09:03:30 So welcome everyone. Hello to everyone.
09:03:34 And this is our third session on plus modifications to layering. So let me share just a few slides for the introduction.
09:03:57 Is it. Here is it.
09:04:02 So the, the session is devoted to the impact of other lunches and jobs on layering.
09:04:16 We will have, let's say, two, two and a half or two and plus something tokes a few slides presented by Yusuke capers who got who already gave a talk to India.
09:04:22 The first weeks were state cases are shown to be driven by artist structures exhibiting stagnate staircase, like features are produced by our parents jobs.
09:04:42 And in the same kind of spirits.
09:04:44 We asked, Nate balmforth, who kindly agreed to make a review of zero physics three real physical three dynamics, the review of, always, and you will see that they they share and so some common features with this, the jobs.
09:05:03 The jams. And then we'll have parts. The end. Speaking about traveling spreading and an explicit nonlocality.
09:05:11 So, a few points for the introduction.
09:05:17 As shown in this picture here where you can see versus time in the y axis and the radial direction the right direction of the confinement interconnect basketball or the direction of the propagation of the heat.
09:05:35 You can see different structures structures moving regularly, either outward, or inward and structures which are vertical, which represent shared layers and actually what is shown here are the, the, the sheer of the velocity.
09:06:06 in the not in the right direction. And so there's the coexistence of both those events we propagate regularly. And she layers and this coexistence is enabled by the self organization, which in some cases, exhibit this kind of structure which is a separation
09:06:15 in space, and hence leading to staircases.
09:06:20 And in a sense we reach here, one of the open issues of the generation of first take cases in this context.
09:06:32 We have fresh your layers. On the one hand, we have avalanche or turbulent propagation, or tablet and spreading, which is stopped at seven locations.
09:06:44 And we have at the same location where avalanche has stopped, and the flow she develops. We are the profile steepening meeting to the the staircase to one of the issues is where to start the loop in between the three species.
09:07:02 What is the coach or and the consequence of the others.
09:07:07 And, which direction, actually.
09:07:11 So they are showing in that respect staircase invalid avalanches are sort of unsorted chicken and egg problem.
09:07:20 There are some evidence. Well, not exactly related to share layers, but to transport buyers more generally more broadly.
09:07:29 There are some in interaction between avalanche like events and transport buyers.
09:07:35 The width of the transport buyer has been found in numerical simulation to fluctuate under the impact of other answers. This can be related to travel and spreading.
09:07:46 There's some experimental evidence on show that turbulent transport barriers are eroded due to avalanches which impact on them.
09:07:59 But on the other hand, if we look at the, ton of self organization of avalanche and waves.
09:08:08 Due to the, the time delay in the first grade and the response that was shown by Yusuke acres well, then you you end up with a ticket equation that features some solutions exhibiting avalanche jumps and four roadways in your physical three dynamics.
09:08:31 There's. They are spontaneous structure formation, or shock waves, which Stephen up in turbulent flows.
09:08:39 So in each case, there's a kind of suggestion suggestion that. Sorry.
09:08:49 That staircase structures or layering might be the consequence of the organization of of trauma violence like transport. So some open issues I see are the flu in UCLA or stop avalanches, which then would be this distinct beast, or are they a manifestation
09:09:10 of the avalanche complex dynamics, such as charms for role waves are the shear layers reinforce or eroded by avalanches two phases of the same coin in a sense, avalanches trapped in sheer layers.
09:09:28 And what kind of introduction of mechanisms between the Sheila years and the other engineers, the role of Percival non-local processors.
09:09:41 The sheer sun does it payroll, and so on and so forth. So I think that I stopped here.
09:09:47 Those are questions that might appear during the discussion, and we will start so by the few slides by us okay just before he if faulted sleep.
09:10:00 Two hours am so thanks a lot, too old to the three speakers and especially to use a.
09:10:09 Let me stop sharing the screen and give you the peace.
09:10:18 Okay, so maybe I can share my screen.
09:10:31 All right. Can you see it.
09:10:37 Yes, perfect. Okay so, um, maybe I can start know.
09:10:44 So, well thanks for having me here. Again, this is 2am right now eh path.
09:10:52 And I was just, I just received this email yesterday asking me if I can talk in this discussion session so I have just started preparing this slide that during the daytime.
09:11:05 And I'm giving this talk in this midnight, so forgive me if I'm a kind of incoherent.
09:11:11 But what I want to talk in this short contribution is.
09:11:28 And then I just want to elaborate some other remaining issues that I think it's important to understand.
09:11:36 So I'm here let me start by summarizing what we have talked the other day.
09:11:46 So here what we have interested in was this avalanches and jumps, and we have talked about the simplified models to understand the behavior of these avalanches and the jobs.
09:12:02 And here we are interested in was this evolution of the deviation of the temperature for the marginal, and this bloke some point, how they're evolving time and space, and their dynamics was described by this simplified his boss equations.
09:12:20 And the difficult part was to understand what the flood says.
09:12:29 And one way to model this heat flux was to invoke the joint reflection and symmetries.
09:12:35 That gives this simplified the simplest form of the flux to describe our bond dynamics.
09:12:42 And if we use this to set of equations we can get the burgers equations to describe our lunches in the framework of the shop, tablets and things like that.
09:12:53 And what we have discussed in the other days talk was extension of these models to include this time delay effect here, the instantaneous heat blocks that plasma Kelly's can deviate from the main values dictated by disjointed reflections and literary
09:13:12 argument and applause nice respond.
09:13:16 These floods is titled The main values in the final stand tall, and this was introduced based on the analogy to the traffic flow dynamics.
09:13:26 So in the context of this traffic flow dynamics. We weren't just in in the codons evolutions.
09:13:38 And that evolved, due to the car flocks, or the speed of the cars and the car, and the speed of each individual individual car can deviate from the mean speed, which is determined by the equilibrium or surrounding cause the fear cost speed deviate from
09:13:52 the surroundings. The drivers can respond in time tall.
09:13:57 This is kind of unreliable, the drivers response time.
09:14:02 And if the plasmas can quickly response to the main valleys. There's no jobs, we can have some propagation of the blogs have shown. Yeah.
09:14:12 But if you have very very long time.
09:14:16 Over the plasma takes a long time to adjust its heat blocks to the main values.
09:14:23 The weekend had the jam formations and shown here, this is time evolution the policy initiated in the systems and it propagates.
09:14:30 It starts growing like this.
09:14:34 And then the question was how we can relate this john paul nation to the formation of the corrugated staircase like patterns, and to get that we need a multiple jobs to get the multiple jobs what was important was to have some injections.
09:14:50 This is again the analogy from the traffic dynamics.
09:14:54 We can keep injecting cause until we get the multiple congregation because it's a show here.
09:15:01 So we can do the similar, we can do the same in the plasma as well. we can keep injecting heat into the systems.
09:15:10 So this is one initial conditions. And once we initiate these policies they propagate the past jumpstart forming.
09:15:21 And as time goes by, we can get the multiple jobs forming as shown here. and this is where the deviation the temperature from the marginal so by adding some of the values, we can get these staircase like patterns so I guess the Avalon given jobs, provide
09:15:38 a one way to look up information on the staircase or layering process.
09:15:44 So that's kind of a summary, my talk in the other days, and I just want to elaborate some few points that I think it's important.
09:15:56 And the one point is a forcing to get the multiple layers and to get the staircase like patterns.
09:16:03 It was important to have the right forcing into the systems.
09:16:07 So, here let me give you some examples. We don't do that, right. Do not right. Do not do it right.
09:16:14 We just inject the polls and let it go without any further forcing, we only get the one single job of showing. Yeah, there's no multiple jobs for me.
09:16:25 Even if we try with this initial conditions. If we do not keep forcing in the boundaries.
09:16:34 When we just get the single jungles showing here, it just destabilized by the shared so.
09:16:42 So to get the multiple layers. We had to keep forcing it into the systems, and that's one way to get these multiple layers and the staircase through the avalanches and John's.
09:16:57 And of course that's kind of simplified.
09:17:00 The real avalanches is kind of the stochastic entities, and to initiate Babylon is in more realistic manners. I think it's better to have some sort of a stochastic forcing.
09:17:16 So maybe it's important to implement noise.
09:17:20 Or there could be some better way to do so he have any ideas, I'm happy to hear your opinions on this, so I think it's important to
09:17:31 to further analyze these.
09:17:34 The difference in terms of forcing for this junk formations.
09:17:40 And I think there's other important ingredients that I missed to discuss in the last talk on that one is like the
09:17:53 dependence on the target as intensity itself.
09:17:56 The model was kind of oversimplified, we have just in today's the response time, and that's kind of the parameter in the systems.
09:18:06 But in reality, these are response time can be a function of the time that's intensities.
09:18:21 But if the plasma close to the margins and the tablets is not so active very might take a long time to push the system to mean value so we can get the jobs, and to get this type of behaviors, this dependence on the tablets intensities maybe we need to
09:18:40 have some response time which is a function of bank density on the tablet itself.
09:18:46 And at that point, we probably need some couple of equations for the congregations and tablets and density or tablets dynamics, and I guess probably working on this and I had this comment from him.
09:19:01 it's complicated, that just one line something of this type of research, but I think it's important to push that direction as well. And if we want to start doing it I think it's important to understand what kind of feedbacks from the quarter gauges, the
09:19:16 tablets dynamics.
09:19:18 It's important to understand that feedback loops. And of course, there could be a multiples. One would be the congregation could this to the me cheers and that can impact the underlying timeless dynamics.
09:19:32 They can also enter the mixing skills through the line scale like dependence, I think that type of the important there.
09:19:44 There's another point that I want to make here. One thing came to my mind is that, well it's nice to have the jobs are volunteers and the jobs and formation the staircases from there.
09:19:57 Well, that could be some difference in the tendency of the Germans, depending on the nature, the turbulence, correct me if I'm wrong, but the important ingredient here is the marginal state.
09:20:11 And once we have the marginal states the region talk about the deviation from there, how close we are to the marginal state. And then if we are close to it, then we can talk about some diverging response time or long response time, it's nice to have these
09:20:28 type of stories but I think it's kind of clear to have this type of story for the turbulence, with a clear marginal state.
09:20:36 What does some different types of turbulence, without any explicit threshold, or marginal states such as some destructive type turbulence in those cases, I'm not sure this jam is so effective to happens.
09:20:54 For that, I think. Once that's done better.
09:20:58 And the last point I want to make here is, is there any relevance in the broader context of this type of stories.
09:21:08 This is really an extension of the show, turbulence, or burgers turbulence.
09:21:26 So, if you have some layering type behaviors in shock turbulence I think the job, models may be a one way to look at these layering behaviors there.
09:21:27 So if that good happens in broader context, such as GFD was a master physical context I think this is one way to approach, and the female some examples that I just, I'm happy to hear your input from there.
09:21:44 So that's it for me basically.
09:21:47 Thank you for listening.
09:21:50 Okay, thank you. Okay.
09:21:56 So we have some time for discussion.
09:22:02 Please raise your hand.
09:22:09 I don't see.
09:22:13 I don't see any
09:22:16 me that just yet.
09:22:19 Sorry I don't know to raise my hand, this this is the pundits here.
09:22:23 I just want to make a comment which is tangential.
09:22:29 I come from a very different community which normally does homogeneous so tropic turbulence.
09:22:37 And over the past decade or so we have built up very detailed theoretical and numerical understanding of, you know, multi scaling dynamic multi scaling so this formula that is still on the screen Tao epsilon goes is epsilon two minus epsilon see to the
09:22:58 minus alpha.
09:23:01 There are key dependent times characterized by dynamics multi scaling exponents.
09:23:09 Of course, those are
09:23:13 times you get out of correlation functions.
09:23:16 But here you're asking for response functions, which is much harder because you don't have a fluctuation dissipation theorem.
09:23:25 But since you're in Japan I would like to tell you that Takeshima Matsumoto in Kyoto has been looking at response functions in the homogeneous isotopic turbulence context, using more complicated versions of fluctuation dissipation which are associated
09:23:45 with the names of Hurghada and Sasha if I got that right.
09:23:52 So, as I said, it's not directly related to what you're talking about. But there is a lot of activity and related fields which you might or might not be familiar with.
09:24:02 Thank you.
09:24:04 Yeah, well, thanks for the comment I actually did not know about that work but maybe I'll look into that. Yeah.
09:24:13 The name sounds familiar to me so maybe I can talk to them in some physical society meeting or things like that, but thanks for, I could even send you his email messages.
09:24:25 Yeah,
09:24:32 don't see any of your quickly is it by shot. Yes, it's, it's me I, I missed the part of the talks that was interruption here. somebody called me my day.
09:24:46 So it's now. Technically, question.
09:24:51 Right.
09:24:53 Yeah, I have a simple, simple one, exactly the slides that you are showing now, if I understand correctly, this delta t is.
09:25:22 So to say like on a local traffic density, right.
09:25:16 In, Internet, and I, when I look at the sword equation right and take the operator D cheap was one of our toe and applied to the first equation.
09:25:29 Then I sort of get rid of few Of course, it's one scene, but the dx is concerned, so if they start with.
09:25:41 So, compact distribution of cars, then it's the first equation doesn't conserve the number, necessarily, depends of course some kind of distribution with a but it doesn't conserve mathematically, if I integrate the next from minus infinity to infinity
09:26:03 assume that no cars at infinity is and the number of cars will will decay.
09:26:16 No as this time Joe.
09:26:16 How do you think interpret this result is kind of follows from these three little equations, right away I guess.
09:26:24 Yeah, I think it's better to walk on this past equation as it is.
09:26:30 In that case, it's just a car density evolving DT the divergence of the flat so if you integrate this over space of I guess the car density is conserved up to some boundary values of the planets.
09:26:44 But if there is no boundary wireless if, if there is a zero at infinity.
09:26:49 So I started I look at this initial problem. I seen it find it distribution delta t.
09:27:08 dk. It was infinitives and integrate and then I first equation tells me that I get zero. So it conserved quantity, but if I apply it, try to eliminate qV is fine it fields to kind of paradox here, which I think, I think we go, that's too deep into the
09:27:24 detail the technical details he may meet maybe you can discuss further.
09:27:29 This is Misha with with you. Okay, yeah, it's not so I propose that we, since we're already running, not hate that.
09:27:41 Thank you very much, you should care for your, your token suggest we move to a band for, if you could share your screen.
09:28:01 How's that, that's fine, thanks.
09:28:05 All right. Well, hello everybody.
09:28:09 I'm going to tell you about robots.
09:28:12 So, let me just give some credit to my collaborators at the bottom for help with some of the theory and experiments.
09:28:21 So here we go. This is going to be a review of role wives. And first of all, I'll talk about the sort of the classical turbulent role wave. And there's some illustrations here.
09:28:34 So on the left, you see a spill away from a damn. This is very fast turbulent motion, and you see the crest of waves highlighted by the front end of the water.
09:28:45 You see waves created, and they lengthen and they traveled down this turbulent water course so those are always the first sort of technical discussion of them that appeared in the scientific literature shows over 100 years ago.
09:28:59 There's anecdotal mentions of them in literature predicting that by a long way.
09:29:05 The example on the left is from a concrete channel very close to where I live, and we were passing, one day and it was full of water, and we took a nice movie.
09:29:21 The phenomenon that you see.
09:29:22 So, you see these credits of the role waves again, so that the the channel here is a couple of meters wide.
09:29:31 I didn't get down there to tell you how deep it was.
09:29:35 But that's that's the phenomenon of the turbulent roll wave.
09:29:43 And it's been reported a number of times in the literature. And it's sort of about 100 years old.
09:29:46 Okay, So, I'll now mentioned sort of the typical way that people have model this. So, rather than thinking about something in the field you normally think about doing this in the laboratory and you can actually do experiments on roadways so is an experiment
09:30:06 that we did, essentially with a piece of guttering.
09:30:10 It was quite long, so the the channel itself was was 10s of meters, the width there is 10 centimeters and it's a couple of centimeters deep. So you put water down there and it forms these nice jumps, these are the role waves in the laboratory, and the
09:30:26 classical way to try to model that is using the St Bernard equation so you have the water depth, and the water speed is the sculpture the geometry and so forth, there's some angle of the channel.
09:30:38 So we've got conservation and mass conservation and momentum and to simplify things will just assume that there's no dependence upon depth or the coordinate across the channel so it only depends upon the coordinate down the channel, we've got, he has
09:30:54 the fluid depth that's the conservation and mass equation. And we were assuming that the velocity profile is essentially uniform, with depth. So, the flux is simply given by h times you and conservation of momentum is the second equation here.
09:31:11 The only novelty is the drag, so we're assuming that there's some sort of a turbulent drag and by dimensional analysis you simply write that down in this particular form, where there's some coefficient as an order one number.
09:31:25 As the cheesy coefficient. So this is a popular model in hydraulic engineering.
09:31:31 Okay, so you can explore that particular model there is an equilibrium solution, in which you have flow of uniform dex that's basically moving turbulent at a terminal velocity that's the that's the balance between the gravitational driving and the drag
09:31:51 is moving uniformly down the channel. And you can perturb that so Jeffries did that in 1925, you have a mean depth and I mean flow speed, and you look at small perturbations with some wave number and growth rate about that.
09:32:02 And if you perform a simple traditional linear stability analysis, then you find that this flow can be unstable and the critical parameter that dictates whether it is so is the fruit number, so if you divided by the square root of gh square root of g
09:32:20 is the wave speed associated with surface gravity waves. Yeah. So if that's bigger than two, then these small perturbations will grow. And that's what people have rationalized the roadway phenomenon by.
09:32:35 And what people also do to help with the computations and the physical interpretation of things is that they add a viscous term, do you do to you by the x squared term into this equation, that helps because it sort of regularize is shocks.
09:32:49 And if you look at the linear instability here, then it doesn't cut off at short waves.
09:32:55 If you add the viscosity, it gets rid of short waves. So that's just a useful thing when you're actually exploring things in the nonlinear regime.
09:33:06 So here is a computation with the same model, and you starting here with a uniform flow and you see these roadways growing and reaching finite aptitude.
09:33:17 This is the amplitude as a function of time. This is a logarithmic plot so you see the linear growth and saturation, and at the end of this particular simulation, where we find is that we've got these for roadways.
09:33:38 Alright so here we've got a simulation and started out with this background flow plus some small perturbations, the roadway instability kicked in the roadways grew and saturated.
09:33:51 So we can look at a little bit more about the detail nonlinear dynamics. So, what happens if you wait long enough, is that after saturation you get these roadways but if they are too close together, they usually go through some sort of a course learning
09:34:05 process. So they they approach one another, they collide stick together. And in this simulation here, what happened is that they all collided and merged into a single roll away.
09:34:17 So of course nothing happens when the role waves are somewhat close together. If you start with a simulation with a single roll wave in it that's in this case it's periodic so it's really a roll wave train that's widely spaced.
09:34:31 It's the read here. What happens is that the back of the role wave it decays almost to a flat state, and the flats date, that's the equilibrium, all over again.
09:34:45 And that's subject to the same instability that created the role wave in the first place. So what happens is that when you have a very wide wave train as a secondary instability that kicks in, which is the new role waves like to grow in the spaces between
09:34:58 the role waves, they generate more role waves. And so you get a wave train with smaller spacing. So there's this competition between coarsening when it's closely spaced.
09:35:13 And it's sort of a secondary instability that nucleotides small role waves when it's more widely space. So that's the nonlinear dynamics that's captured in this st Vanette model.
09:35:25 And it's sort of roughly matches what you see in these types of fluid experiments.
09:35:33 Okay, so if you will think about roadways. One of the things you can ask is whether they are convective or absolute. So, essentially, that means that we've got this flow that sweeping past a fixed station, and the traditional linear stability analysis
09:35:50 establishes that, if you will, going with the flow. Then these role waves will grow, and therefore if you put yourself in a periodic domain, which is what's happening in this simulation on the left here is that I start with some small disturbances, and
09:36:05 they continue to cycle through the domain, and they keep on growing and saturating and then you see that normally new dynamics that I mentioned on the previous slides.
09:36:15 But if you now change the boundary conditions so it's no longer periodic. So now we've got some sort of an outflow condition at the bottom here, what happens is that the background flow, simply sweeps out the initial disturbance, and the role waves grow
09:36:33 as they're going with the flow. But once that initial disturbance has been conducted out of the domain, the flow settles down, back to the equilibrium state.
09:36:45 So, what this is indicating is that the roadway phenomenon is convective but it's not absolute and effect station, the instability will not grow. It will only grow if you're going with the flow.
09:36:59 And you so you can actually establish that more definitively using Briggs this method with the linear stability problem. So, one of the conclusions is that you have to have a source for always upstream if you are going to see these things and effect station.
09:37:16 So for example, on the right here at the upper boundary we've got a little wave maker, that's continually seeding small role wipes. So in this way you see the role waves continually growing from the top boundary.
09:37:29 Okay, so that's the nonlinear dynamics and the sort of flavor of role waves as a convective or absolute instability right so now I'm kind of done with thinking about the turbulent problem.
09:37:43 I'm going to switch gears a little bit and talk about the laminar problem. So we're all familiar on rainy days with waves that form on Windows or on the roadway, there's a picture of roll waves on a road.
09:37:59 Now the difference here is that this is a relatively thin film, and it's not to be an emotion.
09:38:05 But nonetheless, there's a close analogy between these types of waves, and the turbulent roll waves on turbulent screams. This type of problem is sometimes referred to as the puppets the problem.
09:38:30 after the turbulent one. So in the 1940s and beyond. So these are waves on falling fluid films. And the scale is much smaller everything's much slower surface tension could be important, but I'll ignore that detail.
09:38:36 So what you can do in this particular context is you can start with the NaVi stokes equations.
09:38:43 When we began with the same finance model over here we already had to do some make some assumptions and parameter ization of drag for the lemon in case you need to do that you can start with an obvious stokes equations and you can look at its uniforms
09:39:01 flow sheet flow down and incline, and you can do a traditional linear stability analysis of that again. And what you find in this particular context is that there is a critical Reynolds number above which you expect a sort of a long way of instability,
09:39:16 these are laminar roadways. And that goes back to Benjamin and you're in the 50s and 60s. One of the things that people do, is it they try to advance beyond this long way theory that's relevant to the onset of instability, you try and push it up a little
09:39:31 bit further and some sort of crude away is by looking at vertically average models. So you basically taken an obvious stokes equations and you integrate across the thickness of the film to come up with an equation for conservation of mass another one
09:39:45 for conservation of momentum. And with some assumptions about what the velocity profile is across the film, you can then derive these equations which looks somewhat similar to the same equations.
09:39:59 So we've got the usual infection of massive momentum. There's some coefficients that come in here which are related to what you assume about the velocity profile.
09:40:08 So, this is essentially doing what von Karman and Paul housing did for boundary layers over 100 years ago.
09:40:16 Once more, we've got the gravitational driving, and the drag now comes directly from the lemon of this cause stress.
09:40:24 It takes a slightly different form from the cheesy one.
09:40:27 Anyway, you can then explore falling fluid films this way and you get similar results to the turbulent case.
09:40:35 Okay, So, those are what you might call traditional types of roadways turbulent or laminate. So now I'm going to tell you all about a gallery and analogs of this.
09:40:47 So first off, I can move away from water, I can look at other fluids, and for mud for example was often observed as the formation of mud surges so this is a month series of mud surges for the Yellow River and the photograph here doesn't really give you
09:41:09 the impression of what the material is but this is mud. It's a dilute slurry of clay.
09:41:17 And here is a picture of a another such slurry in the laboratory there's another flume where you see these roadways forming on it.
09:41:28 Let's see, Here is a short video taken.
09:41:33 I'm not sure, for which TV channel but it was, it was provided to me by Christophe and say, which shows a series of surges in mudflows in the Alps.
09:41:48 So this is somewhat larger fluid is clearly a lot thicker.
09:41:55 He's a mud surges, so what you can try to do is if you're willing to assume something about the Constitution of this fluid ie the reality, and it's it's a fluid that's going to your stress it's got a viscosity that depends upon the sheer right.
09:42:09 It's actually has some aging, it breaks down and heels.
09:42:14 So, if you're willing to assume some sort of a model for that and you can repeat some of this analysis for role waves, and you can build some theory of this phenomenon.
09:42:27 Moving on, we can go to other fluids so now this is doing the same thing for a corn starch suspension.
09:42:35 And here is a laboratory experiment with a corn starch suspension custard.
09:42:40 And what you find for custard is you get Rolaids again, these are relatively slow and very large amplitude.
09:42:50 So some people call with the material, ooh black. So this is the material that it looks like a fluid but if you hit it hard enough it reacts like a solid, you can run on it but if you stand still your sink into it.
09:43:05 So, the mystery for roadways and customers and you look at the phenomenon in the laboratory, you can roughly balance gravitational access acceleration with the viscous shear stress, and that's gives you a guide as to what the rentals number of the flow
09:43:23 up to be.
09:43:25 Have you assumed that there is a typical viscosity that characterizes this material. Now for the capital problem, what you find is that you see these waves on falling fluid films when the Reynolds number is devoted 10.
09:43:40 And that's consistent with theoretical modeling for cornstarch what you find is that the effective Reynolds number that you seem that you had the way you seem to get these rentals numbers is remarkably small, which leads you to wonder what on earth is
09:43:54 the reality that you're observing via the roadway phenomena, and some known results for other types of fluids. One is sheer thickening materials or share thickening is what you commonly associate with cornstarch or tested.
09:44:11 When you do things quickly it responds very solidly.
09:44:15 But if you try to put simple models of sheer thickening into this role wave theory, we find that that's actually stabilizing goes the wrong way.
09:44:23 And you also know for Polly Merrick fluids that visco elasticity can be destabilizing, but that's a problem because these materials is cornstarch material doesn't seem to be very viscous elastic.
09:44:35 So it's not at all clear from this phenomena what it is about the reality of cornstarch, that caused this instability at relatively low rentals numbers, and some recent work on this, where there's, there's been some connection with recent recent sort
09:44:52 of results on what the reality of contractors.
09:44:57 So at this point we've sort of, we're going away from Newtonian foods have gone non Newtonian. We're trying to see what the roadway phenomenon is telling us about the material, And people have done the same thing with granular materials so this is sand
09:45:11 or glass spheres. And here you see role waves on a moving layer of grains to particular experiments, so this is again an instant where you're trying to look at this phenomenon, and understand something about the reality of the material, because that's
09:45:27 the main unknown.
09:45:30 Okay, so we've gone from Newtonian fluid and non Newtonian foods and granular materials. These have all been free surface flows. You can also look at bodily fluids.
09:45:42 And so, I suppose this is going either towards an industrial context, or to something more relevant to everyday life so whenever you get a pint of Guinness you see waves to propagate from the froth into the fluid.
09:45:56 And people have drawn an analogy between this to face phenomenon and Rolex. And, in fact, this, this makes it into art installations. I was at the Royal Tyrrell Museum in Alberta recently, and they had this series of channels where you have bubbles going
09:46:14 at them. And the whole point of the display, was to display the slugs that you see here which which are apparently and an analog analog of role waves, according to these particular papers.
09:46:26 Okay, so you need an even have fluids. Here are some instances where people have drawn an analogy with role waves in traffic flow. And this is an experiment with vehicles that people have also looked at observations from crowds so this is a marathon.
09:46:43 And what you see a density variations and it's and the formation of the density variations, the analogy, that's the analogy with roll waves.
09:46:52 You don't have to have a free surface you can have fluid flow, that's in a wall, as long as the walls are somehow formidable, you get roadways. So, there is a thread of literature in the physiological world that about flow down vessels, collapsible tubes,
09:47:11 so it could be air flow in the longer it could be blood flow in veins.
09:47:18 And the idea here is that you're flexible wall can deformed, and a simple model of this type of configuration is to say that there's a two barrier, and a flow rate, you have a conservation and math equation of conservation and momentum equation, you need
09:47:33 to add a pressure versus area relation.
09:47:37 And you need to somehow model the drag, but this once again is very similar to the same finance problem, and you can look at the stability of flow uniform flow down collapsible tubes and you get a role wave instability, and this has been suggested as
09:47:54 an explanation of corrupt cop sounds. These are the sounds that the doctor listens to when you having a blood pressure test.
09:48:03 What happens is that they put a cover on your arm, it causes the blood flow to stop in your veins, as you release that pressure, the blood starts to go down to collapse tubes, and it forms what people think might be roadways and generates these little
09:48:19 clicking sounds.
09:48:22 And finally, there is a geophysical analog that which is volcanic tremor. This is a ground vibration that's experienced near volcanoes and and it's been inferred to be due to the subsurface motion of magma, and what's what's kind of unusual about it is
09:48:39 that it's very periodic. So here's some signals so these are ground vibration signals, and the, the, sort of the crudest model of this is by somebody called Bruce Julia, which was to say that there are two main chambers under the ground.
09:48:53 There's a conduit between them. And there's the flow of Mang between the two chambers and the walls can rock backwards and forwards. And there's an instability.
09:49:03 So Julian's model, ended up as being sort of a conceptual lump parameter model that was related to some simpler models that people had done for Krakow sounds, but you can build a more elaborate model of this, and you have a conduit surrounded by an elastic
09:49:18 solid the wolves can move, and you get something very similar to the roadway phenomena.
09:49:24 Okay, so I could say plenty more about this but I'm not I'm not going to what I wanted to do is I just wanted to sort of summarize the whole thing. And in these times we're all a little bit staff, the ability to go and work at the board.
09:49:41 So, I see that the chair of the sessions.
09:49:58 But anyway, so here's my Blackboard.
09:49:50 And hopefully you can see the writing on it, so what what is common about all of these problems. These roadway problems in their analyze well, you know you have a conservation of math equation, conservation of momentum equation, the momentum equation
09:50:07 has some dragging it, and it's got a pressure type term, the pressure versus the density variable. So, the density variable for these, these traditional roadway problems is just the fluid that.
09:50:21 But for the traffic, it could be a genuine traffic density.
09:50:25 So you need some kind of a constitutive law for these variables. But these types of equations have uniform flow solutions you can look at the linear stability of them, you get a relatively simple dispersion relationship.
09:50:42 This is common to all of these problems and out pops some kind of a criterion for the onset of instability.
09:50:49 And one of the critical things is that depends upon this drag. So it's the drug that's really at the origin of the role wave and stability and very very crudely you can think about it as far as if the drag decreases with the density.
09:51:03 So for the traditional role waves the drag decreases as the thickness of the fluid goes up. What that means is if you're willing to pile up a little bit of material into a wave and the drag goes down if accelerates and picks more fluid up and keeps on
09:51:20 growing. That's the sort of a crude way of thinking about the role wave instability.
09:51:24 Okay, with that, I will finish, like it.
09:51:32 Thanks. thanks a lot for this really, really interesting talk.
09:51:37 I'm sure it's going to trigger several questions.
09:51:43 Yes, but.
09:51:45 So thank you, is, is quite interesting.
09:51:48 I wanted to zero in on the viscosity issue, which is, it's not hard to see that that's of some importance and utility.
09:52:00 But the question is, what is it to really In other words, you can treat it as a constant, but might not be maybe more appropriate to have at least a piece of it, that is a function of the steepness of the forward face.
09:52:19 So in some sense you might expect as the thing begins to steep in that it may you know it may Ruffin and produce something akin to turbulence so the viscosity would really you know would really be nonlinear In other words, it would it would have the content
09:52:38 of the have a breaking model or something of that ilk in it. And the thing that comes, it's not strictly turbulent but some similarity. What comes to mind on a surface waves you know you'll see capillary waves on the forward and face which spill and go
09:52:58 So, I that was, that was one question and I had another but maybe address that. Okay, well yeah so so that's that's a good question, you know the the onset sort of depends upon which problem you're considering really.
09:53:16 So, there's one end of the scale which is the lemon or roll away from them, and you've got a lemon a film that's very thin, and you start with an obvious stokes equations and you can develop those.
09:53:27 When you add the viscous to it, so maybe I'll just go back to my screen here.
09:53:34 So, in that particular context, the drag
09:53:42 that comes from the vertical, discuss stress, but it's common to add in, and the additional Do you know the D to you by the x squared out those those, those also originate from the traditional viscous turned in an obvious those equations and, and you
09:53:59 can add them if you're willing to work at it and go to higher order, so you can add those in. And those have a very clear origin. And when you go to higher order and you and you pick pick out those terms and put them into the equation.
09:54:11 They have a very specific for, and that they're essentially the linear viscous terms that correspond to the diffusion of momentum, along the film, and they are you know that there may be some dependence upon the fluid debt because of the fact that it's
09:54:28 averaged over the fluid depth, but that's the only non linearity that would come into that particular viscous model. Now that's obviously not true. When you're talking about the turbulent version of the problem because the, the cheesy drank term that's,
09:54:43 that's basically bottom drain but it, it's coming from turbulent stresses. And if you if you wanted to add some kind of a viscous diffusion in here to that would also be some model of a turbulent stress, and in that particular case then you put something
09:54:59 in there that conform to your physical interpretation of what's creating that. And it's true that if you look at rollers and breakers. Then there's the actual wave breaking processes, there are pillory rebels on top of the thing that can be providing
09:55:14 the extra stresses.
09:55:16 So, they will they could be a very complicated nonlinear model for the viscous turn that you add to that particular problem.
09:55:24 Maybe if you go to the other analogs for the, for things in non Newtonian fluid you would hope that the the viscous terms that they come from the related non Newtonian viscous terms that you started with.
09:55:40 In a granular media, the trouble there is that we don't really have a good handle on what the constitutive description is in the first place. And this grain size is sufficiently big that the fact that it's an assembly of grains can be important.
09:55:59 So it's, you know what, it's again it's an issue about what creating this viscous stresses. And I suppose in the bodily fluids case this, there's it's a bit easier because you've got Newtonian fluid to begin with.
09:56:13 But in this traffic flow model.
09:56:17 This is again an instance where you don't really know what the constitutive description is, there's a lot of work on on interactions between particles and flocking and recently.
09:56:32 Recently, but, You know, coming up with the definitive statement for what the fiscal stresses it's completely unclear. So I think the answer is that it just depends upon what the context is for crowds I mean you have the Gelman joke about particles that
09:56:43 can think that Yeah, yeah, yeah, it's that's a tricky question, but the. Thank you. The second question was, that you know you mentioned that disco elastic effects can be destabilizing is that simply due to the fact that the thinking of Oldroyd be enter
09:57:07 a time delay. As part of it, it depends which model that you use to write.
09:57:13 Part of it is that it could be a time delay in thin fluid films sometimes the time delays not that important because it's too thin for relaxation to be, you know, relax is very quickly.
09:57:26 So what happens is that the main effect is actually the nonlinear viscosity, that comes about because of the because it's a pollen Merrick fluid. So you have a viscous shear stress that depends upon the sheer right as a result of that so it just, it becomes
09:57:42 a little bit more like some of those other fluid models, it's not really relaxation. It's the fact that you've got a non constant viscosity.
09:57:50 Right, okay. All right, thank you.
09:57:54 We have.
09:58:04 Next, was nice to have some of the structures, you showed at the beginning were quite spiky in so I was wondering was is instability evolve towards final times you need it in absence of organization, or the quadratic nationalities are sufficient to provide
09:58:16 situation.
09:58:19 And, hang on. So, let me just back and then it's probably the fourth line, it's it's it's at the very beginning.
09:58:30 Yeah, this kind of structure style.
09:58:35 So I think, on the left hand what you mainly seeing is just the lengthening of the original wave trained.
09:58:43 So you might sort of look at this thing and then pick out the fact that new waves could have appeared in the gaps between waves.
09:58:50 But for for the original thing is you just seeing a sort of a process where the original waves the lengthening is they go down and and that that might be because you know the flow itself is developing.
09:59:05 So the picture of one picture if you look at the slide after for instance.
09:59:13 No, no, the one after that again.
09:59:16 What happened is when you're showing your solution. Yeah. At this point I think the wave train is developed some typical wavelength, and it's quite impressive actually this this concrete channel.
09:59:28 It's about a mile long, and you can go to the top, and you can go to the bottom and you see that the wavelength is about the same at the top and the bottom.
09:59:38 And what provides the thickness of the phone.
09:59:59 You mean this The thing is that for the work which which know the phone. The phone, that is a whiteboard, I mean
09:59:58 well.
09:59:59 So, if you believe the model, then this type of model what what happens is that, without any viscous terms, it generates a shock.
10:00:12 Right, so you've got this linear energy which is senior cycle but it forms a shock, the front face is is actually and jump in without any biscuits stresses if you add this stresses, it actually smoothed it out and so the thickness of the front is controlled
10:00:27 by the biscuit stresses that you add.
10:00:31 Maybe I should I should have said one thing that I forgot to do it, which is that this type of model, where you have a density equation in the mass equation, if you throw the drag away, you get something that's very similar to what people use for kinematic
10:00:44 waves kinematic equations, you know they are the canonical models for shock formation, but the shocks there are all to do with the initial condition what divorces the roadway phenomena which comes in these equations.
10:00:57 Once you've added the drag is that there's a linear instability, so the linear instability creates the structures which grow, and then without any viscous terms, they form shocks.
10:01:07 And so in in in these kinds of flow models, well so us I guess you can see the difference here they do have a thickness, and it's complicated because you see the things rolling over right it's, it's sort of a breaking wave.
10:01:25 Yeah, so it's constantly that control this form. Yeah, I mean that's one way to try and interpret it but whether whether it's really a simple viscous dude, doing those kind of tricky to say that goes back to patent question.
10:01:40 Thanks.
10:01:42 Then we have, David.
10:01:50 Um, I was just wondering Neil you had, we've had two talks which both have got the traffic flow traffic jam problem in them, and one of them.
10:02:00 Essentially went to, you know, had a short relaxation time to a mean flow and yours is going to come about through some drag I guess in the to that friction term.
10:02:10 These, these, these related in some sense, are these just two different conceivable explanations of the same thing.
10:02:24 Well, from a mathematical perspective you'd say that there are somewhat different because there are different different terms in the equation but I guess you know that the you know the fact that you've got these two occasions, and there's maybe have slightly
10:02:36 different terms in them. And you need something to trigger the instability in some ways, and it's, you know, you need the wave to continue to grow once you started it, and and I guess there could be different ways of doing that, that stemming from different
10:02:52 slightly different physical effects. So in that context it said it was the same thing, perhaps.
10:03:00 So in the in the plasma, maybe it's a question for the plasma people so I don't know, in the plasma problem.
10:03:08 Where, where this relaxation to a mean flow and, and the jam is invoked through that could that could that also be wrong. Be worldwide. Go on wave type instability.
10:03:20 Instead, and mean, if I could break in. Yes.
10:03:25 It's raining just business. Well, I mean, but I mean, he discovered this, well I mean yeah and let me finish Nisha, and then you can break in.
10:03:34 I mean if you. It's strictly a matter of getting a sufficient delay time leaving or rate to, leading to overturning, and by the way if you if you dust off your whittam, you will see the two problems discuss side by side and a correspondence between the
10:03:54 two drawn.
10:03:55 Okay, and with the, with the, the delay rate for one and the business with the friction, you know with the friction for the other.
10:04:07 And both can reduce to kinematic waves in a certain limit of course.
10:04:12 Yeah, okay. All right, I will dust off my Whitson.
10:04:19 Okay, Thanks again, how the question as well.
10:04:35 Yeah, several that essentially you're not on the distinction that you make between confected and absolute. It looked like it was well for for saying this to the two it looked like an important thing to, if we were to to apply that, for instance too fast
10:04:58 I am interested in or to some other system because the way I understood what you said is that because its collective and not absolute you actually need constant forcing right.
10:04:55 Could you elaborate a bit more I'm not sure if he really understood what you meant by convective versus absolute.
10:05:01 Go.
10:05:04 Let me do that. So, the issue is that you know you things moving along, it's got a face speed in a certain direction the instability. And so if you're sitting there at a particular station, then the instability will go by you.
10:05:20 So, you know, there are there are theories that you can exploit to determine whether the instability is sufficiently strong that it will not nevertheless grow in place, or whether it will just have to go with the flow.
10:05:36 That's, you know the broadcast or the bricks criterion is gives you in the simplest way.
10:05:42 In the context of these role waves. And, you know, using the simple model, you can take the dispersion relation to figure out whether you're dealing with an absolute versus a convective instability.
10:05:56 I wanted to mention something about this volcanic tremor problem.
10:06:00 So, in this volcanic tremor problem you can you can look at a model which is basically like this you have a conduit with elastic walls, you know that the thickness of the conduit can vary, so that and there are viscous stresses in this fluid, that's the
10:06:17 magma, so you can come up with a roll wave instability, but in this particular problem, what you find is that the instability is convective.
10:06:27 So it's not absolute. And that's a real problem in this particular model, because what you usually have is you've got something a bit more like this basic scenario where you have a chamber connected to another chamber, through a finite length conduit.
10:06:43 and it's no good if you have a convective instability because you may start with a disturbance at one and that then it gets flushed out the other. So there's no mechanism to get this thing to sort of rock backwards and forwards in place if you only have
10:06:58 a convective instability so somehow you need the one end to talk to the other end. That's the practical setting of why the roadway phenomena, probably doesn't explain volcanic tremor, because it is convective and you need something that's absolute because
10:07:13 of the setting. But that's exactly the follow up to to the question if we were to, to try to apply that to the plasma to to a plasma to any kind of system where you actually have a boundary, or you're not able to have a.
10:07:30 It's not periodic, and you, you at some point you have to, to hit the boundary. The fact that it's not absolute, and that it's connected, as I understand it means that you would have trouble, having a stationary.
10:07:49 Let me, let me just mention a bit more about the granular problem here. So what's done in the granular problem is that you put a wave maker at the top.
10:07:58 So you've got a basic flow down the chute. And there are fluctuations, due to imperfections, and that bit of noise can be sufficient to see the convective instability.
10:08:12 So if you have a sufficiently long channel, then you will see it.
10:08:16 But the problem is is that practically you may not have something that long. And so what people do is they they put a wave maker in at the beginning, in order to see it.
10:08:25 So the.
10:08:34 The bursary flew right at the beginning that it showed this one. So, so we had to have something that was 10s of meters long in order to see the natural development of role waves there through the vagaries of the flow.
10:08:39 It was almost impossible in something that was only 10 meters long to see it. You needed something really, really long precisely because of this issue.
10:08:47 But on the other hand, if you're willing to put a little wave maker at the top you can see them. And then they develop into nice what wave trains and you can look at the dynamics that way.
10:08:55 But this I think this is a key issue about the phenomena.
10:09:01 You give me well maybe I'll come back later I have a second question.
10:09:06 Yep, there we're running out of time. I mean, last maybe last question by Misha.
10:09:15 Yeah, I just wanted to add your beautiful gallery of anomalous phenomena.
10:09:23 I couldn't resist to tell you this in plasma people rediscover this by look at the environment, the head of the cometary shocks I guess movement that gases, get from the comet in the solar wind gets ionized and drives cyclotron waves and the, the equations
10:09:44 that quite similar.
10:09:58 And that instead really determines just cyclotron instability this other example is astrophysical very of have this pressure gradient near their shocks was shocked like supernova shocks and pressure gradient works as a body force for the acoustic their
10:10:22 did you try to apply something like Gotham's of history is really creation, which is just generalization of ULKDVKDV for two dimensional flow they applied it I think for Capillary Flow over the surface like window and problem people haven't done that,
10:10:40 it's it's largely one dimensional, you know, normally, you've got a channel and so and so rather than having a sort of a sheet flow model which is what's on the screen there, you kind of do an average over the conduit that you're dealing with.
10:10:52 So that's what would be people would do in hydraulic engineering and you end up with a one dimensional model, but for the lemon are falling fluid films people have done exactly what you said.
10:11:02 you could look at a proper 3d falling fluid film, and then think about what happens in the in the other direction to, and people have done exactly what you've said in this context.
10:11:20 And the iz able to describe the merger. I mean, you see you see this beautiful network of young girls other experiments where people have poured viscous fluid down the outside of cylinders and flat planes and they've looked they've looked at the, the
10:11:38 two dimensional patterns that you get and their interactions and to try to describe it by those types of wave equations.
10:11:45 I can't remember exactly how successful they were but there's definitely been attempts in those directions.
10:11:52 Thank you. Very interest.
10:11:54 Okay, I proper so thanks a lot for the extremely nice token the discussion and very nice features. I propose we stopped discussion now and we can come back if we have more time in India and.
10:12:08 So, but if you can share your screen oh my god is actually time, so.
10:12:18 So, always technical problems.
10:12:24 All right, can everybody see.
10:12:28 Perfect. Alright so this is is written originally motivated by jams and trying to address these questions of the fate of the delay and the phase.
10:12:43 And, Well we're part of the way there but there's something else emerged along the way that's finished so on turbulence spreading and explicit nonlocality.
10:12:55 And you might some of you may think you're in the wrong Zoom Room with a title like nonlocality maybe you're in the quantum information program but no you're in the right program and we'll talk about that.
10:13:07 And it's by the way with the entering how from slip and chain law who came to San Diego in the winter of 2020 on a scholarship to work with me for a while and well here's something else came to San Diego in the winter of 2020, but at least we were in
10:13:28 same time zone, even if we ended up working by zoom, but it's a lot easier than different time zones and an excellent student I my dad. So what's this about Well, a lot of explanations some analysis which is definitely.
10:13:46 It's quite detailed and it's.
10:13:58 We had fun, and they description will be schematic for the time, and then we'll talk about some of the results and I won't say just a little bit on simulation comparisons because well we ran out of time and energy and that gets a little boring to a point
10:14:07 so the prologue briefly what's turbulence spreading while it's entrainment and then we had great confusion last week and I would remind everyone in the sense of towns and, you know, he speaks of nibbling and involvement and what we're talking about is
10:14:24 nibbling where the mid size is small compared to the scale of the in homogeneity.
10:14:30 It's the advancement of turbulence into a region which is stable or where exploitation is weak or absent it's intrinsically a turbulence propagation problem.
10:14:40 That's a component of basically all k epsilon type models and in homogeneous turbulence. And it's intimately connected to avalanches. I mean you can't really separate it and I might advertise to Golden Oldies here since the since we don't have a color
10:14:59 picture in those as we discovered yesterday people tend to forget them.
10:15:05 And, you know, just a little bit, both spreading an avalanche involves non brownie and radio propagation of expectation avalanche King is overturning and mixing of neighboring cells coupled through the mean gradient and track something like following
10:15:24 law and card or something like a shock or burgers and all this is quite an old story turbulence spreading comes really through spatial scattering through nonlinear coupling.
10:15:41 It couples through the envelope through the turbulence intensity field and often ends up as a kind of nonlinear diffusion. As you can see both things stem from thin ice also these triads where one legs the profile.
10:15:58 One legs the envelope, very, very subtle distinction it's impossible to have one without the other. Spreading can persistent strongly driven non marginal regimes so you're not necessarily pin to marginal, which effect is more dramatic depends on the details
10:16:18 and a lot of the discussion of the distinction i think is a thinly veiled fade over priority.
10:16:26 For a detailed review of these in an obscure place. There's an articles quite fairly recent 2018 by hom and me and the Georgia. Journal of the Korean Physical Society it's that that's why it's there.
10:16:42 And it's the 50th anniversary issue but it's specifically devoted to this issue of, you know, deviation from Vicky and transport, by the way the status status quo of modeling these is some kind of nonlinear reaction diffusion equation with all sorts of
10:17:03 bells and whistles potentially in the in the diffusion that's why I asked Neil about nonlinear diffusion defects, and various nonlinear saturation and in homogeneous grow things but it's it's a glorified k epsilon model by any other name on nonlocality
10:17:24 you're not in quantum land I mean we as you may have figured out, we have an obsession in this business with the flux gradient relation which historically has taken to be local and, of course, isn't.
10:17:37 So the question what's the relation between the intensity say the flux, and the low, and the gradient and you know so instead of a finicky and thing you might have a relation between the flux and the gradient at a different location and by the way, Ed
10:17:56 Spiegel was writing stuff like this down for advanced mixing link theories in the 60s so please no fights over novelty here, and likewise you could speak of non local growth, which will see is maybe not a bad thing to speak of the real question is is
10:18:15 the nonlocality explicit or is it due to fast or local front, and most importantly, what's the physics going on and if there is a nonlocality what's the scale.
10:18:27 I find the local versus non local debate, kind of dough at times because it's it's meaningless until you specify the scale, the status quo and I'm sure they'll be disagreements is that I think people recognize the dynamics are non local but rather weekly
10:18:47 so the scale is uncertain and the physics is controversial and there's really no clue. The rest. So that brings us to the analysis so I like simple models, a very simple model known to some of our colleagues is the Dharma a model, which really should
10:19:07 be the model, by the way, peloton touch air wrote it down first.
10:19:14 And it's for trapped die on drift waves and maybe you don't want to know what that is but because basically because you bounce average you reduce a three dimensional problem in space to two dimensional and that makes it interesting but because of the
10:19:31 precession resonance you allow kinetic effects so it's a beautiful little model that you can play with.
10:19:38 And if you want to retain well the ret, not if you the POS own equation and there is the you can treat the electrons and the circulating ions is as Boltzmann, which simplifies it it's a crude approximation, but at least makes for a simple closed model
10:19:56 and the polarization scale which determines the zonal flow is now set by the banana with the same has a rich spectrum of scales from the gyro radius to the banana with the mode scale the spectral scale and the profile by the way for those not in the,
10:20:16 the, not familiar with the jargon the fact the field, magnetic field after the plane of the page is in homogeneous means particles drift off it and they drift back and forth and they form a banana shape and that cheerful word goes back to tree Yes, I
10:20:35 I think 1967 I associated with Birkin gelei of, but maybe more general.
10:20:44 Anyway, we're gonna you know in the end, it's a philosophy equation, okay and a glass of equation and in some form is a natural for playing games with PV right the velocity of equation is is a conserved phase space density and what's PV it's effectively
10:21:02 something conserved along particle trajectories.
10:21:05 So, not quite that simple but it's a good way to think of it so you can construct moment equations and you come up with an equation for the vortices at which should look the little like things you're familiar with.
10:21:19 There is a breaking in the. When you go to the fluid, of the, the conventional PV conservation due to the fact that drift is a function of energy but it's only linear.
10:21:35 you still, you have an additional source term for the potential and stuffy.
10:21:59 The zonal flow is as before right again you will know that the zonal flow evolves differently from the, from the fluctuation and that's because of the effect of the electrons and so forth and this message still is in doubt, but the equation is simple
10:22:04 enough so what we have is PV conservation with additional effective production via curvature. So it's important for those of you outside the business to know curvature modifies things a little bit.
10:22:20 Think of curvature is boy and see if we want to calculate transport and we'll come back to more on that at the end you want an intensity field.
10:22:30 If we want to calculate transport and we'll come back to more on that at the end you want an intensity field. How now this is important what's actually locally mixed is PV, right. So, if you want an intensity feel the way to get it is to construct the
10:22:41 potential and stuffy evolution, and then work backwards from potential and stuffy to intensity that's kind of important because it's it's PV that's mixed not potential.
10:22:54 Right. So what you do is you form a two point correlation equation for PV evolution and then you operate on that with two greens functions to invert it so in Michael McIntyre speak, enter the game of PV and variability which is very important here.
10:23:14 So, and you cannot go often follow, you know, games we played before I refer you to the classic by Tom do pre and 72. By the way, do pre died this past year.
10:23:31 One of the recent losses in this field.
10:23:35 And bingo you have a PV evolution equation, and you have the usual set of sources but you have In addition, this is the fluctuating potential and stuffy but you have of course the curvature acting as a drive there and those of you who are with us last
10:23:51 week might recall the work we did with assure Yvonne and it's a similar idea exploiting the if you're going to do mixing exploit the use of the thing that's mixed.
10:24:05 Now, in contrast to the work on two point correlation of do pre at all over the years what we're interested here is not the two point correlation but the intensity field or the envelope.
10:24:17 So what we're going to do is ultimately take the radio separation to zero and calculate the correlation as a function of the in homogeneity and radius and the separation and why, or, or five, and that departs from the pre game.
10:24:36 So you do a little algebra, right, and you can do here, you close this by a two point kind of quasi linear calculation I can't really get into that.
10:24:49 And you find basically evolution of the potential vortices the correlation of the form of this sort, where you have scattering and radius and shall we say correlated or relative scattering and why.
10:25:06 Okay, you'll note, by the way, exploiting the symmetry. You can average over the centroid and why so only the relative scattering and why is important but you do, because of the in homogeneity, you do need to take seriously the radio evolution.
10:25:26 So you arrive at this point which is an amusing equation to deal with and now you have to get to the intensity field so you operate with that with the Greens function.
10:25:38 Now, the point is the scaling thing the Greens function in this model is easy to identify it is set by a factor of times the inverse banana with so again in McIntyre speak the banana with is the scale of P of inversion, and this automatically defines
10:25:57 it as the scale of nonlocality right because you integrate over something of that of that size or the cut off rather have that integral will be imposed by the decay of the Greens function.
10:26:10 The ultimate spectral evolution will involve the Greens function involved with the PD correlation equation so that leads to certain implications, about the scale.
10:26:22 So, you do a little more algebra and there's quite a bit of algebra here which I won't bore you with and you arrive at something you really want which is the potential evolution equation but now it's a little lively it's integral differential because
10:26:38 of the inversion.
10:26:40 And you'll see appearing both a smearing due to the inversion over the nonlinear diffusion which is straightforward from the quasi linear type approach.
10:26:51 But that also smears other things and in particular smears the production term due to the curvature which gives you a D localized growth. Okay. In the limit of the banana with going to zero, you get back to the same old thing, right, the same old thing
10:27:10 being something like like this here.
10:27:13 And that's sort of what you get from the Euro sticks, but the point of all this is the heuristics isn't really right. Okay. And one other point the heuristics is usually in for in the form of potential of energy.
10:27:29 And I would say that's the wrong variable to play the game and better to work with it and stuffy due to PV conservation and then ultimately, even in the heuristics you got to worry about the inversion to the potential, and that's something that hasn't
10:27:45 been faced before.
10:27:48 So that brings us to the answer of sorts, which is kind of the the spectral evolution equation, and you ask what's new here, what's new here is you have non local nonlinear diffusion the car, the convolution of the Greens function with are familiar friend
10:28:09 and you can keep a correction if you so desire. Not really important.
10:28:15 The other interesting thing is you get non local growth, and quite clearly the nonlocality appears from the convolution of the Greens function again going from PV to potential with, with the growth of fact with the basically with the heat flux that drives
10:28:36 the whole thing, and that that is have some potential interest. So that brings us to the results.
10:28:44 So the first result is what I said and I rewrite it here.
10:28:49 Just for emphasis.
10:28:51 And what's new and important the non local growth is important, can we understand the physics behind it. There's explicit nonlocality in the problem right and that this really is due to PV conservation.
10:29:06 The other thing that I think he is least in this model the range of the nonlocality is quite modest right it's, it's basically a factor times the banana with.
10:29:18 There's also.
10:29:20 There's also a new wrinkle into the sheet in the scattering or diffusion, but we'll see that that's that's not as important as the non local growth.
10:29:34 So here's a surprise for you amazingly my neighbors a General Atomics had a point in their paper in 2005, I mean in the middle of this fight over turbulence spreading and nonlocality that's gone on for years.
10:29:52 They made a purely, you know, a simple, you know observation that you could fit a lot of the data, taking a local model with a growth that had an exponential nonlocality stuff on it, and there of course there was no physics in that it was a curve fit.
10:30:11 but events have shown their curve fit, I think may have a physical basis, and we may have the scale identified. So here in some sense the point you know the the message is, it's not a bad curve it okay and the physics and the scale at least in this model
10:30:32 is determined the non local growth is the dominant in fact you're going to have to take my word for that I got tired of making view graphs. at a point but we obviously looked at switching on and off and that's the biggie.
10:30:47 The non local growth accelerates from propagation so you get the Fisher type of scaling with a correction due to the banana with, and you get a slightly faster speed, the penetration into the stable region from the unstable region is also stronger and
10:31:05 it scales with Delta be over lt kind of like rose star with the banana with, but I mean it's still fundamentally in this model a fisher type of story and it's easy to see why the penetration is stronger because the tail of the influence function can connect
10:31:26 the stable and the unstable region quite simply, so the bigger picture. This is yet another example of the utility of PV, which can become stronger and stronger in my view, nonlocality it's not really used in the, in the plasma community, as well as it
10:31:45 might be.
10:31:47 The nonlocality is explicit, but it's modest. It appears to me that the conventional wisdom of weekly non local dynamics is on target in some sense but now we have a physical mechanism and a scale arrange something that I think people will agree with
10:32:08 weekly non local does not mean quasi linear evolution equation is a crucial intensity evolution is a crucial constituent for calculating the flux is still.
10:32:22 But it's important to distinguish I think the different pieces.
10:32:28 One can dispense.
10:32:31 You could argue, I'm sure, GM will say this well if you're going to do all this why, why bother with calculating the potential correlation why not calculate the flux evolution.
10:32:42 Okay, and go for jams and of course that's what we started to do okay and that's actually it's done but maybe not quite ready for prime time.
10:32:53 But, I mean, they're not so many surprises on the jams there. Okay, in the sense that you're going to the delay is a sensitive business. There can be perhaps some nonlocality in the delay which is interesting to me a more interesting direction rather
10:33:13 than flogging this continuing to flog this this dead creature through the fleet, as it were, would be to maybe explore a problem with a larger banana with and watch air stairs one in the face is exploring the non local problem for the energetic particles
10:33:38 and in particular, again apologies to the people outside of plasma.
10:33:44 But the energetic particle mode, I think, which is a kind of electromagnetic trap die on with a touch of L famous coupling in it which is heavily dependent on trapped ions is, it's an important beast when a burning plasma, and it, It is an ideal candidate
10:34:05 and it'll have a big fat banana with a much larger scale.
10:34:10 And that I think is pointing to. I think what I think we should be doing is exploring this nonlocality question in the context of a, you know, in the, in the energetic particles where I think things will get quite rich because, among other reasons there's
10:34:31 less of a pinning of the turbulence right as in the low frequencies your pin to resonance surfaces. And with the EP is not so. So that's all folks. Thank you.
10:34:47 Thank you very much bus.
10:34:53 Yeah, I already see Misha.
10:34:58 Ready raise the end this.
10:35:04 Michelle, can you unmute your microphone, the usual suspects, yeah thank you I I didn't see him. I thought I am not the first one, it's it's fascinating story I, when I kind of trying to understand what maybe naively what what is the cause of this propagation
10:35:28 of instability domain into, into the stable, don't wait, if you have a some sort of interface between the two it can identify or demarcated.
10:35:53 equation or the turbulence propagates and change the medium, and after we should start more unstable so we'll see it so can you actually distinguish between these two.
10:36:07 I think in in this kind of in this kind of thing is clearly the ladder, I mean the physical, the physical analogy.
10:36:15 The clearest, you know textbook analogy it's inland I haven't listed so you must have read it Nisha is the wake spreading right where you have the wake invades the stable region around the boat.
10:36:28 Okay, it's kind of it's quite analogous to that.
10:36:33 And hence, hence the subject of, you know, the idea of nibbling right you know, as, as a kind of gradual erosion. The point of course here is that well in the in the wake of the thing is neutral.
10:36:49 Here the, the surrounding region can be damped so the really the question which is precious little answered is the depth and penetration right and that's the figure of merit in these games of avalanches and scaling is how how the penetration depth scales
10:37:07 with the parameters of the unstable region. The other thing which Robin Heyman in looked at was if by some chance you had a sub critical problem and he didn't answer the question of the physics of the sub criticality is what kind you know what kind of
10:37:26 propagation dynamics, you can get and there is you might guess you go from more of a fissure type structure to more something like that fits unit mumo type structure you exploited by stability.
10:37:42 Okay, so, I was a little confused when you started from last if I thought if you go directly from wassup and let some particles to be neutral, the stable domain name might you be you, you couldn't do that.
10:37:57 I mean we did that because it's quite frankly it's easier to do this. All right, and I think it's it one has a clearer sense of the physics, working with the moment equations and we were interested in touching base with the old older formulations but
10:38:15 it's a good point.
10:38:17 If you you know I mean, as you know, we, you know, I've done a lot of calculations on the evolution of two point correlation for the distribution function.
10:38:27 So in principle you could formulate this problem at the level of working from the distribution function directly and then God help you trying to connect with the answer it with reality or physical physical observable at the end, what that would introduce
10:38:47 that's new, that is I think maybe what you're driving at is of course the resonant particle effects. So in addition to nibbling Allah Townsend, you could get in a sense penetration of the stable regions by ballistic modes.
10:39:04 Okay, which then by the charing cough process could excite things. Okay, so that would be an additional effect.
10:39:13 Do you get, you can even stay within the linear framework, basically right well I mean you could for that but you couldn't Of course for the nibbling.
10:39:24 Yeah, yeah.
10:39:28 Thank you.
10:39:29 Yes.
10:39:28 Okay, I think dm was next.
10:39:33 The the nonlocality of growth that you're showing is the same thing as the nonlocality of time delay. Right.
10:39:41 Not really. Although it's related. It's related. I know I wouldn't, I'm sorry I wouldn't quite by on to that statement yet it's related, of course, but I'm not the same thing.
10:39:54 But what plays a key role is the boundary older caught off of the brain function, the course but here you know here you, you know, except and I wouldn't advertise this for the Lh transition where you're sitting right on the separate tricks right and that's
10:40:09 that's an interesting point right if the banana goes through the separate metrics. What does that sound like, sir. That's a familiar story too, right it's probably older than you are right that's orbit last one more time.
10:40:25 No but excluding that the scale of the banana excluding a barrier, certainly, you know, and certainly VLH transition, excluding that the scale of the banana is generally small compared to the variation of anything else so I don't think we have a problem
10:40:43 here.
10:40:46 But, but to have an important. Well, to have a major effects of nonlocality of the growth or nonlocality of what you would call Thank you intrinsic nonlocality implicit explicit explicit nonlocality you need to have boundaries of the cortex of the brain
10:41:07 function that are large, possibly, and the larger the better, right that's, That's why I pointed EPM is the way to know. Absolutely. So, can you can you think of situations, especially well you know where this is heading, especially near marginality when
10:41:26 one of the, absolutely. You knew this coming, where the boundary was not for instance rule, are your or delta be here, but would be your intrinsic avalanche skill or you you know mezzo amazing skill and emergent Mrs killer.
10:41:48 Some ganda Rhines or whatever.
10:41:51 Do you see a situation, well I mean the. If you want a set of parameters so you can write a proposal for either or West funding Don't ask me but I think that that's quite possible right and one could hear it would be interesting what I could think of
10:42:12 is. Are there other effects the getting the story like that scattering on the banana or a bit or something but what what you say is quite possible right if you, if you, you know, it really comes down to what physics and there's the PV and version, can
10:42:34 speak right that's the, if you ask what's the conceptual point that's it. So if you can find an effect that kind of strings out the width there in the PV inversion it amounts to in your language, em, is the PV in some sense inversion operating near a
10:42:56 critical point, then you'll get a big nonlocality.
10:43:00 Okay.
10:43:06 Okay, there's an
10:43:11 interesting comment and the question. In fact, the comment is that apart from the case of first is which is your views mixed up, I agree with you.
10:43:22 There is also the case of the edge where you could have a relatively large safety factor, and still a decent temperature the top of the pedestal and therefore a banana which could be quite high there.
10:43:34 Absolutely, absolutely. And I mean and bear of course i mean i i don't even like to utter the words orbit loss right but you also have that thing right you'd have, know your your inversion with overlap the separate tricks right.
10:43:50 We're talking about, and then really I mean, you can imagine what's going to happen at a certain code we both know we'll see it you can count on that, that.
10:44:01 Okay, so I think that's, that would be a very interesting development as well. Yeah.
10:44:08 And just maybe to bounce from the common exactly the same occurs in the in the very cool.
10:44:18 Could that be a way also to populate the.
10:44:23 The quote by this nominal effect. Well I mean look one thing my, you know, one of the reasons we need bright young people is because they don't, on the one hand they don't listen to us which is good, but they also work at it, you got to have both one
10:44:41 without the other isn't good but, you know, dancing, how does both.
10:44:46 He went off and looked at some of these old classics that some of us are really fed up with in particularly the lens simulations from 98.
10:44:59 And also on the other on the other side of the pond something be loved in France Macmillan at all right, similar exercise.
10:45:10 And one of the things you know I said to him. These are old simulations and you better be careful on thinking anything like this for it because of the is the runtime sufficient.
10:45:24 Well, it looks like actually it is they get around in the simulation run they get about 800 I on bounce times and those simulations they're quite there, you know in their collision less to the point you can believe any code collision lyst structure trap
10:45:41 die on effects are quite, quite fair game, and of course that's never invoked. Of course everyone points to Rosenbluth and Hinton which is a trap die on screening on dissolved flow but never the trap die on mode itself.
10:45:57 The other comment is even the averaging time in those simulations is about 100 bounce times. So I mean I think this is very relevant to the core.
10:46:09 Okay.
10:46:11 But I would encourage you guys. Our colleagues and cataracts, are the group to really nail this to the wall by combined operations with Jews Ella and your dorm a code that's one of the reasons I wanted to give this little talk to energize you know i mean
10:46:31 that that would be the way to to elucidate this matter.
10:46:38 Yeah, but maybe because I haven't taught on the way to disentangle this effect from other kind of physic that could lead to to spreading. So, would you.
10:46:56 Have you thought of a way of unambiguously from a simulation,
10:47:04 say that this effect is at work, and not others.
10:47:11 Would you.
10:47:11 I think that's it. As you know yeah maybe that's a tough one because it particularly well in in the pic codes you don't have the privilege to turn different things on and off so usually in GZOA you do I mean, what I would do is you do have some control
10:47:32 over the plus own equation and I believe in Jews Allah, and I mean what this is saying is the scale in the process own equation is determine the polarization is determining the scale of the spreading or the beast.
10:47:50 You know the non trivial scale. So you could scan the sensitivity of the spreading to the scale of the polarization.
10:47:58 I mean would be one thing to do the problem with that is some other things may change and I just don't know.
10:48:09 Okay, so uh, I see your point that the, as you mentioned is certainly tough so we have to think about why use ways to to address this issue.
10:48:21 Well, thanks a lot to the other speakers I don't see any other.
10:48:28 Any other questions so maybe I have since we stopped, we had to stop the earlier discussion.
10:48:34 If there are some urgent questions to any, any toke I see that you should care is still, Still alive, he's connected I don't know where he whether he he's a Steeler, the week.
10:48:53 No, no more can it can I ask it. Oh, yes, please. All right, let me get ahead of dm for once I burst in I'm learning the sort of a question to kneel and Yusuke a to, I mean, there's no you know there's little doubt that you can play these games with the
10:49:14 row with the role wave and jam model what's, what's and they mean I was very interested to see that the corresponding effect was well studied in the role wave problem and coarsening is kind of what this workshop is about isn't it when you get down to
10:49:38 it right.
10:49:39 The question is about the train right and I mean, in other words the striking things about the staircases is the regularity, and any, any thoughts, I mean it sounded Neil in your talk, that the train effect may be do related to the drive of the system
10:50:02 or something that mean could you could you comment further and particularly what might set the, The distance between crests and things like that.
10:50:16 Well in the St Bernard model there is a sort of a little window of stone operations. The windows and fruit number right.
10:50:27 And so, I guess I don't I don't know how the window.
10:50:32 So, the window I was referring to is the window of typical separations between waves that changes, presumably with the parameters of the problem, like the fruit number I don't know, I don't know how it changes in that way.
10:50:49 But, you know, for a given fruit number you can you can look at the nonlinear dynamics and there's this competition between nuclear nation in the in the wide gaps and course name for short gaps.
10:51:01 And that leads to this range of separations but it's it's despite a lot of separations that that you know would fit in into the gap that, you know, as you get bigger and bigger domains you.
10:51:14 You would fit a certain number of role waves in there, but there wouldn't be steady, it wouldn't be a steady wave train they'd move around quite a lot.
10:51:24 I guess what's striking about the observations of that concrete channel was that these waves separations, there was well pretty well defined.
10:51:35 Even in that that turbulent water costs and I'm not sure why that in that particular example the separation was so well constrained. That's what cut what promote the question right yeah yeah it seemed as though, there'll be a far greater spread of possible
10:51:53 separations based upon you know the simple model. So I don't know why in that particular example, it seems as though it was a relatively clean separation.
10:52:05 And, you know, that's, that's in the same banana model way you can kind of understand where the secondary instability comes from that limits the separation, and other problems.
10:52:17 There's no limit to the separation between the structures and the current elite equation. There's no similar secondary instability, somehow, you need to be able to access the original instability or another instability in order to limit separations.
10:52:33 So, in that sense it's definitely problem specific.
10:52:40 Yeah, I don't have any good answer to your question I'm afraid.
10:52:44 Interesting. Thank you.
10:52:47 Please My name was rose in milk soon grocery boys, you should just start from linear instability. Look at the maximum growth rate wave number does you go away.
10:52:59 Later when non linearity kicks in.
10:53:03 Certainly the wavelength of maximum growth is normally not in this table band.
10:53:08 Normally normally if you kick off with the unstable linear mode then it will subsequently nonlinear the course and into a wide awake trained
10:53:18 in plasma similar problem in plasma normally remembers that maximum growth rate and then you just develop linear waves and Stephenson, but it it maintains its separate separation from the inherited from the linear phase of evolution.
10:53:40 Yeah right after Of course if you have dispersion if you have distortion, like your drive, not drive unstable your drive. Katie viewers, for instance, then you get variable separation so that you don't have specific distance between your roles or shocks
10:54:04 or whatever. Yeah, I can imagine that there's plenty of systems so it's the linear dynamics exert some kind of control on the nonlinear wavelengths that you've observed, but there's no guarantee that the non linearity would would do that for you.
10:54:18 And in that these roadways problems. Typically, you know, the most unstable wave train is is is is too close together, and it causes.
10:54:29 Maybe they Misha, Misha.
10:54:32 Misha, Misha. Yeah, yeah, yeah, maybe we can give the last word to to em if you don't mind and then you can interact with either by email or via slack.
10:54:44 I'm happy to have the last one but I guess the UK had a follow up, because he was also mentioned in tax question to pants. Yeah, well, I was about to say for the damn case, this most unstable web number for the damn growth, they kind of do a good job.
10:55:00 We're talking about the spacing.
10:55:05 And for the simulation part and the choruses.
10:55:10 we do find it that much. With this simulations maybe it's because the way we posted, we just post, the boundaries so we just inject things there.
10:55:22 Maybe we will, if we force, entire regions or some, we have some distribution of the forces over the space, we might have the coalescence might happen, but that we haven't tried, but it could have been, I guess.
10:55:49 Okay, since short 1 million. Yeah, actually, quite close to work but just just said, but maybe with a different take, it's actually directed to you, and based upon what you discussed with the Canadian equation.
10:56:11 I was wondering when you were showing that there was an evolution principle free energy to say that would in the by model, which equivalent to two can heal the odd.
10:56:25 There's this evolution equation that you know in the end you just get one john.
10:56:30 Would you have such an evolution principle for this model or four generally those role wave problems.
10:56:39 I think that. So there are two equations that I know of that are that nice variation of structure, there's the real Ginsberg land and there's can't hear you.
10:56:52 And they both because of that variation or structure, they both have to evolve to the longest spatial scale, but for the, for the blind the ball z or the Y slab bottle, it doesn't have that, and it's simply a numerical observation that, it seems to cause
10:57:02 them all the way to the end.
10:57:05 And certainly for the St. Vincent model it doesn't have that none of the models that I mentioned today and have that structure to it. So, there's no guarantee that they're going to keep coarsening.
10:57:16 It's very special, I think, looking at it is, you know, from the perspective of all the models that you can think of. It seems that there's something very special about real ginsburg land down and con Hillier, which force you to go to the longest spatial
10:57:31 scale, any perturbation is going to destroy that because you know you know the interaction is where you've got these structures, they've got these exponentially small tales, and they feel each other through the exponentially small tales.
10:57:44 So that's extremely delicate. So the minute the minute that you know you screw up the exponentially small tail, you'll stop it happening.
10:57:51 So I think that's more generically what will happen.
10:57:56 I was under the impression, sorry that's the end, I was under the impression that you started your talk on Newton Hill you're saying that there was a systematic derivation of your earlier blow into 10.
10:58:14 You can reduce it to can't heal yet, but that's a narrow window, and your way out outside that window. So if you know if it was a property of the system that it had to cause them to the longest scale you'd hope you to be able to prove that by extracting
10:58:26 some kind of a lap and I functional. And I don't think you can
10:58:40 do. Thank you very much.
10:58:35 Okay. Well, time to drop the session. Again, thanks a lot for the talks that discussions, and see you in one week.
10:58:47 Bye.