Miles and Howard to spectrum I just forgot the names but Garret Munch which is known in oceanography and also there may be attractors of different forms which are multiply reflect from the sides so yes here it's plenty. It would be interesting we haven't done this. The first what we will do is biharmonic which is this corresponds geophysically and astrophysically to situations with tidal excitation. For example in this situation you have a loom which is rotating around a planet or some object and if it rotates in one direction there will be nothing just some internal waves but in another direction it's finally good. Just to show this you will have just a second excuse me. I'm sorry it wasn't here but okay sorry I haven't prepared but instability here looks very very impressive. If you want I can show you it's just on a different video. Excuse me yes okay thank you.