 I invite you to talk in this very, this place, very exciting conference, thanks to the organizers for organizing this meeting. I'm going to talk about particle pair diffusion, in particular a new theory since Richardson's original theory 1926. And I work in Saudi Arabia, King Fahad University in the Department of Mathematics. Let me begin by, this is going to be a basic outline of my talk. I'm going to go over some of the classical theory that is based on Richardson's locality for pair diffusion. And then I'm going to introduce my new non-local theory. And then present some results from simulation method, results, and then conclusion. So let's have a part one introduction, Richardson's locality. Now, first of all, let's just talk about this few snapshots of turbulence, as we know, we're all practitioners of turbulence, very one of us complex sciences that we know of. And it's a kind of mixture of statistical picture together with coherent structures and so on, and a lot of exotic details. And that has consequences, particularly for diffusion. On the right is some classical result. These are actual particle particle trajectories from Van Dott way back in 1984. So one of the big questions in turbulence is how does turbulence diffuse particles? Now, in 1921, we have the famous Taylor's theorem for one particle diffusion, that's kind of pretty well known. But there was a second famous paper, 1926 by Richardson, in which he addressed not one particle but two particles. So we have the one particle theory of Taylor 1921. But then we have, say, pair diffusion, like that. And how do these in turbulence, how does this, so this is called pair diffusion, this is one particle and this is two particle. So Richardson, this is 1926. And Richardson more or less pioneered this theory, this field of pair diffusion almost single-handedly. And later on, of course, this was all that theory was embedded within Kolmogorov's K-41 theory. Now, K-41, of course, pertains to the inner range of turbulence. Let's assume that this is a turbulent spectrum. Most of the energy, of course, is contained in the large scales by quite a long way. This may contain only a fraction of the all the energy in the turbulence. But of course, we know we have a rate of any dissipation. There's a kind of cascade process randomization. So in this inertial range, it's also far from viscous dissipation. We can assume it's statistically stationary. It's homogenous and isotropic, so which allows you to develop scaling arguments. You have the famous five-thirds spectrum. Of course, this is after 1926. Richardson had no knowledge of this as such. And the questions, the key questions we want to address is what governs pair diffusion process? And in particular, what roles, if any, do local and non-local processes play in the diffusion process? And in particular, the importance of intermittency and also related problems of how do you actually model these things? Theory is one thing. But we'd like to be able to simulate these things. That's another important issue in itself. So let's go back to first principle. Let's go back to Richardson 1926, in which a very classical paper. It's only 100 years, in fact. It will be 2026. It will be a famous centenary of this famous paper. What Richardson did was he collected a number of data from various sources, mostly from geophysical sources like volcano ash, spreading of volcano ash and balloons in the atmosphere. And he parted this data. And he presented the pair diffusion theory in the form of the, what's called the pair diffusion diffusivity. This is not the one particle diffusion. This is literally the pair diffusivity, relative diffusivity associated with pair diffusion. And what he said is consider, well, consider this picture here, at any particular time, this is your pair separation, as it were. And he made a number of bold hypotheses. One was that he assumed that the pair diffusion itself is scale dependent. So earlier here is the modulus of the pair separation. And that should be the second important, couple of other important hypotheses embedded in this. One is that he assumed that there was a single power law that governs the entire pair diffusion process on all scales. In other words, this power here is a single power. At priori, there's no reason why he shouldn't have the power itself to depend on. But he made the hypothesis it should be a single power. And the next hypothesis, he actually gave a number to this, which is four thirds. And which he obtained kind of guessed from the data. This is where the data came in. And actually the four thirds, I'll show you in a moment, this is actually equivalent to a locality. I didn't use the word locality in those days, but it is actually equivalent to locality in a way that I'll show you, very easy way of showing it. So this is the basis of Richardson's locality theory. Now, why should the pair diffusivity be scale dependent? Well, it's not that difficult to see why. If you have a very small separation of particles, then the energy at that scale is going to be small. So the rate of separation is going to be small. But if pair separation is that much, then you have bigger energy containing. So it's not unreasonable to hypothesize that the pair diffusivity should be scale dependent. And in fact, it should be an increasing function of separation. So long. The other thing, well, let's go on. So this is Richardson. And let me just show you, that's very important, actually, to understand what you mean by locality. There's a little bit of a issue as to what you actually mean by locality. Locality definition I use is that something is local, the pair separation process is local. If the further increase that separation depends only upon energies that are containing energies of that size, a little, you have to think about it. In other words, in fact, if you, if you now anticipate Kolmogorov 41, you know that energy content in energies of that scale is given by the minus five thirds law. And from this, we can get, obtain a scaling for the pair diffusion directly as being proportional to L times Yorba. Yorba is just proportional to square root of L, Yorba, and that gives you immediately four thirds, which is what I said. That, oops, yeah. So this four thirds is in fact equivalent to locality, which is actually quite remarkable. Because in doing so, Richardson was actually anticipating Kolmogorov K 41 16 years later. So we could almost say Richardson predated Kolmogorov rather sort of in a roundabout way. But it is actually very important to note that this kind of argument does actually anticipate K 41 16 years ahead of its time. And in fact, in 1941, Obokov also showed that this four thirds, which is, which is the power law for the pair diffusion, is actually equivalent to the pair separate, that's so called TQB law. And together, these two are known as the Richardson Obokov regime, locality regime. In fact, it's become, and since then, it has become like folk law. It's, although it's, it's not a law, it's an unproven hypothesis, but becomes so much part of the background it's now referred to as the Richardson Obokov law. In fact, it's not a law, it's just a kind of hypothesis. The other thing I wish to emphasize is that this theory, and it is a theory, is valid for asymptotically almost infinite Reynolds number, that is for very large inertial sub-range. Now, how large the inertial sub-bridge has to be is still an open question, it's just very large. Let's leave it like that for the moment. Now, this is how things stand at the moment. And since 1926, daily 400 years, this has gone more or less unchallenged and more or less every theory of pair diffusion is based around this idea of locality, that is till now. And so, let's now look at it again. So, I'm going to, I revisited this whole idea recently. I've got a new picture coming up and let's start with the 1926 data set again. Now, what I note, first of all, for a long time, and I think everybody thought the four-thirds that Richardson drew was actually a best fit. In fact, it's not a best fit, it's just a guess. The best fit, if you do all seven points, is actually 1.248, which is the red line. So, that's a little bit. In fact, Richardson and Stomer later on did note that you could really have any power between 1.3 and 1.5. And of course, there's a lot of error associated with, in the sense that these data points are from different sources, there's lots of buoyancy in the industry and so on. So, this power law is invested in approximation. Furthermore, I noted, I don't know, if you look at this point here, if you go back to the original paper, this comes from molecular diffusion experiments. It's got nothing to do with turbulence at all. It's laminar flow. It's in fact some molecular diffusion. So, I thought, well, let's remove this point. It's an outlier. Forget that. Let's see what happens. And all of a sudden, this happens, the black line. So, these six points and these fall nearly on, it's nearly a perfect fit, except for this point. And you get a slope of 1.567, which is actually beyond even what Richardson and Stomer suggested. So, at this point, we then are entitled to ask, well, is it really a locality? This seems to suggest something beyond locality. So, at the very least, what it suggests is that at least locality is not necessarily the best model. And there's certainly room for new thinking in this field. And what about other evidence? There have been lots of observations, tracking experiments, DNS. The problem with all of them is either that the errors are too large. For example, Tatarski, 1960, he observed comets, tails in the atmosphere. But the errors are just too large to be decisive. And other things like measurements in the lake, the quasi-two-dimensional approximations. And the problem with tracking experiments in DNS is that we cannot get a large enough subringent. That's the DNS. The better DNS can do the data is maybe one or two orders of magnitude. But we need really much larger than that. So, in other words, things are still, we've progressed not much further than 1926. And it's culminates in this very famous statement by Salazar and Collins, 2011, that there has not been an experiment that has unequivocally confirmed the Richardson-Oberkov scaling over a broad enough range of time and with sufficient accuracy. So, the summary of all this so far is that there is room for new thinking. And I will hope to persuade you that there is, in fact, an alternative theory. And I'm going to, therefore, approach this whole subject from first principles. And in particular, I'm going to retain some of these ideas. Some of the ones, the bits I'm going to, the assumption I'm going to retain are that it's nearly Reynolds number. That means very large enough subrange. We retain the hypothesis, Richardson's two hypothesis of scale-dependent pair diffusion and a single power scaling. But I want to generalize the spectrum, not just to K minus five-thirds, because I want to see what happens when we have, for example, intermittency correction. Because we know that in 1962, Komagoro himself corrected the minus five-thirds to a correction to take into again intermittency. And I would like, and unlike the original work, I wanted a little bit more rigorous than just plain scaling arguments. See if you can. I'm going to adopt the method of Fourier decomposing the relative velocity itself. And we know this is the sort of a bachelor argument, bachelor, the Fourier decomposition velocity. I do now, I now look at the relative velocity of two particles here. And I decompose this also in Fourier component. If you take the scalar product of this with L, L dot V, and take the ensemble average, this then becomes, this of course is actually the definition of the pair diffusion itself. And homogeneity means we can integrate also overall space to effectively get rid of this X1 dependence. And we need a closure hypothesis so that we can repose this in terms of L squared and the average of A k and so on. Alpha, beta are the angles between these vectors. And now we appeal to isotropy. So we integrate over shells so that we can effectively get rid of the alpha and beta dependencies. And finally, we therefore obtain a form for the pair diffusion directly as integral k is now the modulus of the wave number. And where do we go from? So now we make the central physical hypothesis of this new formulation is that we assume the existence of a local neighborhood. And I, let me write this down. The idea is now, is this, that we have a pair diffusion size L. And let's associate a wave number with this. And I assume in physical space, there's some kind of neighborhood called the local neighborhood. Outside of this is non-local. In wave number space, this amounts to saying, let's say the pair diffusion is here, kL. And so in other words, this is the locality, local neighborhood associated with that separation. This is the non-local neighborhood. And this is what, in other words, we can repose this integral as a sum of two parts. One is the local, comes from the local neighborhood here. And this is the rest. In other words, the local neighborhood consists of those wave numbers which are less than, well, k star is this here, we should put here. It's an arbitrary, it's an arbitrary size. That's not, for scaling purposes, it's not important to determine k star. Although if you want to measure, you'll have to make some kind of, oops, where are we? Here. So here it is. So this is your L, this is k star, which is the designation between local and non-local neighborhoods. And this allows us to pose the pair diffusion as a sum of two parts, local and non-local parts. And the only thing we need now is a form for the energy spectrum itself, which I said we can use a generalized k minus p. And we look at the range between p cos one and three and over a wave number range, say k one to k eta. And this then allows us to explicitly write the pair diffusivity in this form. And then after a bit of, and there's a little bit of algebra, it's not important. But the important thing is that when you work all this out, putting k minus p in here, we obtain this, that the pair diffusivity is the sum of two, essentially two processes, a local and a non-local one. And indeed, when you work this out, you'll get that the local power range is the correct locality scaling. So when you put p cos five thirds, we get four thirds as we got from Richardson. But then there's a second component. And this also shows what Richardson's hypothesis amounted to. It amounted to basically ignoring this a priori from the beginning. And the difference really is that I haven't about, I haven't ignored this from the beginnings. And this shows very clearly that we, that the pair diffusion process is the sum of two local and a non-local process. The non-local power law is always two. Okay. So what about the other, the second Richardson hypothesis of single power law? If we now say that KL is a single power law, in fact, we get this. So that the manifestation of the pair diffusion as a single power means that this must lie in between the two asymptotic limits. So then in other words, gamma p is between the local exponent, local power law and the purely unlocal. So in other words, it's between one plus p over two and two. So this is actually the main outcome of this theory. And we can obtain some specific results as follows. That is that if we were to say in the simulation go from p cos one, which is the pure local limit, we would get sigma one to the power of one. And in the opposite, which is a non-local limit, we get sigma of two. And then we would see a smooth transition between them as single power laws. In particular, for the commagora of five thirds, we would see that the power law is bigger than four thirds, but less than two. This is the basic result. And if we look at the commagora with intimacy, which I say about 1.74, we would note that the power is much greater than four thirds, but still less than two. This is as far as we can go with theory, to get an exact precise numbers for gamma p. We have to do experiments or simulations, but this is the main result of the theory so far. And finally, if we can go to, we can actually convert this into a generalized commagora, obokov hypothesis. So this is equivalent, say, to the mean squared separation to some power chi. Now, note that the gamma p, if you look at the pure local local theory, then gamma is linear in p. That means small changes in p produces small changes in gamma, but chi is actually non-linear. So we can expect bigger changes there. In particular, if we look at p cos 1.74, we get the difference between three. It's quite large. And of course, a key question is, this is only purely on the locality theory. What will the non-local theory give us? So that's the question we need to address through simulations. So this is the theory. What we now move on to simulations. The problem that I said is that DNS is out of the question many decades away. So we look at a Lagrangian model, kinematic simulations, which is basically a Fourier series with the amplitude proportional to the energy spectrum in this form. And we can also have an unsteadyness here. That's not so important at the moment. Let me skip that. However, one limitation, it's not dynamical. So there is some, but it has the advantage that it can generate very large inertial subrange, which is suitable for our purposes. It's not perfect, but it's it's interesting. We've got the moment. So let's see what happens. And of course, we can integrate the what we're doing. KS, we actually generate the entire trajectories by integrating the flow field through any suitable method, Runge-Cutter, Adams-Bashford, whatever. And we harvest this. The randomness in KS comes from choosing many different fields and the directions are chosen isotropically in each different field. So then we get an ensemble and then we can do statistics on this. And the nice thing about KS is that the energy spectrum can be made arbitrary. Now, ideally we would like to get this kind of spectrum, K to the fourth or small and five thirds or minus p for large. But in fact, we do a trick here that for, we know that for very large scales do nothing but sweep. So we don't actually need the large scale spectrum. So we can actually simply zero this in here. So we all we need to do is to generate a very large inertial range spectrum and off we go. I'm going to stick to 10 to the six. Seems to be reasonably good. That's the maximum we can do at the moment. And let me go on to, let me just say it's been validated in the past in 1999. Am I doing for time? Let me, yeah. So 1999, we published this paper comparing KS with available DNS at the time. It was very, quite good result, very good matching. And in fact, we even got the fourth order flatness to be very well matched, although KS is not actually specifically defined, designed for this. So that was reasonably good. Although this was for low runners number, I have to say this was a, anyway, so let's get on to the results. We've used KS to generate a large inertial range and we looked at trajectories and these are the sort of key components. So we look at 10, inertial range of 10 to the six, we use about 200 modes, 5000 flow fields, eight pairs of particles per realization, about 40,000 trajectory, which is the largest ensemble I've ever come across so far to date. They're using the fourth order bashforth method, fixed time step and so on. And these supercomputers, so let me just give you the results. Main result is here. And if we note this, this is exactly as predicted by the theory. If we go back, for example, where is it? There's a theory. I have seen it here. This was the theory to prediction. It was go from one to two with a smooth transition all the way through. That was the theory. This is the result from KS, which more or less exactly matches the theory. We can't go to P because that's a singular limit. We can get close to it though. It goes roughly from one to two. And this is the same thing on the right except compensated by whatever the power. So it's flat wherever the power, in fact, that's how you obtain the power. So wherever it's flat, that's the value of gamma we take. And this summarizes the result. The blue dashed line is what we would expect to get from pure locality scaling. And the black dots are the ones that we actually did. And the ratio between the two is on the right. This is the cyan line. And the maximum deviation is roughly where you'd expect Conmagorov things. In particular, for the Conmagorov case, five thirds we get the scaling of 1.53, which is already larger than the four thirds, Richardson. And with intermittency 1.74, we get 1.57. Now, this is almost exactly the same as the 1926 revised data. I think the closest is so close. I have to say I'm a little bit cautious. There is so much error that I think it's a little bit coincidental, even if I say so myself. So I wouldn't read too much into how close these two things are. But nevertheless, it's interesting. It's remarkably good agreement. And at least it suggests that the theory is reasonably plausible. Okay. And finally, the generalized obokov. We can convert this. And this is the blue line is the, again, locality. And this is the obtainable. This is the right. It's the same, except we're looking at a smaller unit. And indeed, Conmagorov, we get 4.21, which is already much larger than, I said that the, it's much more sensitive. It's nonlinear, so it's much more sensitive. And indeed, for the intermittency case, we get 4.65, which is much larger than T cubed. Okay. So the summary of the theory so far, and is this, that the limit of asymptotically large inertial subranges, we obtain this from theory and simulation, combination theory and simulation. It's been observed both in 1926 revised data and kinematic simulations. And just as a note, that if we assume that the integral length scale is, take ad hoc value, 10 times larger than the largest inertial range, then we get roughly a minimum Reynolds number of about 10 to the nine, which is why you cannot do experiments and DNS at this moment in time. We have to look, rely on observations in the atmosphere, if we can get a hold of them. So this is huge. We're talking about geophysical turbulence and larger. So finally, take home messages. Let's repeat. Nonlocal theory has been developed for pale diffusion. We've seen that it's governed by local and crucially nonlocal processes. At the heart is this. The main physical hypothesis is this, that the, we assume the existence of a local neighborhood and nonlocal neighborhoods. Both of them are significant. This is the specific results obtained, repeats the previous slide. But a word of caution, still only a hypothesis. Kinematic simulations is not dynamical. It's not perfect. But so we still need experiments and or DNS one day to validate all this. But I think perhaps the most interesting thing is the last one that excites new thinking. Whatever you say, it's, there is new thinking and new methods as well. And it must have implications for general theory of turbulence, I think. In fact, I haven't mentioned this here, but I've already extended this to inertial particles in the stokes. I've got a poster out here. So you'd like it to come and look at the poster. And that also needs to be modified. So I think this is, to me, the most interesting, most exciting part of this research is it does actually offer a new kind of way of thinking of, for pale diffusion and for the general theory of turbulence. And I'd like to thank you for your attention. Thanks very much.