 Good day. Yeah, I can get some. Yeah. Good day. I'm sweating. I'm sweating. I'm sweating. I'm sweating. Okay. Thank you. Just shake hands with our director. Hi. I'm here. Thanks for coming. Okay. Thank you so much. I want to talk about this topic. It's supposedly on, but it's far enough. It's on. Snezan suggested that I speak about this topic. And I think after a while she got a little nervous as to whether I would in fact live up to her expectations. But then she was committed to me already, so it would be impossible to change it. So, what I want to talk about is the program that has survived or existed for 10 years. And 10 years, as you know, is a long time. It's a long time because many of us who have been coming to this program have changed jobs, some of them, some of us that is have changed fields and some probably even their spouses or whatever. But it's been a long, it is a long time. And it's also, it also happens that some of the colleagues who are associated with this program have disappeared. There is Steve Arsag, Padma Shukla and Albert Peters, then Oleg here and then Leo Katnoff. All of them were associated or came to this program at one time or another, but they have left us and gone somewhere. So, it's a long time in some sense. So, it's a good time to ask questions like what's been accomplished by this program in the last 10 years. I think it's a reasonable question to ask. And beyond that, beyond the people who come to this meeting, what's been accomplished by the larger scientific community which professes to work on this subject that is turbulent mixing and beyond. And what may lie ahead for the subject and for the program itself, I must right away confess that it is foolhardy to talk about such things. Therefore, I start with an apology to all of you in some way. At least to some degree, these comments that I'll make reflect my prejudices, my shortcomings, and I'm sure I'll offend all of you in one form or another so you have to bear with me and we'll move on with that understanding. But first, I should maybe remind ourselves to get us on the same page as to what the past activities of the program were and what its goals were. So first of all, how did the program come into being? In 2005, Snezana made a proposal for the TMB program held at ICTP in 2007. Usually, it takes about a year, a year and a half. And at that time, her thinking was something like this. You've heard these words from her before. Turbulent mixing takes many forms. Steady and unsteady, single phase, multi-phase, subsonic, supersonic and hypersonic speeds at very small scales and very large scales. And because the subject is rather diverse, the communities that work in these topics are very diverse as well. Some work on very basic problems and some work on very applied context and they have traditionally worked somewhat disjointly. They have not really talked to each other all that much. And so part of the goal for this meeting was to bring these communities together on the belief that they could benefit from each other. In fact, yesterday I was talking to Bruce Remington and he says he comes to these meetings mainly because in no other meeting does he meet people whom he does normally not meet at all. So probably not just him, but many others could make the same statement. And the goal of the program was to bring together these communities working on turbulent mixing and beyond. The and beyond part is quite important because not every talk here is about turbulent mixing clearly. Anything that has some connection to mixing, if you simply put up a first slide saying you work on mixing and then do whatever you want to say it still somehow works in this case. The only thought is it must be interesting and connected in some spiritual way to the subject. And to accomplish the goals that I set forth here this came about just about a year or so after the meeting was actually proposed. So the first thing is I embrace mixing in all its richness. That is don't simply say I'm going to be interested only in passive or mixing in isotropic and homogeneous turbulence and meet regularly and raise money for it of course. And the scientific program these were the goals through the understanding that simple problems one increases one's intuition that will be applicable to very complex problems. You can't simply say well you're interested in very complex problems let me just not worry about the basic stuff you do. Likewise it was very important to have a predictive capability for complex problems. And of course the idea was to publish things in some comprehensive way in cohesive venues and strengthen links between experiments, simulations and theories between applications and basic work. And build community expertise educational and outreach programs and organize mechanisms to share research tools, data analysis and visualization and finally this was spelt out explicitly, prepare a white paper for the community project for the US national academies to launch a decade study of non-equilibrium turbulent process. So that was a very specific idea. So when I make an assessment of whether the program is successful or not it's against these goals that I will try to make an assessment otherwise you can go all over the map in making statements. So the first meeting was held in 2007 and here is a group photograph that was taken at the time and I want to point out you will probably find some of your own photos. So I want to point out to some important people if I can. There is Snezana here. There is Joe sitting here and there is Victor Loa from whom I got this picture and there is Susie who really ran the program at the time. That's the important people here. The rest of them don't matter. So here is a full list of people who are present. So it was a huge list and I'll pass for a second to see if you can actually find out where your name is and not very long. Then it was pretty successful in terms of number of people and the quality of people that came. The second one even more so because it didn't last for one week actually it lasted for almost two weeks and there were ATR, the invited speakers and here is the full list but I'm not going to let you read it. That's two years after the first one and two years later there was another one and at this point we had already realized that one of the things we should do better is trying to induct younger people into the program because usually it's okay we get accomplished people let's say and then that's not enough so younger people are essential so we said well we should have some tutorials etc. for the younger people that was done and there was a list again. So it was pretty clear from all this that many people of distinction came to these meetings and they made contributions which I will tell you later on. And also different people participated in different ways. Some partied like Joe Nimla there others were very busy listening to lectures maybe you will recognize some of yourselves there again lots of people and here is another set here is another set and all people having dinner and some people dancing to music that was being provided here by Joe some were dancing even without music you can actually see this here so that's a very interesting and lively set of meetings and on top of it besides the support that ICTP gave Snezana in particular was able to raise money from all these agencies some from the US some from Europe some from Japan Russia all these places although I must say that it may look very successful at this point that it was the program was able to raise a lot of money but in reality it was a bit hard it was like pulling teeth out of a patient you accumulate some money from each of these people so it ultimately boils down to something significant I don't know why it should be that we can't support science better but that's the way it was nevertheless the program always had enough money to support its participants and at that point I got a little worried because we had so many programs as three programs already run by the time so many participants we could always say every meeting had about 200 people but in any given lecture there were only about 40 people or 50 people some were going to Basilikas to see all this stuff like Itamar for instance but some were simply not here because they didn't see anything interesting that was happening in the rest of the sessions so it was not like a regular conference like many conferences put together in one so we thought we could take a break and Snezana instead organized with some help from me a meeting in the Plasma Physics meeting of the American Physical Society in Denver in lieu of the program in 2013 we just wanted to think a little bit as to what might be happening or what should be happening this meeting was very successful there were many invited talks and tutorials and so on I here list the participants so at that point we thought that meeting here in ICTP was very valuable and the concern we had that is a lot of people with very disparate interest were coming to this meeting it wasn't clear that we were actually succeeding in putting them all together and so we thought a meeting more focused would be better and that's what happened the subsequent year 2014 and this one was the fourth meeting and deliberately the focus was limited to limited to mixing in rapidly changing environment particularly in inertial confinement and things like that and as a consequence there was significant participation from the US national labs and it was very interesting meeting where we learned significantly about the kind of problems that people in ICF have of course this meeting you know that's the numbers that Snezana provided a lot of people thank you all for coming here but I know there could be more people in this meeting for instance and then in terms of visibility well I googled turbulence turbulent mixing and beyond I didn't say how many thousands because I got different numbers depending on whether I googled here or in New York city and so but I'm sure you will google them as we keep talking about it and of course there were many publications that resulted 250 publications believe it or not and there were all different topics like the one I have listed here and many of them appeared in physical script in these issues to which many of you actually contributed and again thank you very much for that so that was a very significant number and the authors the editors of those issues I have listed here there were other issues of the proceedings of the transactions of the Royal Society and also people who edited that are listed here so everybody in some sense or the other has contributed to the visibility of the program but I must ask to be a little bit rigorous how much of this work is new I mean in other words things that were going on already sort of gets going anyhow is that the case or is it something different and did this program generate new collaborations that otherwise would not have happened I think this is a very reasonable question to ask and I will tell you a little bit about how I came to these numbers of course I haven't read all the papers some of you may have read all of them but I did look at which one was original which one seemed like a lot of review and things like that so a lot of the work actually is new and the new collaborations that you would never have thought possible a few of them have appeared since the time the program came into being but not too many people who write together are already people who are collaborating with themselves and so that was a significant part there were a few new collaborations that I am aware of myself and then all of this is fine in terms of what went on logistically and so on but like Ithamur would ask what are the results what are the results of this whole thing a lot of activity a lot of papers a lot of things but I would like to spend the rest of the time on what I must also say again I could go off in many different directions and so I take as guidance the focus of the results of the round table discussion which said these must be our goals our goals must be there were believe it or not a real consensus that there were many experimental facilities new ones that are required in order to study certain problems well have that happened or not new measurement techniques had to be invented have they happened or not it may not be necessarily by the people who are here I am just taking a bigger view of the community that is interested in turbulent mixing and related problems and I am asking whether in fact these things have happened since then in the last decade advances simulations have to be used in synergy with experiments and theories we have to aim for basic results some of the ones you can't simply you have to come to some conclusion about some problems that's what I mean by specific and applied research and basic results have to be tied together better this is the standard by which I will try to ask myself and I will say what in the last 10 years has happened in these topics but of course the choice of my topics is entirely my own and so you should excuse me for that as I already said in terms of facilities Everhart built a fantastic facility in Göttingen a closed circuit wind tunnel like this using sulfur hexafluoride as the working fluid and it can be pressurized from 1 millibar to 15 bar so that's a huge change in pressure as a result the density changes and the Reynolds number can change over several lots of magnitude and using traditional grid he could get up to Reynolds number of 1500 micro scale Reynolds number and if he used active grid it would be about 7500 this is about 1.5 times more than what was already done some time ago by Tisler and Rebolovich in 1966 there was a wind tunnel in Southern California that aircraft companies used to use for testing full scale aircraft and before it was decommissioned they were asked to do some basic study in that wind tunnel for some time that Reynolds number went up to about 1000 or thereabouts now this tunnel is working we might ask what are the results as always and one result that actually came about is that in fact you see behind the grid the energy of turbulence decaying like a power law and the power law exponent in the past which did not cover very high Reynolds numbers so it covered maybe up to here you can see pretty much any exponent you can find but what this experiment in particular I think pointed out was over a Reynolds number range of about 3 decades almost it is approximately 1.2 and of course you should also know that blue squares were my own experiments that was done over a Reynolds number but had the same exponent the point is this is exactly what Safman had predicted and in fact not only was the decay law the same as Safman's prediction but also other things like the growth of the length scale etc I came out to be the same as what Safman had so I sort of tend to think it is entirely likely that the physics that goes with the decay of turbulence behind the grid is pretty much contained in the theory that Safman made so I would say that is one of the important results to come out of this another this is taken from this paper another interesting experiment with which I had something to do therefore ICTP and this program had something to do with Kapio near Balania this is a huge pipe that has been built and this is the same tunnel where Mussolini had his has the idea of making aircraft to prevent bombing from the Allies of course he didn't build more than two aircraft actually but anyhow that was in disrepair and using that space we built a I didn't have much to do with it afterwards Alessandro built a very beautiful and long pipe 110 meters in length as you can imagine it is a long thing almost a meter in diameter and the reason for building that was you can go to very high Reynolds numbers as in fact was done in the super pipe that which was small in diameter about a 0.13 meters when you go to very high Reynolds numbers the scales of turbulence which scale in some proportion to the big diameter will become so small that you cannot actually measure them so in fact the idea was that unless you really invented new techniques for measuring small scales which I will talk about a little bit later on you had to really build some big experiment and this has been built but not not really nothing very important has come out of it yet I shouldn't be careful when I say that so we'll see, we'll see there is a lot of promise in this facility likewise in super fluids there is this beautiful facility that was built by Philip Roche in Grenoble and it's called a Shrek and it's like the French washing machine that you have all probably heard about so two discs which rotate in opposite directions creating a turbulent flow and it can go to Reynolds numbers of the art of 10 million which I think will be remarkable you can use either a liquid helium or super fluid I think there is a big expectation on what might be possible those of you who remember Patrick Tabling he had one facility like that but it was smaller lower Reynolds number and things like that and another facility is this convection facility again built in Göttingen and this now goes to really numbers of 10 power 15 it can be pressurized up to 19 atmospheres use a sulfur hexafluoride for the working fluid I must also point out however we had a facility here which we went up to Rayleigh numbers 10 power 17 and this is still some ways to go but it was using helium and it had its own issues which we had to deal with so that's I think produced a number of results yet in my opinion very conclusive and there is this Taylor Couette apparatus very beautiful in both 20 and University of Maryland again goes up to Reynolds numbers of that large and you should compare with what Sweeney had some years ago almost there but now it is much better instrumented and everything than before finally almost finally I want to without showing you an experimental facility one of the problems in which real progress has been made is rotating convection surprisingly it seems to me that the big picture about the rotating convection seems to be clearer than it was before and I produce a graph for you which plots Nusselt number here and Rayleigh number here there were no buoyancy no rotation whatever then it would follow this line and of course we don't really know what that line is if you ask the experts in the asymptotic state but for what we are going to talk about it doesn't seem to matter and in this regime when the rotation is relatively large and the Nusselt number doesn't follow this it will follow some power law like this and then afterwards when effectively the Rayleigh number becomes larger so the buoyancy is larger relative to rotation it follows this line so I think this is a beautiful set of experiments which have happened over time in all in the last decade or so finally Rayleigh Taylor which is one of the very important problems for this community there are many issues about creating experiments and experts here would know and others would have to just believe me but I think one of the ones about which I am very excited is the apparatus built in the University of Arizona by Jeff Jacobs and what it has is a tank here that sort of accelerates down which has two fluids there, different densities miscible or immiscible and then it just basically is controlled acceleration and you can measure all the properties that you are interested in so that's my way of experimental facilities but one thing that I thought was very important which in some way this conference was instrumental although I am not sure whether Bruce would agree or not if he is here the National Ignition Facility has sort of opened up participation to a large number of others who are not exactly already in that community, I think it's a very important development although still it's very hard to get data and things like that from them it is still a very important step in how the collaboration has gone on so much for facilities but what about measurement techniques now there are always improvements taking place and for example item one and two are improvements pressure measurements which were very hard to make now Yuki Suji in Nagoya has a really reliable pressure measurement device in fact simulations by people like Toshi Goto and P.K. Young have been compared with the measurements and they all hang together and I think we know what the pressure spectrum is like again miniature velocity probes for example there are others too who have been doing this and that seems to have worked very well but you may remember not that long ago there were very few Lagrangian measurements now of course there is a plethora of people who do Lagrangian measurements and very beautiful ones and there is good synergy between simulations and experiments on this course I think that's a very important development and furthermore if you are into super fluids before this 2007 began my graduate student Greg Buley had developed the tool for visualizing super fluids especially super fluid vertices but they are always intrusive because the particles were too big and this was developed even more by Dan Letrop but now a way ago in the University of Florida is developing non-intrusive measurements well they are not exactly non-intrusive but more or less basically you ionize certain atoms of helium and then use the ions to track the fluid itself and it seems to have worked pretty well in terms of the data it already has produced another thing that I find myself interested in is that there are fantastic tools now on seismology that is you just observe something on the surface of the object like the sun and then you infer what is happening inside the sun especially the convective region so they are all new measurement techniques there may be others that I have not listed because I am ignorant but by and large some development has occurred but I must say that in relation to the development of the experimental facilities the development in the measurement techniques has sort of lagged behind it has always been an issue in the field and I think it has remained that way in the last 10 years as well so let me skip that so let me now talk about simulations what has happened in simulations as you know the ability to simulate has just grown tremendously as a function of time here the number of floating point operations you can perform is parted here on leadership machines and you can draw a sort of line like that through the Reynolds number of the of the simulations and then this is the kind of asymptote for the the operations per second as a function of time you see not that long ago I actually remember these times when we were only here now it's many many hours many people use tens of petaflops now without any problem but now what's come out of all this I must say these are the people with whom I collaborated but there were a number of others in this audience who have done similar simulations so what's happened obviously the larger capacity to compute has led to larger and larger sizes of computations for example PK Young computed a box which is 16,384 on the side and remember that Steve Arshog not that long ago computed 32 cube calculations and that excited the whole community at the time now I think we have gone way beyond that and many things are possible now this also means that there's a proliferation of computations now everybody and his brother doing experiments considering it's harder who would want to do simulations and as we do more simulations more problems have come into the fore and one of them is the resolution people always thought that direct numerical simulations meant that you compute the colmograph scale and everything above but actually turns out as you go higher and higher in Reynolds number smaller scales than colmograph scale come into being and become important for many things and therefore your ability to really go to resolve bigger and bigger boxes using the same resolution as before just somehow gets diminished somewhat and that's one and the other is higher the Reynolds number you have got to somewhat go longer in terms of time in order to get the asymptotic state so to speak people who are used to doing two integral length scales or three integral length scales and saying well we have reached the final state it's actually not true it becomes harder and harder as the Reynolds number goes up and then forcing methods we still are not sure whether the forcing the way you force large scales has any influence on inertial and anticipative scales actually that's an important question at some level it doesn't but at some level it does the deeper you go more important these problems become and then this configuration of boxes periodic boxes is certainly not the right one for rotating and stratified flows and also our real problem is there's so much competition for the computer time and leadership machines if you're doing materials or bioinformatics or fintech or some other problems it's easier to tell them that it's very important to give time but if you say I'm going to do turbulent mixing it's a bit harder and so you have to be inventive in how you actually sell yourselves so that's in terms of simulations but let me say now what new results what new results and I'm going to say a little bit which I'm not going to cover there has been enormous improvement in the fluid particle interactions and there are many people are involved in this I've listed a few of the people and they have made especially important progress in terms of computing clouds which Goto talked a little bit about yesterday and so an interaction of turbulence with micro particles even without considering the internal structure of these bubbles it still has been very important development let's however not forget that we still don't know for certainty that Richardson's law of diffusion actually works I mean there are many people who have tried to simulate it but either the scaling range is not large enough or in measurements again the same story so there are basic issues like that still remaining even though I would say the tremendous progress has occurred in terms of ability in terms of capability but finally you have to ask well has this problem been solved and I think we have still some ways to go in matters like particle turbulence interaction and also if you talk to people like Cambon he will tell you that there has been enormous improvement in the EDQ and enclosures ability to study fluctuation interactions these are the things I will not cover and so with apologies to people I have already mentioned here and then considerable progress has taken place in predicting RT flows and RM flows and validating numerical results and these people said just list them as a representative of a group and various models have been developed for stratified effects and compressibility effects ability to RT flows etc that's a very important practical development and also there is a lot of work that has gone on in controlling RT instability which in NIF for instance but I will not talk about that either likewise I will not talk about theory of fondant turbulence which Ithamar touched on thermal convection both experiment and simulation magneto convection, turbulent boundary layers which Joe Klivicki talked about this morning rotating flows turbulence modeling large scale motions in the atmosphere like madden julian oscillation very important development but all of that I will skip but I will only talk about a few sample ones mostly motivated by personal prejudice one of them is the dissipative anomaly and here I think a really important development has taken place in my opinion and this has to do with the impressive work on Ansagar's conjecture let me explain what it is about Namia Stokes equation tells you that the energy dissipation has turns like this viscosity multiplied by the square of the velocity gradients so you would think by something like that that as viscosity goes to 0 the dissipation goes to 0 but that's not so in turbulence now that's hypothesized by Taylor in 1935 if you read his paper you don't know where he got it from in the previous sentence there is no indication that the next sentence is coming and once the statement is made the next sentence doesn't have any connection with that statement so he obviously knew what he was talking about now he said that the dissipation normalized by some length scale divided by velocity cube is just to make it normalized it's about a 1 in a turbulent flow instead of going to 0 it goes to a constant of order unity and this is called the anomaly anomaly because the symmetry that was violated by this parameter here even when that goes to 0 the symmetry that is a conservation of energy is not restored so that's the anomaly and so one way to do that is to say when new goes to 0 this this d the limiting value d which I call d star is of rd unity which is what restating Taylor so this is the experimental thing that I made up many years ago now and it says that as Reynolds number goes to higher and higher values that is viscosity goes to 0 in this axis the dissipation rate that was all over the place comes to be a constant and since then of course many beautiful other experiments have been made and many simulations Kaneda in Japan made these wonderful simulations which actually showed as many things and then later on others have carried on even further so that's the dissipative anomaly one doesn't exactly know how it happens that as viscosity goes to 0 well why should the dissipation remain constant of course you might say well there are boundary layer like structures and then the singular perturbation problem all this stuff but it really doesn't tell you what's happening what I saw there conjecture was that well it's really got to do with the let's say the development of some kind of singularity in the Euler equations and I will describe that and the development here is that in fact that conjecture has been proved so let's see what that means Ansaka's conjecture which he wrote up in the Italian journal and it has two statements one of them is that for weak turbulence of course this is not exactly Navier-Stokes turbulence weak turbulence that is it satisfies the I'll tell you a little bit on that later weak turbulence weak solutions of Euler if they have this holder type behavior for velocity differences that is the difference in velocity between two points at the same time go like some power to the difference in space then depending on the value of this theta here different things happen if theta is less than a third energy is conserved there's no singularity everything is perfectly smooth and this statement has been proved well to some degree Ansaka himself has proved it by others like Ayang, Robert Dushan Constantine, TT all of whom are familiar to this community there's a second so that's proved now there is a second part of this statement that if theta is greater than a third then dissipation is possible dissipation is possible in Euler equations now and how it comes about it's some kind of singularity as I said there's a detailed mathematical structure to this and this is all proved by these people and so but what still remains to be done which is the important part is that do they have anything to the Navier-Stokes solutions with finite dissipation as new goes to zero I mean the problem we talked about and the problem Ansaka had in mind which has been proved now still there is a disconnect between the two of them so we still haven't gotten to answering the entire question but part of the question to be answered so I would say no to that and what also remains is that if you take a line cut of turbulent dissipation it has huge fluctuations like that and there's no indication at all as to whether such things actually happen through the Ansaka mechanism so that's one very important development in my opinion a very basic thing so let's go to the next one which has to do with the 2D turbulence compared to 10 years ago now I think we pretty much understand it seems to me what's going on the important contributions came from Bob Cricknan and George Bachelor most progress has been made through numerical simulations because very hard to do real 2D turbulence experiments and a review can be found here basically if you take the whole range of spectral excitations in the flow and you excite at some wave number there are two types of cascades that take place in 2D turbulence one of them is the so-called direct cascade of enstrophy that is enstrophy the square of vorticity goes down to smaller and smaller scales whereas you have the inverse cascade of energy that is energy goes from small scales to larger scales becoming larger and larger now what do we know about this taking the first direct cascade of enstrophy this was the theoretical result and if you take this correlation it is proportional to r through this flux of enstrophy and the power spectrum has this this roll off with some corrections that Cricknan predicted and it's an intermittent phenomena higher order moments depart from Kolmograph and here the phenomenological level progress has taken place and it's not exactly conclusive the way you want it so that's my assessment of where we are on the other hand if you go to the second problem which is the inverse cascade of energy where the prediction is that you have a five thirds spectrum and that the velocity difference cube is proportional to r through this energy dissipation rate which is three halves coming there and furthermore higher order moments will go like P to the power 3 and this remember is violated in the traditional 3D spectrum that is if this is true then there is no intermittency in that region and in fact these in velocity differences are closely Gaussian with very small skewness about non-zero skewness because it has to satisfy this but by and large I think we sort of understand that this is a non-intermittent process and you can predict these things and actually find them in simulations I think you might say that at least one part of the problem seems to be pretty much understood another one which I want to say a little bit about because I've been involved in this Sasha Mikdo some years ago said well velocity differences are not the right quantities to look at but instead the circulation around the loop and you can take a loop like this or a loop like that or any other shape as long as the area that the loop subsumes is the same then for example for things like this they all somehow mixed up I don't know why but anyhow if you take r as the square root of a times b for a rectangle like that then you calculate the circulation around this contour and take the pth moment of the circulation around a contour whose size is r which is the geometric mean of the two dimensions it has these exponents and these exponents he said might have more interesting properties we had already tried to see about them very long time ago when the simulations were still not as anywhere as much as they are today so now we have big simulations and you can do this and you can sort of ask yourself what's happening so using 8192 cube simulations what we found is that if you compute these exponents these exponents approach the colmograph value with increasing Reynolds number furthermore the linear that is to say you are all used to this notion that if you plot here the art of the moment and take any say zeta p for the velocities this is the colmograph and we have these non-linear relations like that departing from colmograph but what we find is that they are linear here like that and if you do it at a lower Reynolds number it is like that but if you do higher and higher Reynolds number that's how it goes I don't know whether it will ever go to the colmograph value or not but you can see the result of this in the next slide and what you find is that for example art of the moment here and the relative difference from the k-41 that is this difference divided with the colmograph value whereas for velocity differences you might see this kind of non-linear and substantial numbers for the circulation at 8192 cube simulations it's like that and so this linearity in itself whether or not it actually approaches colmograph value or not is already very important if it is linear it means it's like a monofractal in some way and of course in particular if it goes to k-41 it's sort of basically most important thing we have always been looking for for a while so anyhow I don't know whether this is a settled issue or not at the moment but that's how the situation is next thing I want to talk about is passive scalar mixing in which tremendous progress has been made I borrowed this from Gotov basically this is the wave number here and the energy spectral density if the frontal number is unity then it has the spectrum which is exactly the velocity spectrum you have the inertial range with a 5 third slope and you have something here which is not exactly known and if the frontal number is very large then what happens is the diffusivity gets moved postponed to higher and higher wave numbers so you have another power law here which has minus one power and then some behavior here if on the other hand the diffusivity is large then it already acts on fluctuations before viscosity becomes important you have this region which is a minus 17 thirds and then another region there it's all very interesting it has been around for a while but only in the last few years have things been clear now for frontal number of odd unity certainly there is a roll off at 5 thirds or thereabouts and in fact velocity even though the velocity may itself be somewhat non universal it already is the case that the passive scalar becomes assumes its form but the obuque of Carson cons that is the not the exponent but the coefficient sitting in front it somehow still is not clear whether it actually is universal or has some dependence on forcing and shear and likewise if you ask for higher odd exponents now I think it's pretty clear that velocity differences actually have this departure from Kolmogorov 41 and all that but it's not clear what the status is on the universality or otherwise for the scalar and the scalar dissipation range does not appear to scale on classical Kolmogorov obuque of scales for large frontal numbers what do we know we know that minus one power law which in fact was disputed only years ago it now is clear that it does appear and the asymptotic behavior the value of frontal number that's a minor point the bachelor constant the constant sitting in front is again not universal because it depends on forcing and mean shear perhaps Reynolds number and the scalar gradient skewness which was a big issue for a while because it's supposed to be 0 if it is locally isotropic but it is always constant of odd unity and now it approaches 0 when the frontal number goes to infinity or very large values but there is no theory for that and by some measure that in this region there is no intermittency on the other hand just as Godot said yesterday there is a agreement of the viscous diffusive region with Craikman's result rather than bachelor's result but it also has very interesting consequence and the consequence is that because the Craikman philosophy is rapidly oscillating fluctuations it may in fact say that will be that might be non-universal so low frontal number again there are many interesting results and low frontal number convection is a lot of interesting results one of the important things the frontal number goes to very low values then there are structures which remain for a very long time and so you have to average over much longer period of time than necessary so that's then you would think is possible all will be done in a few minutes so you don't have to stand is what I am saying the RT mixing very interesting things do we know the law for mixing on growth is it always like the square of time actually it appears to be the case only when you don't have boundaries you don't have stratification you don't have many other complications so it's really not as universal as one might imagine and then given the unsteady character of RT flows is the turbulence that develops there really standard turbulence or is it somewhat different and Snezan and I took a crack at proposing an alternative which I will not go into but basically it's a question that one has to sort oneself out for and then there is the compressibility effects which can never be ignored in practical RT flows and there are many studies now on compressible effects of compressibility on turbulence for example here as a function of Mach number where compressibility goes up you have the dilatational pressure as a fraction of the usual pressure and you can see that the dilatation effect just takes off and therefore in fact pressure has a very different connotation in highly compressible flows and what is interesting also is how the probability density of pressure changes as a function of Mach number in fact this is an interesting result nobody has explained if you take the pressure spectrum in incompressible turbulence on the right hand side it is almost like a Gaussian on the left hand side it is almost an exponential and there is no explanation for this at all but if you increase the Mach number that is compressibility goes up this side too approaches a Gaussian so it is essentially symmetric and that is something nobody knows an answer for so now with my last slide so I should now say how have we met these specific goals that I talked about in terms of the program in terms of the communal development etc so first embrace mixing in all its richness and organized meetings and regular intervals raise money for them I would give a check mark against it so it I think has done well the second one act as a vehicle for scientific progress that is qualitative understanding for its own sake is something we have to emphasize and the problems I just mentioned are really a result of some of this outlook and I would give a check mark against that as well and develop predictive capability some of the authors I just casually cited there have done wonderful work in terms of simulate in terms of being able to predict I would give a check mark against that as well and published major results in some collected fashion I told you already how many publications etc very respectable journals that's also fine synergy between experiments simulations and theories between applications and basic work I would give a question mark I don't know that this community intended has really worked hard to enhance this collaboration or not well we try everybody tries to say well it has to work together and synergy all that stuff but not much that I know has happened and build expertise and mixing in educational outreach programs this conference arranges some tutorials and things like that so I would generously give a check mark to that notice that the thickness of the check mark here is not the same as that anyhow organized mechanisms to share research tools, methodologies data analysis and visualization we basically haven't done anything we haven't done anything because we don't have an organization behind this thing it's done by shoe string operation you know Snezana working very hard and trying to raise money a little bit here and there and we have a lot of people who are internationally interfering with that basically it requires a group of people and there is really no group at all that is devoted to this big problem and it has to have a numerical strength to it not just one person can do all of it and certainly this ambition to prepare a white paper for launching a decade study of non-equilibrium processes simply hasn't happened at all which I just mentioned so if you want to really make progress on this it's really not done by one person it is done by a collection of people devoted to this and you know it takes a village to make something like this happen and this program has given a lot of impetus to many things but still it has many more things which could accomplish which it had thought would be possible to accomplish but has fallen short of. So that's my assessment of where we are in the last 10 years or so I want to appreciate the work that all of us here have done, Snezana in particular I don't know where she is in this collection and also the support that ICTP has given, thank you very much for that and of course again thank you for your patience in listening to me, thank you