 Okay. First of all, I want to thank the organizers for inviting me and let me give this talk to you. The work I'm going to talk about is called Single Particle Dispersion in Stable Strategy for Turbulence. I'm Nicolás Zuchowalski. My advisor is Paolo Minini and this work was also done with Mark Rust. Okay. So, Single Particle Dispersion has a fundamental role in geophysical flows, at least as it offers a unique insight into dispersion of pollutants and mixing of nutrients in the ocean and suspension of cloud droplets in the atmosphere. Particle Dispersion has received significant attention in homogeneous isotropic turbulence in the past years, but in stably stratified turbulence it has only been studied recently and the work carried out is rather limited. Stably stratified turbulence is different from homogeneous isotropic turbulence as the stratification suppresses the vertical dispersion, but its effects on the horizontal dispersion are less certain. And in recent years measurements of traces in the ocean and the atmosphere became available. So, what I'm going to present to you are models for the parallel and perpendicular dispersion of single particle Lagrangian single particles under the Boussinesq incompressible approximation. So, these are the Boussinesq equations. If we linearize these equations in the parallel direction, the solutions are waves that oscillate in the parallel direction and the frequency of the waves is the Brunt Weissala frequency, which is a measure of how steep the density background profile is. So, from these equations we have two relevant dimensionless parameters, the familiar Raynox number, which in our simulations was around 10,000 and the Frut number, which is another measure of the stratification. And typical values from our simulations are 4 times 10 to the negative 2 for n equals 4, n being the Brunt Weissala frequency, and 2 times 10 to the negative 2 for n equals 8. So, a lower Frut means more stratification. So, we force our simulations, our flows, with two different, very different mechanical forceings. In one set of simulations, we applied a Taylor-Reinforcing, which is a bidimensional forcing consisting of two counter-rotating eddies with a shear layer in between. This generates a coherent flow structure in the largest scales. And this flow structure will be very important in our results, I will show you in a few minutes. Because in this shear layer in between, there will be a development of winds, of net winds, that will carry particles with it. And in the rest of the box, there will be no net wind. In another set of simulations, we applied a random forcing. And in this set of simulations, winds will develop all along the box. And thus, we will have very different large scale properties in both sets of simulations. So, we performed four runs with 512 cube grid points. With n equals 4 and 8 for both forcing methods. And for both, for all simulations, we injected 100,000 Lagrangian particles. These are fluid particles. And so, at each point their velocity is the same as the fluid. Okay, so in stratified turbulence, we have the development of layers in the parallel planes, parallel to the stratification. You see here, what you see here is the density fluctuations in a parallel plane of a stably stratified turbulence simulation. And here, some trajectories in the parallel plane, or we can think of as the altitude of the particles. And what we see here is that the dynamics is dominated by a wave-like motion. And that particles oscillate in layers which they do not live. In consequence, there is little dispersion in the vertical direction. And so, here you see the Lagrangian energy parallel spectrum. And when we computed this, this is from our simulations, we computed this spectrum. And we found that it was similar to the Garrett-Mann spectrum, which is an empirical spectrum for internal wave energy in the ocean. This is an Eulerian spectrum, but Lien took this spectrum and made it Lagrangian. And we see here this line, this violet-blue line. Here is the parallel spectrum. And so, we see that it's flat for frequencies shorter than the buoyancy frequency, the Brunn-Guy-Sala frequency. It has a peak near the N, and then it goes fastly towards zero. And this is what we found in our Lagrangian spectrum. So, here you see the energy contained in the largest studies. Here, the energy contained in waves near the Brunn-Guy-Sala frequency. And then here, the energy of the smallest, fastest enemies. And from the parallel trajectories and the Lagrangian spectrum, we proposed a model that consists in wave linear wave superposition with random faces and the amplitude of the waves derived from the Lagrangian spectrum. What is this? We can think that the altitude of the particle consists of a sum of waves with random faces and the amplitude of the waves derived from the Lagrangian spectrum, with this coefficient, with the exponent of the frequency here taken from the Lagrangian spectrum. So, there are no free parameters in this model. So, here is the parallel in a particle dispersion. We see that it's ballistic at short times. It has a plateau for intermediate times, and then it grows slowly due to thermal diffusion. The Taylor-Green simulations show a deviation from this behavior due to over-returning. And our model correctly fits this ballistic behavior and the plateau. And this is especially important for short times for the ballistic behavior because instability stratified turbulence used to be modeled similar to a machinist isotropic turbulence, but waves can show that behavior as well. Our model does not take into account the thermal diffusion and the over-returning, and that does not take into account that and cannot fit the longer times in the simulation. Another way to study these waves is by computing the probabilistic density function of the waiting times. What is a waiting time? It's the time it takes a particle to cross its mean altitude to consecutive times. And so, we can think of the waiting time as half a wave period. And by computing the waiting times, it's another way of studying the periods or the frequencies. And we see here that the simulations are in blue and red, and the model is in green. And it's in good agreement with the data. And another thing that we can take from this is that the PDFs of the waiting time is not exponential, meaning that the particles have memory and thus waves carry information from their initial conditions for finite amounts of time. Now that we saw that particles oscillating layers that they do not live, let's study the dynamic of the perpendicular trajectories. So here you can see the density fluctuations in a perpendicular plane in a stably stratified turbulence. And here the trajectories, the perpendicular trajectories of some particles in Taylor-Green simulation. And we can see that most of the particles are trapped inside eddies. And some of them are affected by the wind in the shear layer in between those two eddies, I said I was talking before. And here you see perpendicular trajectories of particles in the random simulation. And we can see that particles are affected in different directions. The colors correspond to different heights. And so different layers have different wind directions. And so particles are affected in different directions. So we want to model the single particle perpendicular trajectories. You know we have two components. We have trapping from the turbulent eddies and the drift caused by the wind. In principle we can think of the trapping as similar to homogeneous isotropic turbulence where it has been modeled as a continuous time random walk. And to this model we add the drift. So let's start with the continuous time random walk model. We can think of that for each step a particle is trapped in an eddy with radius r for a time t with a certain velocity u. This gives a central angle of motion and thus a displacement delta r. And so the probability of a time t is uniform between zero and the Eulerian time. The probability of a given radius is this one is taken from the Conmogorov theory of turbulence. The probability of the velocity is taken from the Lagrangian velocity data but can also be assumed to have a Rayleigh distribution. And well this gives a central angle of motion theta and a displacement like here. So for each particle its motion will be its total displacement will be a sum over these trapping events. And to that we add a drift that we model as a bimodal Gaussian distribution to fit the observed Eulerian velocity perpendicular velocity. So here we see the horizontal single particle dispersion for our runs in black for the simulations in red for the model. We see that the model is a single agreement with the data. And the result of single particle dispersion is ballistic at short times for both sets of simulations but then it shows a deviation. In the random simulations as the winds are more relevant and present all along the box and generates a coherent abduction for our particles. The ballistic behavior continues at all times. And for the Taylor-Rin simulations after the Lagrangian turnover time which is the mean correlation time between single particle trajectories it scales as t to the 0.08 and 0.9 which is still different from homogeneous isotropic turbulence which scales as 0.5. What seems to be happening here is that the competition between the deities, the trapping and the wind generates a scaling in between isotropic homogeneous turbulence and ballistic. Okay, so what I just show you is the mean value of the displacement and I think more information would like to model the PDF, the probabilistic density function of the single particle dispersion. That is to say what's the probability of having a particle at a distance r from its original location at a given time t. And so we did this for the model and the simulations. We see here the simulations are the solid lines and the model is the dash lines and there is a reasonable agreement between all curves and we see that the PDFs are non-frightly meaning that we cannot simply model the dispersion as a continuous time random walk. We need the drift. Another thing that's important here is there are two separate regimes. Before the Eulerian time the trapping dominates the motion of the particles and after the Lagrangian time it's the drift that dominates the motion and this gives very different PDFs for both sets of simulations for the Taylor ring here and for the random here. Okay, this is for just one time. It's a snapshot of the particle dispersion but if we take all the snapshots and put them together and we color the lines of equal probability we get these which are the set on terms of the probability density function of single particle dispersion. In blue we see the simulations and in red the model here for the Taylor ring simulations and here for one of the random simulations we see that our model is in good agreement for all times and so this is not just the mean value but it has all the moments of the PDF and so the importance of the model is that with models like this one can estimate the concentration of quantities in a given flow without resorting to simulations with difference in the initial concentrations which are much more computationally expensive and we see here that in the random case the wind sets a net displacement of the PDF which our model correctly captures and so the main importance here is that for example if one knows the large scale behavior of a flow from a large scale simulation and can assume the small scale turbulence given distribution let's say Riley for the velocity fluctuations then using a model such as ours one can estimate the concentrations of particles so to sum up the simulations and the perpendicular model show that horizontal dispersion is not universal as it depends on the forcing method the parallel model shows that vertical dispersion is explained by weights and it is fundamentally different from homogeneous isotropic turbulence both models have no free parameters the perpendicular model takes as input the probabilistic density function of the Lagrangian velocities and of the Eulerian wind and the parallel model takes as input the exponent of the Lagrangian parallel energy spectrum and the horizontal model presented here could allow the probabilistic prediction of the concentration of quantities to be transported by a flow so that's it thank you in which set of simulations there are two sets of simulations which have different forcing in the Taylor Green forcing we have two rotating eddies and the forcing goes down as we go to the center of the box and when we say layer it's not an exact layer but it's the layer in the middle and in the... how many layers are developed depends on the Brunweissala frequency here larger Brunweissala frequency you have more layers more clear layers where you can see this tendency of the particles we did not try to compute the Osmiel scale I think we talked about that at the beginning of this work sorry, can you repeat the question? oh, yeah, yeah I think it was comparable I don't have the numbers here I think the Osmiel scale was a little shorter yeah, exactly that that's why you have this behavior of the plateau sorry? this... oh well we are saying that right now how to break that but it's not up to the forcing but to the anisotropy of the box and also the forcing method you can see here that the Taylor Green is not the same as the random forcing but the random forcing dissipates less energy so there is much more energy accumulated in the larger scales here and we see the plateau as well