09:03:15 Alright, so a little bit of preliminary stuff so welcome everybody and this is the last meeting of the working group on plasma dynamics applications of layering, and it deals more or less with evolution near marginal stability and your marginality I shouldn't
09:03:40 say what exactly that means i think is one of the things that's to be determined.
09:04:06 And we have, well we have a lineup this morning. First we have Mr marginality himself, who's been
09:03:58 reminding us of this issue throughout the program so he'll have his, his 15 minutes of fame as Andy Warhol would say, and I have a brief communication and so it's six slides for without the the drivel on, actually it's a quite It was a question raised
09:04:19 by Boris Ryman during the blackboard lunch, which I think is interesting and it's something we haven't quite we've overlooked.
09:04:29 And then our third talk is Peter Reid who's going to talk about Peavy staircases, and you can decide how that relates to marginal marginality yourself.
09:04:42 And then if as if any time left haha with for general discussion so few questions to set the tone is layering intrinsic to marginality and, or is it only near marginal marginality.
09:05:01 We have two things now that are always advertise near marginality layering and avalanches and these two are in some sense the antithesis of each other, right.
09:05:14 So, who wins, do they both win if they both win. Isn't that a ringing endorsement of the jams model, I think, one could make that case.
09:05:28 That one of transport for for marginally overlapping sales is layering a simple consequences of that.
09:05:36 And I mean, again if we're talking about you know stiff systems and sort of hovering near threshold that's a relevant question. And that's what I'll, I'll talk about.
09:05:50 And then my favorite, which I would hope everybody addresses at some level is how near is near marginality right Can anybody quantify these magic words, and can any theory actually predict how near to marginal.
09:06:09 You know they can the theory predicts the domain of its own validity, which is always to me that the proof of the pudding and these things.
09:06:18 So I think that's enough preliminary stuff and we'll turn it over to Mr marginal himself dm.
09:06:31 Mr module that.
09:06:34 I'll try to give you
09:06:37 a flavor of work up that that really was done during gimme. She knows PhD, he defended a few months ago, and that was especially the first three of us who are here.
09:06:55 The idea was to test aspects of model reduction.
09:07:01 Comparing as you'll see, hierarchy of of models that are quite well established in, in plasmas.
09:07:13 And what turns out is that well near marginality kicks in, and that's where a lot of the models that are based either on questionnaire approximation, or on the separations of scaled.
09:07:38 I'll be trying to describe. So, first premiere on the reduce modeling as it's done, usually in plasma physics so you will know this now we have an open system with a slow forcing that drives out of equilibrium to heat women to matter 40 cities forces.
09:07:55 And we have a large variety of fast relaxation prod processes through the instabilities with them.
09:08:09 Always reacts is very often reactors and you try to drive it and then you have a strong flux. So you hover in the vicinity of something that is well that directly connects to but last point loosely defined
09:08:28 the marginal state.
09:08:30 And here we are in any situation very often where the marginal state above which we hover around which we hover is different from the linear stability.
09:08:45 So there is this gap in between.
09:08:48 That's usually associated to the preeminence of mean clothes.
09:08:55 And this is also the region, where spreading where avalanches France, are found to be.
09:09:07 Well, at least active, or probably active everywhere but especially active here and that's the only region. So, so far as we can tell, where staircases might exist in between linear stability and next to this nonlinear stability which which, that's how
09:09:29 we define here the near marginal. So I'm going to compare a hierarchy of models so first of all, we're going to resolve and have as reference slugs driven approach of the primitive equation, that's less of person.
09:09:48 Next quality electrostatic.
09:09:49 And because that's usually given very impractical because of cost for any band is very exploitation, or interpretation of data.
09:10:00 No very. There are various degrees of reduction.
09:10:04 And to popular reductions are the questionnaire approximation based on the quizzes linear approximation. And the other one is still based upon the plus of Maxwell system, but it's what's usually the jargon.
09:10:24 Revenge still be on slide one.
09:10:28 Yeah.
09:10:28 So you don't see, now we're seeing the title page. Oh really. Yeah.
09:10:37 I thought was a very long introduction but I actually
09:10:45 that is viewed can see
09:10:54 em sometimes it's a problem if you're sharing the application rather than the screen.
09:11:00 Do you see it No, now we see something like premiere on reduce modeling exactly so I'm going to keep it this way, is it fun with you. Yeah, sure, if you share the screen it'll be okay or so I think it's the application show that gives it will do.
09:11:14 So, I was actually well let me stick to this and he said works.
09:11:21 I was discussing this one.
09:11:23 So I was saying was illustrated here, you have this.
09:11:31 So, forcing, and the fast relaxation process, so you all were in the, in the vicinity of something that's the emotional state that was was mentioning, and that state is often very distinct from the linear stability.
09:11:47 And in between you have here this region where that's where we find staircases that, that's where it's all flows that are meaningful, and also especially active.
09:11:58 And we're avalanches.
09:12:01 I can wb criticism.
09:12:03 So, I was going to, I was saying I was going to compare those three approaches.
09:12:12 The reference is going to be the slugs driven less of person system, and the tubercular reduction the questionnaire approximation which I'm going to discuss further and the one based on a skilled separation, which basically just states that you still
09:12:38 resolving the glass of personal system, but you do not do that in a fog driven way so what that means is that here is for instance the plus of equation with swaps and collisions that you want to resolve.
09:12:58 So that's me to either reference.
09:13:01 What I mean by the scale separation is if you expand between a background distribution function and the stock trading one simple the Hamiltonian.
09:13:14 What you dropped. Is this part here. so basically fluctuation fluctuation cannot go back and influence the mean. So that's what's usually dropped in those regions driven models, and you still keep the phone on in.
09:13:35 And you resolve this equation here.
09:13:37 The advantage of that is that you have a fixed ground so you gain a lot in terms of computational power, but you have enforced. Still separation and chilled back reaction on the moon.
09:13:51 So in order to show you what that means in practice.
09:13:56 This is for instance you flux Regent time.
09:14:03 Well, every dot here is a different flux, with respect gradient at a different time of one simulation so you see that within the computation, you you explore large variety varieties of the folks read into the main what the gradient driven approximation
09:14:24 does that you pitch a background gradient, and you cover in the vicinity of that, what that means is that in time dynamics would drive you away from the, the chosen background, which would be the straight line here, so you have adaptive sources and sings
09:14:46 that go against the dynamics to actually enforce the proximity to to choose an equilibrium.
09:14:58 The third thing here is the equation so you all know this but I just wanted to show just to make sure that we're all on the same page that's what you what's usually and how it's usually done in as well so same thing we keep expanding between me and the
09:15:17 fluctuations. So what's done is, we dropped this one.
09:15:22 Those two those two terms here.
09:15:25 This is linear in.
09:15:27 In, the fluctuation.
09:15:29 So, each for your movies and they can function.
09:15:34 And so you can write the perturbation in space and time as a linear operator acting on on the actual potential and the background distribution that is still kind of complex because it's frequency dependent product of things that can vary on different
09:15:56 scales, and to have the consistency in the first place between the separation between a mean and the fluctuation you further say that the main profile evolution is a very slow, as compared with fluctuation evolution, so that thing you rewrite as linear
09:16:14 operator acting only on the potential fluctuation and you put the background, out of that, that operator. So when you read the heat flux, the integration that you need to do over time is a fast integration, meaning that it's on those skills that are first
09:16:42 as compared to the evolution of the of the background, profiles.
09:16:46 So that's how it's usually done. So you end up with fluxes that you're usually quadratic function of your, Your fluctuation in was I'm going to show afterwards.
09:17:01 This is the approach that's massively done for for plasmas.
09:17:09 So the question becomes how do you fit, how do you choose the spectrum for the fluctuation so it's usually an inverter going to show its own like this you have a double power low, that, that basically fits the spectrum.
09:17:26 You keep only the unstable mode so you have the open question of what if you were to conserve also the dance moves because we've seen that they can have an impact on on the rental stresses.
09:17:40 Since your theory does not predict directly.
09:17:45 The the spectrum, or at least the theory that's retained does not is not based on the second order movement.
09:17:55 Right, so the way it's done is that this becomes a fit on reference nonlinear runs the reference nonlinear runs are those ones. Those ones with scale separation.
09:18:10 So, you run, lot of news, General kinetic with scale separation runs, and you pave the space.
09:18:21 And that's how you fit your your closure.
09:18:24 So it means also that we inherit the shortcomings of the.
09:18:31 These models. And it's also not a self consistent theory in the sense that you need external data. And also this spectrum is a static, and it should.
09:18:44 Indeed, depend or respond to to the evolution of the free energy and also.
09:18:52 So that's the context.
09:18:55 So we are going to compare those three.
09:19:00 Those three models. And what we wish to test all the assumptions first of the land linearity of the turban fluctuations. The choice here of the saturation rule.
09:19:14 And also, the assumption of locality on the overall scale separation so these are the three things that we can, we can test. So the way we are going to do this, we're going to run.
09:19:27 First, the full fledged model as preference extract the equilibrium, and run the other two with exactly those parameters, and as proxy. Look at how well you can match the clock says how.
09:19:45 What's the properties of the token fluctuations, etc.
09:19:50 So, so we we have three numerical approaches, and we're going to investigate them in two different visions.
09:20:00 So here are the two different regimes that that we're going to investigate. One, we call it this train drive the other one in your module so this is.
09:20:14 Well, this is a debatable.
09:20:16 Of course, terminology but it makes sense, at least in so far as and strong growth means that the profiles, degree jumps into the free energy source, which is shown here in red is way above the linear threshold for instabilities and hearing read also,
09:20:38 the, what would be an estimate of the nonlinear stability so the new marginal state.
09:20:48 So I'll come back to how this is this is computed in the wild.
09:20:53 The second regime is this one.
09:20:57 It's the blue curve here so the the region's, as you see, are above the no stability, on average, but they are at all slightly below, What would be the norm the norm threshold.
09:21:12 So how is this nonlinear threshold estimated. Well it's estimated through this approach here, so we we take the.
09:21:26 All of the results from the full fledged songs driven case. And we scan. So we take the flow patterns, we take the this year with a to all of the other parameters.
09:21:40 And we basically stand the gradient and find with the amount of sheer, etc. that is found by the phone's ringing case at what for what regions, do we start having a non vanishing flux.
09:21:58 So that's how this is defined.
09:22:01 So, If you accept those definitions. We basically look into two regimes as frankly driven regime, and a new module one in your module is where we have this charities is all of those kind of phenomenon is Dr.
09:22:21 It's is more of a diffuse of nature.
09:22:25 So let's jump to the results.
09:22:30 Here are here's what you get.
09:22:33 Compared to three codes, so the reference here is in black.
09:22:40 That's the plugs driven computation. So we take those profiles and we run the, the one with a scale separation, these are the dots here.
09:22:50 And this strongly driven case.
09:22:53 And that's the quiz the linear take on things. So overall, when we strongly driven. The, the conclusion is, you have a reasonable agreement between, between all three models.
09:23:07 Okay, you have very strong sensitivity to share to stiffness etc in the, in the question linear model but that's second order first order.
09:23:20 Those are quite good women's, things change when you go in your marginal, and your model now you have this amount of flux, that's predicted for the system, and you have zero flux, as you can well imagine because it's at or below the nominal threshold
09:23:41 we need the threshold for which the, the model with a scale separation theory of flux.
09:23:48 And if we were to very strongly increase the.
09:23:53 The by 40% that that we outside any kind of arable.
09:23:57 We don't really recover the amount of stocks.
09:24:03 And same thing for the linear model. We under predict the flux. So, there is a specific problem.
09:24:13 Close to marginality that that's interesting to try to understand.
09:24:19 Further, so is it an inherent shortcoming of puzzling it doesn't seem to be to be the case because both the questionnaire, and the one that skill separation display the same kind of problem, or is it inherited from this gel separation or this locality
09:24:38 So, In order to to test this.
09:24:45 The first thing you to to compete or Cuba numbers.
09:24:49 Of course.
09:24:53 So,
09:24:53 Cuba numbers.
09:24:55 Well, when they are below one you're on safe grounds for for president approximations. And if we go back to this first integral here.
09:25:08 It means that the non linearity is mystery means, which so that the, the, the, the turban cells.
09:25:22 The, the fluctuations. On your dinner, and the determine seller, not disturbed by you made it known in heritage. So, defining that number in the usual way, we can have, we can basically think of several ways to continue to keep the numbers from from the
09:25:40 full speed data. So, as particles transit longer fuel lines, the, the truth the transverse the drift, I did a magnetic frequency.
09:25:54 So if you have a correlation length here lt de drifting magnetic frequencies, you compare that to basically the time.
09:26:06 The measure of the time over which you will explore the structure, which is given by the city, you can coin a number in that we can also coin to go to two other two numbers is increasing, that you can bet your luck Rajan coronation time with the random
09:26:31 walk estimate from the equals be dressed, also. So, depending whether you look at it, regionally or probably leave and have two different numbers. That's why here you have for each new module and from grade three different jobs.
09:26:49 So it's three different estimates for the Cuba numbers.
09:26:54 You will conclusion is that you you consistently get Cuba numbers of order one.
09:27:01 So you're marginally with emotionally valid for marginally better it's a marginally safe grounds for for busy, you know, but interestingly also you get to a systematic systematically Juba numbers of a few units.
09:27:19 So larger in the near marginal case, which is probably saying something like no no if you were winning verse to guess the city, and probably also emphasizing that spreading or, in the sense that nibbling is but puts it or avalanches front etc or probably
09:27:41 not relevant to the overall dynamics in this region.
09:27:48 Okay, so this is for q2, numbers, and now let's test whether or not the linear assumptions, hold. So, what you can predict is the face of the,
09:28:07 of the fluctuations between electric potential and, and pressure.
09:28:12 So, we complete the phase of those we compute the ratio or proxy of the ratio of the amplitude of pressure, over, over electric potential This is all predicted by in the context and you can also investigate.
09:28:31 Depending on the mood, number, which mode number contributes to the flux.
09:28:38 So these are busy curves but what that means is that the thick lines everywhere. They are the frogs driven computations.
09:28:47 The, the one with the circles, they are the question in your calculations, and the arrows or the crosses here, they are the joking at this jar separation.
09:29:02 So, within a factor of two, roughly, while the dinner, assumptions seem to hold qualitatively quite well.
09:29:17 And surprisingly olds who seem to hold better and near marginality. These are the magnitude ratios don't fall from threshold. This is something that I personally not understand the overall conclusion of that is that well, it's a it's a good conclusion
09:29:32 for with you smiling in the sense that the linear aspect in the nonlinear regime, or kind of relatively robust, provided you know not too picky on a factor to factor for overall.
09:29:48 If I recall the factor here district 20 is much much more than tool for.
09:29:55 So, there is something that is. So the issue that still remains is with the turbulence intensity spectrum, and the closure.
09:30:07 So, coming to the conclusion.
09:30:11 What we did is to compare compare three popular
09:30:17 choices for a model reductions donated clothes Riven with a scale separation and quizzes in there.
09:30:26 They're all conclusion is, was totally driven when you're far above leaner and meaner thresholds.
09:30:33 You have a reasonable agreement
09:30:37 between the old three approaches, things go south, when you're close to module, stability, meaning that you're above, okay the official that you're at or below the nominal.
09:30:49 And you can have their very large under conditions of the flux is the interesting thing is that the linear aspects of the fluctuations seem to hold relatively well.
09:31:03 So, The problem is with the closure on the spectrum of of the fluctuations.
09:31:23 So, in order to to move, and that and also looking at the numbers, it means but we already were suspecting that spreading and flow pattern to Qt because those two things are the first phenomena that are wiped away when a suite is killed separation.
09:31:34 And the way forward is to improve the closures for the, for the saturation rules for the cuisine you know models.
09:31:43 So, it means that you must account at least in your margin analogy for some amount of turbulence spreading launches and the secondary secondary patterns.
09:31:54 So, you have several possibilities, either you train your, your closure on sunscreen and simulations, that's going to be somewhat practical or if you want to be fully consistent and have a theory.
09:32:11 Well then you're probably headed towards coupling to another equation for the turbulence intensity or seriously considering what's not done in plasma as the higher order accumulate and expansions.
09:32:29 So that's, That's the end. Thank you very much.
09:32:37 All right, thank you VM, very interesting. So, questions, gentlemen and ladies.
09:32:52 I'll wait for others so.
09:32:54 Okay, go ahead. Greg, I can wait a minute or two, I. Thanks for your talk is very interesting, especially the head to head comparison.
09:33:07 I, I guess I you know I tend to be a literalist so you'll have to forgive me for that but I, you know, to my way of thinking that the qL reduction holds only in the limit of that temporal scales separation that you described.
09:33:19 And in that limit, then the fluctuation equations, I mean if the fluctuations can be unstable then you know we have problems because the fluctuations are just going to blow up with, while the, the slow mean is not reacting.
09:33:32 So I guess the case I've been advancing is not a new idea but it's, you know, the system has to tuned to marginal stability and that limit and, in some sense, and by using that criteria and you can figure out, you know, that's your closure in some sense
09:33:45 you can mathematic, I mean that's what I talked about, I guess. When the early part of this meeting, when I was discussing strongly stratified flows. Now, how well that agrees and performs compared to DNS when you're not strictly in that limit, you know,
09:34:00 that's another important question and I'm all for trying to extend qL, but I guess I don't understand.
09:34:08 I mean, in some sense, I would have. You're saying qL doesn't work in near marginal stability conditions and I would have thought it would be sort of exactly the reverse that that's what, when it's set up to work or it has to be that way almost but I
09:34:24 think the point number two here is an important thing.
09:34:29 I think the way you described, well, what you did in your talk is quite different from what's a popular approach, at least in plasma as people in plasma is the, the take a pragmatic approach and fit this thing on on many many runs that have computed this
09:34:54 thing.
09:34:55 So you inherit actually the shortcomings of this model, where you have actually performed a steel separation.
09:35:04 And I think that this is what's blowing blowing up here.
09:35:11 Because you have, you see that near marginality you have problems.
09:35:34 Comparing to to the DNS for the primitive equation for which you have performed a scale separation, and the same kind of flux is obtained for the questioner but as you would expect because it's actually trained on giving back, the results from the Jeunesse
09:35:41 with the skill separation through, through, you know, those those rules here on the, on the saturation. So I think it's really points towards
09:35:52 an incorrect approach to treating the closure here. Okay.
09:35:59 Okay, thank you. It is it is it. Yeah, I love to still think about it but I think that is the point that how that spectrum is, is, is modeled. Yeah.
09:36:14 Okay, so go ahead Edgar and then I'm next after Edgar.
09:36:20 Oh, yeah. You know, I'm sorry, this will sound like a naive question but it's because I don't work in this area.
09:36:28 But in general, you know, if you do bifurcation theory for Hamiltonian systems and that's your starting point, as I understand it, you always have to include both the growing mode, and the decay mode.
09:36:47 In order to correctly predict what happens, near the bifurcation and the reason is that, you know, the growing modes and the decay modes, you know decay will grow at the same same rate.
09:37:01 And so I'm concerned about the way that you do this problem by, you know, ignoring the the damped modes to me, it's close to the bifurcation.
09:37:12 That should have a bad effect because you may not you know predict the right direction of branching all those kinds of things if you ignore the attempt modes can you comment on that.
09:37:24 Well, maybe a bit, not too familiar with this cuisine approach to be frank, I'm not a big proponent of this we wanted to test it, and because it is personally I didn't think it was very well post.
09:37:44 But that's what would you say is exactly what you see here so you see for a certain number for above a certain number here in total number, everything is put to zero.
09:38:00 What I can say about that is, well, several things.
09:38:05 We've seen and we know that damp moods actually can be can see energy channeling through them and impacting again the Reynolds dresses. So would you say is it probably.
09:38:21 Well, it's very correctly, that can be an issue. I think this is cured in there for stability wise again by the choices they make for the closure.
09:38:31 Because you don't your energy according to a spectrum that is that is chosen.
09:38:39 What maybe saves them a bit is that, at least for those cases, the action.
09:38:46 The mode that really carry the flux.
09:38:50 They seem to be very concentrated where they actually keep all the resolution. So, where the make the cut off.
09:39:02 You see that these are moves, don't carry a lot of a lot of weight. So probably if they were, and that's probably another that they can choose at the beginning of the computation so if this cut off, they would put it closer to hear I think they would
09:39:18 be very wrong in their answer. So they're probably they probably need to be very careful about that additional know.
09:39:28 What about the conserved quantities.
09:39:30 I mean, don't you destroy the conservation if you drop the dump mode.
09:39:39 Yeah sure, but I don't know how digital to be To be honest, I don't know, in that specific was the dinner model how the how well they control things and I could Gerstein that I mean, for quasi linear theory, with unstable modes has been shown to conserve
09:39:57 energy between resonant particles and waves or particles and fields. Right, so, depends what you define is conserved quantities, I suppose, but there's not an issue there.
09:40:11 Okay.
09:40:13 Okay, thank you.
09:40:17 So, let me make, I Edgar added the comment, comment number one would be everybody's favorite damn to mode is the zonal slow, when you think about it, and.
09:40:30 And the reason why it's everybody's favorite it says we say the most minimal inertia. So this is a great argument for including the zonal flow effects and or spectral equation, right, so.
09:40:45 But getting back to some of your other things you mentioned about cumulative expansions and all. Well I politely remind you that they have been in Plasma Physics for an awfully long time I refer to resonance broadening theory and you know, starting from
09:41:01 do pre 66 and onward.
09:41:04 So it begs the question.
09:41:06 Did you look into something like a resonance broadening residents broaden the quasi linear theory for the large Qubo regime. In other words, that would preserve many much of the useful structure of quasi linear theory but his, his tongue very much intended
09:41:26 tailor made to address the, the, the, the large Qubo situation.
09:41:36 Well, the short answer is no.
09:41:38 I mean, you might you might do that right i mean that's exactly when you you would you would kick in, in general, though, you know, I'm curious if anyone has an idea we what we really do in this is we use things that really work for Qubo less than one.
09:41:56 We push it to Qubo equal one because that is as you know the mixing like right where things tend to sit and they mean and we add cure you know it seems to me would be interesting to look at Cobo above one and push down right and see if you came to the
09:42:14 same place, and I've never seen that exercise carried out in in any sort of convincing way.
09:42:23 You go above one gets into all kinds of very different physical pictures you know like percolation and so forth.
09:42:32 I mean, do you have any comments from looking at the data and all on that.
09:42:39 Well I think these are, these guys could probably address address it because they can couple.
09:42:47 Well, should the take exactly what you said before, some amount of resonance broadening and couple, the positive response to a flux driven evolution. They could probably do what you know something to do for the next student right and the.
09:43:09 It does seem to me your bottom line is you cause the linear theory plus a spectral evolution equation, including all these various goodies that we know and love might be a good way to go is that a correct reading know.
09:43:27 No. Well I don't know how that would work in practice, but at least it's it's clearly a we we forward, copying it to.
09:43:36 Well, I think the dynamics of determines intensity field that I think is is a very good way to to advance into to push past those current limitations.
09:43:48 All right so Zach VA has his hand up and then David, what is it's a it's a follow up on your on your comments but I'm resonance brother and we didn't do that for that case but at least in the past.
09:44:14 We checked the scaling of the diffusion equation or the flux, let's say, with with the dominant intensity. And it was the flux was proportional to fight club.
09:44:18 So, if resonance broadening would be important, and I'm a correct to say that you would find a different scaling from that you'd also that you want to look at that in the large Qubo regime.
09:44:30 Now you're you're looking in the small Qubo there's not going to be any problem, right, but in all the cases we actually did of course, we were pushing on the temperature gradient so pushing the pushing on the intensity, but but the point is exactly that
09:44:44 the pupil number, at least in these cases were hardly above one. The system was taught was kind of self organizing your own the state to the rescue workers from.
09:44:55 Well that makes some sense to me yeah but I mean, I mean I was surprised at how much she got above one and some of the cases, in the case of dm there right you know yeah I see your point is just to make the point that it was very tough to get away from
09:45:11 meeting that in fact exactly what you were saying that. That's why they call it the mixing length limit, isn't it, right,
09:45:20 because it because it worked, was also comparing to well, you know better than you gyro and, and all. It was an early work I'm not sure it was, well, it probably was not at all in your marginal regime within your marginal different oh that was.
09:45:41 That's exactly I think that's one that resonance wouldn't to play well for cable louder than one so so what above marginality that is far from not.
09:45:55 All right. Professor Hughes.
09:46:01 Okay, this is not a very well formulated question, damn it just reminded me, listening to pat i mean if I think we're much easier problem. Namely, you know, me and feel dynamos and things.
09:46:15 Then there's only two places where it where it works, and neither of them are useful. So one is small Reynolds number one is vanishingly small correlation time.
09:46:25 And everyone tries to take it somewhere else and it doesn't work, I mean do you listening just what Pat saying you've got regimes where it works you'd like it to work somewhere else.
09:46:34 Do you have reasons for thinking that it that it might I mean in the Dynamo game it's just it's just not working. It's just a waste of time trying to take anything outside of these regimes but is there something different in this I mean they might well
09:47:00 because I don't really understand all your equation. Oh, we understand your, your question may be answered alluded to here.
09:47:02 That's, that's with a question mark here and it is debated.
09:47:08 I would tend to think that your basic operating regime, especially for large volume.
09:47:27 In the sense, defined here.
09:47:29 I'm not sure everyone would agree.
09:47:32 And this has a, an additional caveat that it is done with a simplified response for the election so it's Boltzmann responsible in the elections. And when you put you in a kinetic responsible for the electrons and things are still debated whether the results
09:47:56 those behave in the same way the staircases pattern in the same way, everything that goes on your Muslim stability. It's not yet clear, what, what, you know, what's the real estate there.
09:48:13 So, a lot of people have commented that this new marginal case here is maybe not so operationally relevant. Well,
09:48:26 You know, since we don't have a clear answer now everyone that I guess is entitled to to his own opinion. But I would tend to think that addressing what occurs in your emotional state, as described here, at least with benefit a lot.
09:48:49 Those kinds of models, and the fact that they are not working. It's such a simple framework with a diabetic collections and stuff like that.
09:49:06 It probably means that you need to cure to boot to be put in them. So the way I understood your question is if you actually want those simplified models to actually work in this kind of regime.
09:49:13 That would be my
09:49:16 answer is that what you were asking David. Yeah, I mean I just wondered if you had, if you, if there was something in this game that was that led you to have more hope, and then what seems to be a dismal failure of this and this, this actually is quite
09:49:33 quite a bit of hope because even, you see that that's a puzzling thing. The blue is in your module.
09:49:41 And you see that comparing the circles and the fat ones. The thick lines. That's the DNS and fiction the circle is the linear.
09:49:52 You see that for some reason, here, new marginality the linear aspect of things seem to be quite robust. So if you can cure, find a better way to close your system.
09:50:18 Describe the spectrum of the fluctuations and and everything and probably needed needing an additional equation determines intensity, it will, it will be, it could probably be very interesting step forward as to reliability.
09:50:36 And maybe predictability. Okay.
09:50:29 Let me, let me ask one quick question I mean we have some, Some purists here who I'm surprised having erupted.
09:50:42 But, you know, going back to the, the original work on quasi linear theory, the heuristics sense you know as I say that the key energetics proof is resonant particle kinetic energy density in one dimension of course you know the velocity of versus wave
09:51:03 energy density.
09:51:05 So it begs the question, you know the transport here who's carrying it is it resonant particles or some, is it mainly non resident.
09:51:16 I mean, that to me is is an interest because there's a quit. If you ask what is the era of the origin of the reversibility that allows the game. The the strict answer is chaos Hamiltonian chaos.
09:51:31 So then the question is you all I've all, you know, going back to the beginnings of quasi linear theory particularly near marginal by the way. In other words, low growth rate resonant diffusion is more robust, the non resident diffusion right because
09:51:50 non resident you remember your basic course right like the one I teach non resident diffusion scales with gamma.
09:51:58 Right. So then, so what's going on here is these more resonant particles are non resident particles.
09:52:06 Oh, I'm not sure I knew the precise answer to that. But given that the linear relationship holds and it's essentially a resonant
09:52:22 fraction that that's assumed and it's all media, keep it all the parallel resonating i would i would
09:52:36 i would tend to think it's well the resonance is actually well captured. But to be fair I do not know the answer to your question.
09:52:57 Particles opens the door to all of this stuff like granular Asians and all of that, which is partly that's another issue to think about.
09:52:59 Okay, one more question to Paul on. Okay. Hello. Hello.
09:53:18 Hello.
09:53:08 We can hear you. Yeah, go ahead.
09:53:10 Okay, okay. My question is a general question that can be.
09:53:18 Can you connect the relation between down low and high Google number that it basically led to a unified statement is it possible.
09:53:31 Maybe you have to try that again and not sure you understood.
09:53:39 Listen, I'm sorry. Actually, I am not working.
09:53:44 I'm working on the field or contradictory but you are outstanding. You are working on this area, but I think that is a very general question can we can connect the relation between the low and high goober number is possible.
09:54:03 What would you mean they can connect the relation between low and unified 34 low and high, Google number.
09:54:14 Well, as you know, as it has been said Cuba, you can see it in many ways.
09:54:23 I like to see a competition between non generic teams to kiss the city so when nonlinear early wins you're in the new emotional case to guess the city or diffusion
09:54:39 of your distribution function.
09:54:42 You get closer to one and usually one is the consequence of the other that's when you hover in the vicinity of equals one.
09:54:50 That makes a connection between regimes. I'm not sure what that is what you're asking.
09:54:58 I'm not sure what that is what you're asking.
09:54:59 Okay. If I don't know, I guess, I will send a message to you. Okay.
09:55:04 All right, well as trouble.
09:55:07 Someone. Understood.
09:55:17 I think we better get moving here I know I just looked at the clock I want to make sure there's enough time for Peter, so thank you. So the next victim is me and dm please.
09:55:21 You'll enjoy keeping an eye on the time and handling the discussion, I'm sure.
09:55:27 So, let's see.
09:55:33 I have the right slide. Yeah, Here we go.
09:55:38 So this is a little, you know, what's the word, and a tude guess they'd say in music, right on transport and layering for marginally overlapping cells.
09:55:54 And this is motivated by a question that Boris Raymond asked at the end of the blackboard lunch, which most of you weren't there for. So you didn't hear it.
09:56:05 And in the true spirit of KITP, the director cut off the discussion before we could really get into it, but it was a very good question, And I think it's worth worth a mention and I, well I discovered not me discovered some old goodies I should say Boris
09:56:23 Boris has a relevant paper you can see why he asked from 1987.
09:56:28 So he asked, and I don't remember the exact wording you know David gave an excellent discussion of all the different layering mechanisms and he kind of, you know, Boris is a smart guy right and he he put up, put up rate commented, well isn't there something
09:56:46 else you didn't mention And what about something related to cell structure and patterning or something like that.
09:56:54 So the first thing you think of well in a single cell you have PV homogenization you know on the Reynolds to the one third and later a Reynolds number timescale, and that'll give you a homogenized cell with a sharpie the gradient at the boundary.
09:57:14 So if you like Rossby wave elasticity which is certainly one of the rallying cries of this workshop, you know you have a sharp PV gradient is good for barrier formation.
09:57:28 And that's something that's been emphasized by David dread show and Grecia and a bunch of other people and and this place to the Hamas, the connection of homogenization and barriers, but to that you say well come on it's about layering and that's a single
09:57:48 cell we're not Can you know tell us about layering.
09:57:51 So that begs the question of a cell pattern. And here's a little picture from the user Tango review by the way a useful document for exploring different read what's known circa the time of the different regimes of Qubo number and this is a classic problem
09:58:15 i don't i. The earliest references Moffat but somebody told me it went back to gi Kaler so by the way any of the history experts please enlighten us, but they're at you know you could imagine a cell pattern, and you're going to have boundary conditions
09:58:33 across the box and you each cell has a characteristic V and an L zero the usual crap.
09:58:40 And there are some molecular diffusion D zero and you might ask, you know, what's the what's the average D of the system.
09:58:49 And the point is here you have a picture of marginally overlapping cells. Now I'm biased, of course, but I think the best paper on this is by Rosenbluth head out and I, you could skip the ad owl I can tell you as an eye witness.
09:59:12 just an incredible analysis of this problem. And there's an interesting story behind that paper that's probably best left not for a public discussion, but the physics is rather clear as you know again my favorite question of where's the irreversibility
09:59:28 in the system the reversibility is localized to the boundary layers between the cells right that's where that's where all the action is. So the global transport, you know again you're thinking of some average D is some kind of hybrid of, you know, fast
09:59:47 kicks through the cell and slow diffusion through the boundary layer right that's kind of the physical ID and you notice you've got to transport rates, and at this point in the program, you kind of guess what's going to be coming when you have to transport
10:00:05 rates.
10:00:06 So there's that you know again if you want to see the thing in all its glory read Marshall's paper, but I'll give you the quickie the back of an envelope in the back of an envelope is you say be effective diffusion is the active fraction times, you know,
10:00:23 the usual means square step size divided by the time step. Okay.
10:00:31 And the active fraction and destroying stinks, but you know, I'm not up there with em on these graphic things but the the active fraction is the relative size of the boundary layer relative to the size of the cell.
10:00:46 So the active fraction is just Delta the boundary layer size over the cell size, the time step is just the passage through the cell l zero or V zero the cell circulation time.
10:01:00 So delta squared is d zero delta t, which is just D zero l zero over V zero I mean this is you can make this is a simple exercise. So then putting this all into the effective D and doing you can follow the arithmetic, you get for the effective diffuse
10:01:21 activity of the system the geometric mean of the collision od few 70 and what might be called the cell the few civet e l zero v zero if you want it's not turbulent but if you want to think of that as turbulent you can.
10:01:36 And you can rewrite this as the collision of the few 70 times a enhancement factor that goes as a square root of Peck lane number, and with a good deal more work Rosenbluth exact you know he got the new, of course, got the numerical factor right and all
10:01:56 of that, but I mean that's basically the physics of the story and it tells you what you already knew I hope that this is not a game of simple as simple addition right but it's, you know, it says there's clearly a transition region there.
10:02:11 So,
10:02:14 you know, the interesting thing is that this volume average D.
10:02:21 Right, as I said, is actually results from slow transport to the layers and fast mixing in the sales.
10:02:29 So you could imagine a concentration of injected di and asked what kind of profile, do you get.
10:02:37 And here we have the picture and this is from the original from Rosenbluth paper in 1987 and my oh my what does that look like. Right.
10:02:47 And in the, in the figure caption for the shell he writes for the picture.
10:02:55 Steve transitions in the density exists between each cell.
10:03:00 And you know there's no impressionistic color view graph or anything else you're lucky to get any figure in 1987, I suppose, but I think that's as good as an example as layering is you can get.
10:03:13 And you know, I think. Here we have the Rosenbluth staircase. Right.
10:03:23 And the conclusions by the way are supported by detailed analysis and Rosenbluth and try min and sociological interest rhyme and actually acknowledged Rosenbluth for correcting you never in his manuscript.
10:03:37 So so interesting and or an early example of layering with no moving parts right no feedback loops Reynold stress etc etc. Right, very very simple physics.
10:03:49 So this seems to me to be quite relevant to this discussion which is eternal of what's going on near marginal stability, and you could interpret the D zero the collusion or D here, either as the neoclassical say if you were in plasma physics, or as some
10:04:11 kind of ambient small scale thing so it'd be interesting to look at, you know, some sort of cell averaged effective diffuse activity and what it depends on.
10:04:22 Now the chorus here may be thinking, all this is contrived it's not self organized you set it up, and all that and now you know that's cheating. Right, so bring out the red card or the yellow card or something.
10:04:39 Well this isn't so clear to me right I mean, then this gets too savvy as comment a little bit.
10:04:48 What we've actually got in this business is pin sales at KB will zero surfaces right and I've noticed the, the way of way of putting it into, and those cells are typically small which is just again bringing out our beloved small row star ordering.
10:05:07 So if you have a system that in some sense stiff and other words if the transport rises precipitously as you go above marginal but you have some Dr.
10:05:22 Aren't you going to be sitting right in a state of barely overlapping cells or islands, right, which seems to me to be a natural candidate for this kind of idea and I says a passive scalar analysis and of course has to be revisited but nevertheless it's
10:05:40 provocative. So, it suggests that you know, again, this sounds a bit like savvy as words but you know I wrote, I wrote the view graphs last night. Right.
10:05:51 It's a naturally self organized state, right sitting, you know that supporting a staircase and it's probably going to be sitting, you know in a naive sense to Qubo number order unity.
10:06:05 Right. I mean, in the sense of if you looked at just the cell pattern.
10:06:11 It seems that this comes about, as a consequence of drive and profile stiffness and little else.
10:06:19 And so, why not. And I thought it was an interesting point from Boris.
10:06:27 And the other thing of course given the nature of the queue profile that says that some kind of irregular staircases may be quite likely, and we all, you know, the other day the UCSD plasma seminar was given by Max Austin, who was talking about my current
10:06:46 days the nice negative triangular charity work but we all remember him from the it bees and the zonal flows correlated with lo que resonances in the 3d and others I know jet did it too and we have a European contingent here but it again it begs it begs
10:07:07 then does this fit in somehow relate to that story and could that be just another could that emerged as a large step.
10:07:16 So I think that's probably know enough to try provoke people here but I did find this pic, and I remembered the story that I had forgotten this picture, and the picture from 1987 is really quite remarkable.
10:07:36 I think it says, as good an example of layering as anything else we've had in this program so at that point I'll stop. Thank you.
10:07:47 Thank you very much to like the German football. Yeah.
10:07:55 Well David question is.
10:08:00 Yeah, it looks like you're doing flux explosion.
10:08:10 But I mean it's flux, but not for a single Eddie.
10:08:16 Right. In other words, we usually do the flux expulsion for a single Eddie right at least the way you know here it's a you're interested in, in some kind of volume average transport through an array of cell of cells that are expelling.
10:08:33 Yeah, but I'm trying, but it. Yeah, okay, but so so then the timescale would be very dependent on the properties of for flow right because, oh yeah, no, yes, you know, if they're turbulent, then it wouldn't be a geometric mean of those two it wouldn't.
10:08:50 Yeah, okay.
10:08:54 But yeah, it looks like.
10:08:56 I'm not sure if Nigel didn't do a few cells next to each other back in 1966 I'm not sure, but anyway.
10:09:02 Okay, I was just a comment. No, I would be it would be interesting to know I somebody told me gi Taylor started this industry, okay but I can't find the reference.
10:09:14 So,
10:09:17 during your next. Um, yeah so this is just really a comment, so some 30 years ago, a bill Merrifield and I looked at some problems like this, except that the cells work time dependent.
10:09:32 And we looked at the case of standing waves which also have nodes that you have to pass through only by diffusion and compare the results with what happens if you have for example a traveling wave which doesn't have nodes.
10:09:47 And so there are no barriers of the type that you you were talking about.
10:09:52 One of the interesting things that we found was that there is in fact a resonance.
10:10:05 Were corresponding to the situation where the lifetime within Eddie. In other words, the frequency, one over the frequency of the attic is comparable to the turnover time of the ad.
10:10:13 I suppose corresponds to something like Google number equals one. Yeah, exactly.
10:10:17 So, anyway, so that's just a comment. So some things like that haven't been looked at, I'm sure, I mean, Okay, useful to know thank you.
10:10:28 So you say it's bill Murphy and you Merrifield Merrifield. Okay, and I can send you the paper, please do always happy to see it.
10:10:42 Your next invoice, or it's a it's a comment. So, if he might be a well known story that, but maybe maybe he has in mind, to which was this observation by his Dutch people on produce talk about which, which was shut down then, ultimately, that they had
10:11:03 very highly resolved direct content actual measurements using Thompson scattering, and the so what we would call now staircase and Jacqueline temperature that is original very shops deepening of the temperature and regions where it was very fluid.
10:11:17 And at the time they noticed that the position of the flattening of the temperature correlated with national values of the safety number which for sorry for the jargon guys but that would be the pristine number of the felines.
10:11:35 And then the petition was was basically this one that is islands. That could be localized and the National surface is where we would get these flattening and and steeping would correspond to exactly what you say that these interfaces in between the islands
10:11:54 that is this IP addresses. And then the story went away I don't know, they were they were doing these measurements in the very highly special conditions which there's competing on electrons at the time.
10:12:03 And then nobody raised this again but but it's really for me it's very reminiscent of what you, what you're saying, Well I had, I indeed that crossed my mind as did the choppy profiles and TFTR, which was another variant of that kind of story and, it,
10:12:25 it, you're, you know you're preaching to the choir, as it were, what's interesting is there's a there's a group of, as you know, and there's an industry in our field out studying you know Island transport and all but never really addressed this marginal
10:12:42 overlap state.
10:12:46 in a substantive way.
10:12:47 And there you might have say you the setup might be some, the island with a fast motion of electrons in the island and a little residual micro turbulence in between to provide the kick.
10:13:04 So the geometric mean might be something related to the island with something related to a drift wave, but I've met you know for never seen that laid out in a systematic way.
10:13:20 The other thing on that.
10:13:22 Yeah. The other thing I would say on this is if, if you use the D zero is neoclassical and the D is the turbulent you know you're going to get current scaling out of that quite easily right.
10:13:37 Did you know because you will have a row Fida squared, you get a square root, which is always been for me, a kind of a people regard this as a solved problem and I, you know, but there's always some vague words of appealing to neoclassical polarization
10:13:54 in the gam or those old flow.
10:13:57 But that's I think it's far from home free on that issue.
10:14:07 Okay. Thanks. Boris with your your next.
10:14:10 All right, ready, very interesting that may ask you some questions still don't know zones are just scared kids miss you show this profile is it monotonic or not.
10:14:27 What do you mean by and trying to monotonic.
10:14:35 What do you mean by monotonic, what you see is what you get right the stuff goes in, on the right end and is clamped that the left end, it doesn't go down and up.
10:14:46 If that's what you're asking. So it's not like profile sales table certification density in that article which is not monotonic no no no no it's a toy problem.
10:14:59 So there was no.
10:15:01 So there is no there is no sense to monetize it, do not get anything from. So, by our criteria isn't doesn't doesn't have much though.
10:15:14 it's a toy problem that I think is extremely instructive right toy problems are where you understand things, and then you go after bigger things with them.
10:15:26 All right.
10:15:26 Thanks.
10:15:30 Question. Misha, please. Then I'll go Yeah, very quick question I unfortunately I, shame on me but I haven't seen this Marshall's paper but the question is that it all looks like.
10:15:43 Karma world, what is called karma world for your life to the urban areas, W periodic driver and then you get a bunch of sales but the question that I have is whether you have any sort on on what is the what are all of the average wer TCT important because
10:16:07 might get one self driving rotating in one direction and other in opposite direction and then you will, of course, create some sort of sheer flow between them or smooth transition fabulous thought about this right i mean i thought about it that a given
10:16:39 I was trying to just piece together the basic story I haven't done any analysis and that's obviously I mean this is a this is a somewhat contrived model and the minute you. By the way, it's not as trivial as you say, in other words, you can have a boundary
10:16:43 can, you know, different boundary conditions, but it's doesn't have to be periodic.
10:16:51 But I mean, apart from that, it's something one one could do and then of course if you away, depending on the orientation of the flows, you're going to get here flows, which will, but I think, one could continue the analysis that was done they're allowing
10:17:09 for some minimal sheer flow effects right because that that is effectively going to modify the way I was going to modify the active volume fraction and it's going to modify the timescales for the active volume, but beyond that I haven't thought about
10:17:30 it.
10:17:31 Okay. Makes sense.
10:17:33 Well I had one question or comment I don't know to me what you you show here reminds me a lot of what we heard yesterday from the parallel.
10:17:46 I don't know if you'd agree on that because this.
10:17:50 I think it's interesting little thing is intrinsically scale when you would have from the individual motions who just, you get this big genetic cell of kinetic energy.
10:18:02 And you could probably get several of those in depth could could ask her what's producing the layers between them, but I mean one thing I was, I was, I shudder to say this because we bet.
10:18:20 But I was left I still don't understand what the what the emergent scale is in the DDC right and there was some mumbling about internal waves, but I didn't, you know, I didn't get it if someone knows Please tell me, so there's a good.
10:18:38 This is a natural to get an emergent scale right the boundary layer with that I think that's your point right. Yeah. And it also gives a natural way to understand the to the facilities.
10:18:52 Yes. Yeah. Yeah.
10:18:55 All right, we, we better shut up and let Peter right.
10:19:02 So you chair again.
10:19:04 Yeah chair again so Peter, your Europe.
10:19:09 Hey, thanks.
10:19:10 Let me know I entertainment.
10:19:23 So thanks for to Yannick for the invitation to give this talk, I mean I have to say it was a little unclear exactly what what you were looking for but suggested that I talk a little about experiments lab experiments in a sort of a slightly broader context
10:19:42 in the context of PD staircases which we've heard quite a lot about, of course, During the rest of the, of the session.
10:19:51 So what I'm going to do is, is give a kind of a sort of a little bit of a random walk in this sort of area with sharing a few sort of examples and raising a few ideas which I hope in one or two places might resonate with with plasma physicists I mean
10:20:10 I'm I don't, I certainly don't count myself as a plasma physicist, certainly not in this company.
10:20:16 And so whether this hits the spot, as far as you're concerned or is completely irrelevant I guess it's it's for you to judge.
10:20:25 So what I'm going to start off with I guess you know why do we do lab experiments.
10:20:33 The mean there are all sorts of reasons but here's a few that I think, sort of, hopefully resonate in various, various contexts. I mean, part of it is discovery because when you work with real fluids they don't always obey the rules that you that you
10:20:46 that you expect.
10:20:48 And so one of the, one of the joys of doing experiments is that you do sometimes discover things that you weren't expecting.
10:20:54 And it's very good way of investigating these, these processes and keeping your feet on the ground.
10:21:10 Another strength of experiments is that is that you can you can do lots of experiments usually and cover a lot of a lot of ground in parameter space. So you can actually test ideas or or discover idea relationships I guess between various properties of
10:21:20 flows and the various regimes and in a parameter sweep.
10:21:25 And what you're looking at, of course, is, is very strongly nonlinear effects I mean we're not explicitly solving non linear equations, but we are certainly implicit resolving them.
10:21:34 When you run a complex experiment. So phenomena light wave breaking and turbulent cascades and verifications and so on are things that we can, we can certainly explore.
10:21:47 And I put quickly well down the list the notion that you actually do experiments to verify models and data analysis methods which is certainly part of the game, but is by no means the only thing that we do experiments for.
10:22:03 So,
10:22:06 just to kick things off.
10:22:09 I mean we've heard a lot about sort of making staircases in various ways. What are the basic and getting ingredients that we need to satisfy in order to create PV staircases and zonal jets and such like.
10:22:22 So, I've kind of listed them here in no particular order, but first thing that you you seem to need is rapid rotation which has the effect of through the Taylor problem and effect giving the flow or certain stiffness along the rotation axis.
10:22:38 So the fluid then behaves as though it's a fluid, with less than three dimensions to explore.
10:22:47 And this has the net effect of having
10:22:53 a potential validity that is a materially conserved quantity by the, the sort of the two dimensional component to the flow.
10:23:17 And we usually look look at forms of potential gorgeously like the so called shallow water version which is where you basically have the absolute volatility divided by the the axial axial links, ah, and this also has the, because that's conserved, in
10:23:20 addition to energy in in busy situations, then you that leads to the there's an upscale kinetic energy transfers that are associated with, with all sorts of all sorts of interesting phenomena.
10:23:35 And for PV staircases we also find that you need to be able to force the flow. I mean, we're talking, essentially about force disability flows in any experiment really, in, in external in.
10:23:52 So forcing has to be on a small scale and on a much smaller scale. Typically, then the the sort of scale of what you might expect staircases to take place.
10:24:03 And so, exactly how we do that is, is, is many and varied within experiments. There are all sorts of different ways of doing it.
10:24:30 But the kind of interesting experiences, is that it doesn't seem to matter too much exactly what process is doing the staring you just need something to keep keep the fluid moving on these very small on these very small scales.
10:24:27 On the right, I've just indicated, of course that I mean we we come across PV staircases initially in the context of planets and planets tend to be a spherical shape, but experiments doing experiments, you know terrestrial lab in in spherical geometry
10:24:43 tends not to work terribly well because gravity has this rather annoying habit of always acting in one direction, naming the local vertical. So for that reason.
10:24:55 In the main what we tend to do is to not do most of experiments in cylindrical geometry, with the, with the, the axis of symmetry upright.
10:25:04 And that is, as near as we can get to being analogous to the sort of the planetary situation.
10:25:10 Another feature that we need is some sort of breaking of the symmetry we need some sort of an isotopic external planetary vortices the gradient now on a planet that's associated with the spherical curvature of the planet.
10:25:24 But in experiments again because we have to run in cylinders, then we can apparently get away with it.
10:25:31 With it with a similar sort of end result by using sloping topography so that we, we have the, the depth of the system. And the axial direction, bearing with with radius.
10:25:44 And that gives us an imposed component of the potential border city that has a systematic variation with radius.
10:25:52 And that is equivalent in many respects to to the to the spherical curve of curvature on, on a rotating planet.
10:25:59 And finally, you got you got to have the friction, as, as weak as you can make it in order to get nice well developed
10:26:08 staircases and zonal jets. And this means friction the master be weak on the on the small forcing scales so that we don't inhibit the upscale kinetic energy transfers.
10:26:21 So, the various link scales that we typically think about so there's a domain size of forcing scale.
10:26:29 And the scales that we've seen in other parts of this program the Rhine scale.
10:26:48 and I Satrapi scale which I think Boris Galperin was also mentioned in his talk, which is in some respects akin to the middle of scaling in stratified flows.
10:27:01 And it sort of represents the largest scale of isotopic mixing or stirring.
10:27:06 And then you can even define a friction scale.
10:27:09 Just by equating the idea of returning sort of target overturning scale to the rate of loss of friction.
10:27:19 And in, in, in experiments in rotating fluids experiments. The main source of friction is bottom friction associated with Ekman layers and so we can define a a an admin friction scale.
10:27:39 And so, for getting multiple Jackson staircases what we need is to have the domain size larger than the net any of the any of these other schools.
10:27:49 And we also need the, the, forcing scaled as I said to be smaller than than the, the lb Torah and all scales.
10:27:55 And then what we tend to find is that jets and the staircases have steps separated by something that's comparable to the rhyme scale in practice.
10:28:06 And for distinct and robust cases we also need to have a scale separation between this rines, and I Satrapi scales as well that the ratio of those scales is often referred to as the zone apostrophe index.
10:28:19 And for gas giant planets, we need to, if you want to emulate what what happens on Jupiter or Saturn, then you need to be aiming for skills that are, you know, so most of the indices which are really quite large.
10:28:32 Of all the five or or longer, or larger, whereas the in the Earth's oceans. We also see some sort of latent parallel jets and so on in the open ocean, but they're the, the ratio of some sort.
10:28:48 So most of the index seems to be rather much rather small maybe, maybe a little bit less than two.
10:28:54 So, how do we access the Zola's traffic regime and then of arbitrary so the, if we look at how the how this is the last trophy parameter is defined. It turns out that we actually have to work really, really hard, because when you actually look at the
10:29:10 parametric dependence of this arbiter parameter.
10:29:13 You see factors like beater to the one 10th and such like and the, the pi epsilon which is that which is the upscale kinetic energy transfer rate to the minus one fifth.
10:29:25 So everything depends rather weekly on some of these key parameters.
10:29:30 If we estimate the, the, the upscale Eddie, Eddie energy transfer rate by the dissipation rate, which you can define just by the kinetic energy divided by this this segment friction time scale.
10:29:44 Then you, then you, then you can also parameter eyes are beater again by these other parameters, but all again raised to this one 10th power. So this all makes it very hard to increase this this, I'll be to two values that are large enough to be a planetary
10:30:00 interest.
10:30:02 So what happens if you if you just do the naive thing and just run small experiments and this is how I started out.
10:30:12 Many years ago, playing this kind of game using conical boundary boundary typography.
10:30:19 In small cylindrical experiments so tanks with a typical radius about 10 centimeters. Also, a typical depth, very similar.
10:30:29 This was a rather nice experiment where, where we were rotating, a cylindrical tank with the and heating it directly by eating so we used a week electrolyte as the working fluid past electric current between the outer outer boundary, and a thin wire going
10:30:50 down the center of the tank. So that tends to heat the center of the tank and then we cool, the outer boundary.
10:30:56 And so you generate a essentially a very clinically unstable flow.
10:31:01 And as you increase the rotation rate you increase the the the visa term and eventually you get into a region where the, the Rhine scale is much less or significantly less than the radius of the tank.
10:31:14 And lo and behold what you then start to see is what's essentially visualized here on the right, you start to see wave likes structures, these are essentially recipe waves, and they start to form parallel trains in radius and you start to see this as
10:31:29 an example, with two separate parallel trains, but it's a little bit fanciful to sort of describe this as a staircase of this sort of stage because what you get is multiple, we get but the role the diffuse PV steps.
10:31:44 This particular example is kind of interesting because you notice it's a wave number six pack them on the outside of the tank. And this is Barrett clinic instability in the flesh, and it's kind of interesting that there is this other feature that we see
10:31:59 on a much much larger scale in Saturn circles.
10:32:03 North Pole the hexagon feature that was discovered in the Voyager encounters in the 19 in the 19 1980s that appears to be in some respects dynamically similar services, so that you see this one particular branch of this, the staircase on Saturday that
10:32:28 this very symmetric way number six structure. And one way of increasing the possible zone as traffic index that we can get access to is to run the experiment on a much much larger scale, and I've been privileged to be able to run some experiments on the
10:32:42 Coriolis platform in Grenoble in southern France, which is a huge tank it's it's a tank with a radius of six and a half meters. so you can actually write on it.
10:32:55 As you can see, my, my holiday snap the on the left hand side there.
10:32:59 And so this is a large tank but you can only rotate it relatively slowly, and at the time we could only rotate it with a short period of about 40 seconds or so.
10:33:09 So moderate depth, allows you to get to the numbers of perhaps the smallest for 10 to the minus five which is small but not that small, and then we were forcing it with natural convection in various ways, which form plumes on on scales were banned about
10:33:26 10 centimeters so we sort of satisfying the small scale, forcing requirement.
10:33:31 And this would lead to formation of multiple meandering jets that we could, we could visualize with, with all traces and the scale of the Jets the separation we could actually then demonstrate by by changing the rotation rate.
10:33:49 We could change the effective.
10:33:51 Ryan scale you can see the Rhine scale shown by the size of these areas here.
10:33:55 And as you as you increase the rotation speed or reduce the D rotation period, the Rhine scale would get smaller and indeed the Jets would would get closer and closer together.
10:34:06 And if you actually plot the scaling then you find the effective radio wave number of the jet. In fact scales quite nicely with the, with the Rhine scale and this is this is this is quite, quite commonly seen in many of these, these sorts of experiments.
10:34:23 The Jets appear that I mean it's quite easy to demonstrate that they the Jets appear entirely mainly through through Eddie stresses. So we can actually look at the direct correlation between rental stress divergence, and we can correlate that with the
10:34:39 actual measured rate of change with time of the of the of the zone or flow with it with a fairly decent correlation coefficient and we can even determine buys doing spectral analysis we could actually do it measure the spectral flux and show that there
10:34:56 is this inverse kinetic energy transfer for scales which is smaller than the scales of the of the of the connection that was actually driving with those you can see here but in this this this is a function of a wavelength.
10:35:09 So a negative.
10:35:11 Flux here essentially corresponds to an upscale transparent a positive flux would you see at the very smallest scale shows shows the forward transfer.
10:35:21 I miss In fact, you can then see leads leads to a PV staircase so this shows a time series of the only averaged PV within the very narrow window that we could actually actually measure with with political image dinner symmetry.
10:35:38 And this shows you can you can see there are several regions where the gradient is enhanced the radio gradient is enhanced and as a function of time. You can see that the the staircase, sort of, meandered around, but every every so often you get these
10:35:55 episodes where the, the gradient appears to reverse there's a kind of
10:36:03 excursion of the of the of the staircase. And it kind of sort of wraps around itself and this is essentially the signature of this so this one is Rossby wave breaking kind of phenomenon that temporarily reverses the radio gradient wallet wallet wallet
10:36:17 I'm mixing event essentially takes place before the before the flow then is then restored back to what it had before. If we look at instantaneous snapshots then we can actually see the PV gradient which is what's shown here.
10:36:33 produces this hyper staircase. This is what what Boris was alluding to, before in the sense that you have a PV profile that is locally and, Well, you might think temporarily essentially a hyper staircase with this non monotonic variation of PV with with
10:36:52 radius.
10:36:53 If you take a time average then it's kind of smooth things out but it but in fact if you have to take very long time averages before these, these profiles become essentially non monotonic.
10:37:08 And this, this raises the interesting question that non monotonic PV staircases are not necessarily unstable.
10:37:16 So, this respected it's a little different to the, to the density
10:37:24 stratified flow situation that week reversals of PV PV gradient can actually be consistent with a marginally stable flow so there's my link with the theme for the rest of the meeting.
10:37:39 So, the record for the zonis traffic index in experiments. I think he's currently held by definitely miscarry and her and her colleagues.
10:37:52 If you must say so definitely gave a talk earlier in the program.
10:37:56 So there they use a fairly a fairly large Tango one meter tank, which they then rotate very very fast.
10:38:04 To the extent and they have a free upper surface which is which provides the, the essential topography that gives rise to the to the necessary beater effect.
10:38:14 They have achieved a value of this this index, which is approaching three.
10:38:20 And now the flow is starting to look a lot more Jupiter like so that you're starting to see the formation of equilibrated alternating jets that actually don't meander that much, that they actually stay fairly, fairly, fairly straight and anchored in in
10:38:37 position.
10:38:38 But in long time averages, you still have to look a little bit quite, quite hard to be able to see very, very clear staircases I think this is a fairly long time average that they that they're actually showing here with with any rather weak staircases.
10:38:54 Well, to finish I just wanted to to mention one other experiment that I've been involved with rather recently again on a fairly large scale, using the term lab facility in Turin in Germany and Italy.
10:39:09 So this is a rather smaller experience it's a mere five meter diameter tank.
10:39:15 That can be that can be rotated rather than rather more quickly. And we did some some some experiments that use a very simple, very localized forcing just simply a staring back and forth along one radius in order to to generate a, essentially, a plume
10:39:30 of Rossby waves that would that would feed off feed off of that forcing.
10:39:36 And this actually leads to.
10:39:40 We were able to do some some extremely high quality particle image fellows symmetry to actually visualize the flow and particular the vortices the dynamics of the flow.
10:39:50 So, what we're showing here is the potential multi-city this this to omega plus kill you, over, over h.
10:40:06 With the sloping boundary at a relatively low rotation rate so that there's only, only one only a handful of jets.
10:40:22 But what we can see now is we can resolve the fine structure of the PV as it evolves, and you can see these raspy wave breaking events in beautiful exquisite detail. And this is really showing the this this very strong Philip, Philip mentation effect
10:40:28 produced during these breaking events, and actually leads, then to these these enhanced sort of bug bundles of filaments that then get successively stretched and folded essentially reinforcing this this this boundary I guess which is, is where the there's
10:40:49 Rossby way of elasticity is kind of is kind of enhanced. So here you can actually see this, this, the detail mechanism I guess as to how this, the boundaries of these, these staircases are actually actually being reinforced as a result of this highly
10:41:09 nonlinear sort of multiple filament ation effect.
10:41:15 So, you know, I should just pay tribute to Rome and young man young colleague who did the analysis on this and actually managed to pull together some pretty amazing velocity fields that were in the process of analyzing right now.
10:41:30 So just to draw things to a conclusion.
10:41:32 Just wanted to sort of make the point that these PV staircases in general are relatively straightforward to demonstrate in the lab, but actually getting really, really nice clean persistent sharp robust staircases needs us to get into this so called zones
10:41:49 traffic regime this this fit relatively high zone was traffic index case which is turns out he's actually really quite hard to achieve.
10:41:57 But nonetheless, you can see that, see all the all the sort of mechanisms that we we've been hearing about at this, this meeting.
10:42:06 So these nonlocality stresses drunk driving jets upscaled transfers and Rossby wave breaking.
10:42:14 But there's still lots of open questions that experiments I think are still there's still plenty of work for us to do.
10:42:21 So to explore in more detail is the role of this lamentation and the and the ultimate effects of small scale, small scale diffusion.
10:42:30 This persistence of hyper staircases and hence this this link to to the possible role of
10:42:39 self organized self organized criticality with this.
10:42:46 With these these, these reversing gradients.
10:42:49 And there's a lot of work to do. I think in terms of looking at the dependencies on forcing.
10:42:54 Some of the experiments us forcing that he's actually fixed in space and you will then have the problem that of course that's not not exactly how it works out in the big wide world or in or in other planets and the extent to which that is an important
10:43:10 issue I think he's still a somewhat somewhat open question that we're only beginning to look into.
10:43:18 The other aspect is, is that although the flow that we generate in these experiments, typically, ends up being highly Barrett tropic, at least on large scales, the forcing is often often highly Barrick clinic.
10:43:32 And so exactly how Barrett clinic processes like Barrett clinic instability or, or, or convection and so on.
10:43:40 Can can interact to produce these.
10:43:45 These staircases and are they are they always staircases that are themselves.
10:43:57 Get independent order the staircases themselves have have. Non, non trivial vertical structure.
10:44:00 So there's all sorts of all sorts of questions there but I think probably I've gone on long enough and so I should stop to allow little time for some discussion.
10:44:12 Right, thank you very much Peter fascinating. So questions, ladies and gentlemen,
10:44:22 dead silence.
10:44:27 People where's our zone is traffic to out
10:44:34 or Yannick, you got a question.
10:44:36 Oh, hi.
10:44:55 Burton nice Sophie, very interesting. And I had a question regarding what do you mean by what you mean by rusty Wait, breaking so we I think that we we saw what you saw in the last movie, what you what you were referring to the PV I think it was PV that
10:44:56 was visualized in the movie so we kind of so those structures emerging from
10:45:04 some regions. I was wondering whether this recipe with breaking down did well, how that depends on the strength of the the staircase of the PV of the PD staircase.
10:45:23 In other words, you see more raspy wave breaking in the vicinity of the the Jets, when they get stronger, or when they are week.
10:45:36 You have some, or it's, it's not meaningful my question would be, to let me know.
10:45:43 Yeah okay so the, the, the, the, the Jets at least the pro great moving, moving jets tend to line up with the, with the concentration of filaments that you see in the in these.
10:46:01 Actually I can probably go back and just let me go back and just just run this again.
10:46:07 But that wasn't the right thing to do.
10:46:14 Let's,
10:46:14 let's try and see if that will run it again.
10:46:17 Yep.
10:46:20 So,
10:46:22 if you look very closely, you can also see the, the velocity vectors that are actually shown here so so we'll see when that when a sort of substantial filament occurs.
10:46:34 I mean the term breaking is, is simply by analogy with, with the breaking of waves as surf on a beach so that you can see. Essentially, sort of like a cusp of, of the, of the PV essentially being drawn out, and then essentially overturning because there
10:46:55 isn't really an analog to, to, to axial gravity acting in this sense so exactly why the, the, the, the wave of overturns is is is is to do with it with the detailed sort of strain field of the of the of the surrounding flow.
10:47:15 But if you get I guess it's not pretty, not, not, not shown quite as clearly on this as I hoped it might have done, but you can see with this with this filament here that I'm trying to move you.
10:47:26 You can see that the the flow was essentially moving along the boundary with it with these filaments.
10:47:32 And so as the filaments essentially get folded over and compressed, you're essentially accelerating the, the pro grade jet aligned with, with those filaments and then there's a kind of well mixed region that is in between these filaments in in a multiple
10:47:54 jets situation then you would you would essentially have sort of parallel channels of these of these, these breaking events with jets, sort of, lining up as the boundaries of your of your cells.
10:48:12 So in a sense, this is a little bit along the lines of the situation that I guess Pat was talking about in his in his previous talk.
10:48:22 It just is that the cells are not sort of nice little round things that they're, they're kind of sort of Angular Angular chambers, as it were.
10:48:33 I hope that gives some.
10:48:35 So the, the size the size of the, of the brain, the size of the broken waves is the width in between the two layers or in between one layer and the boundary.
10:48:49 It's, it's the width between the layers, primarily so so this this this beater effect this planetary volatility granted the stronger it is the harder it is then for for
10:49:03 radio excursions to, to, to, to, to get far enough before they're essentially turned round by by the by the by the sort of the volatility us is that there's Rossby wave elasticity.
10:49:17 So the, the stronger the stronger the this beta term then the stronger is the elasticity and hence the that the shorter the shorter the distance that you can then actually actually get get excursions taking place.
10:49:31 And so that that is essentially the, the kind of.
10:49:37 That's a sort of a partial explanation as to why that's this rain scale that essentially determines more or less the the the spacing between these. These jet structures.
10:49:55 Okay. So another question on breaking. I mean, I was preparing my summary for the barriers group, and it seems to me breaking is tick Rossby wave breaking is arguably the most critical issue in this business.
10:50:17 And going back I was looking at the early talks in the group and in the meeting David ritual emphasize this repeatedly, because the linear theory adherence.
10:50:28 I think if you catch him on when they're in a good mood, they all admit that some fix up is needed and the fix up is related to some model of breaking, which of course has been used for sir ocean surface waves that's been realized for a long time.
10:50:48 So the question is how, how do your experiments constrain the model. In other words, this in other words you could imagine different things being put forward, starting from like, what is it really cool oh right that grad PV equals zero where you would
10:51:08 have the beta and something else that I mean what else can you say about what how can you met looking to constrain breaking models from the experiment.
10:51:23 Yeah, that's a good question. I meant I must confess I haven't, I haven't thought in much detail about this.
10:51:30 I mean it's partly because we we've only been only very recently I think managed to get the experimental data of sufficient quality that we can actually start dissecting exactly what happens in in the experiment in terms of actually tracing these, these,
10:51:45 these, these, these, these film entry structures. So, I mean, maybe I'd like to throw that question back to back out to the floor of anybody who knows a little bit more about exactly what these what these breaking models are capable of predicting as to
10:52:01 whether we can actually measure something that is, is, is useful.
10:52:08 All right. Brian you had your hand up.
10:52:13 Yeah, I'm a quasi linear models form exquisitely accurate to the point where you look at the cloud track wins.
10:52:26 And you can lay the quasi linear model results right on top of them. In fact, the s3 key results which are actually covariance infinite ensemble, fixed points solutions are the nonlinear term responsible for this purported raspy breaking phenomena.
10:52:53 It step to zero.
10:52:55 In those, so that's an analytical proof that exact correspondence with observations has gotten setting that mechanism to zero.
10:53:10 Okay. So are you saying that it's irrelevant then.
10:53:18 No, it's assuming the conclusion. You see, there's a invulnerability principal between potential for TriCity and velocity.
10:53:24 So, if you want to form a jack, or form a form some sort of a velocity structure, it won't look like the analytic fixed point that you see on on on in the planet, but it will look like a local area of concentration of velocity.
10:53:42 If you go in and stir the planetary vortices it, but that's assuming the conclusion. That's it. That's exactly the, it's almost exactly the same thing as saying, I want to make a jet and I'm going to do it by putting the body for sin, which looks like
10:53:58 a jet. Would you consider that a theory for jet formation.
10:54:12 As form a jet, but it's more like assuming the conclusion. You have an inverted principle here you go in and move around planetary participate by some macro scale, mixing, you're going to produce a concentration of velocity.
10:54:16 But that that's not the problem that you were that you were supposed to solve. He was supposed to solve the problem of asking, what is the mechanism by which jets arise out of turbulence.
10:54:31 Not Can I go in and make some macroscopic mixing, which is equivalent to a body for us and get a velocity.
10:54:43 Well we do, we do these experiments for all sorts of different reasons and that particular experiment with the, with the, with the imposed steering was was just one way in which we can force flows that there are other experiments that you can do where
10:54:58 you just you just induce convection all over all over everywhere, and just, just watch what happens, sort of emerging emerging naturally that's that's asking around the different kind of question that's closer to the question that you're, you're, you're
10:55:19 raising Brian. But I, but I think once you once you actually see the, the jets that form spontaneously.
10:55:24 You'll see the mechanisms that were also demonstrating with that. Very, very simple steering process which will also also be active. And it's important also to read, to, to, to be clear that the, the recipe wave breaking process is one way in which the
10:55:42 flow is is actually generating structures that are very, very small. And therefore, capable of being acted on by by diffusion and so therefore are essentially the, the way in which the flow develops irreversibility.
10:56:02 Now it seems seems to me that that's not something that linear theories are going to be going to be able to deal with it in any detail. And I think if you if you're going to try and link what what real flow real flows x actually do you have to.
10:56:21 I mean, I mean it may be that the cozy cozy linear approach will eventually give you the right answer.
10:56:27 But trying to understand why it gives you the right answer I think is is is also a very good motivation for doing these kinds of experiments.
10:56:37 Okay, well I just remark I violently agree with the ya issue but GM has a question. So, well actually it's triggered by the Brian's remarks, so it's probably more question to him since we have four minutes of discussion, what way are you saying that the
10:56:57 body falls through an array of rods will necessarily concentrate velocity, do you have in mind, the backward nature of the was the waves and hence concentration of momentum or why why the necessary conclusion that you were alluding.
10:57:21 gradient beta. So, if you do a macroscopic form a macroscopic overturning.
10:57:26 What you're going to do is take that background beta and mix it.
10:57:33 That's it. That's your Order, order one perturbation to the system.
10:57:38 Once you've done that you can then invert for the velocity that that mix it you just finished doing accomplishes, and that will be a concentration of a jack, it will make a jet.
10:57:50 But that's that's, it's the same thing as if you put in a body force with the shape of the jet, you just did it. You know, one step removed, so it was sort of hidden.
10:58:01 You did it by macroscopic Lee moving the planetary for TriCity around, I mean that that's a, it's a, it's equivalent to saying I've made a jet by putting in a body force in the shape of the jack.
10:58:15 And I'm just saying that that's just not the question that you think you're answering I believe the exam example of this, you know, is in the stratosphere ue sharpening of the, of the polar stratospheric jet, and there you've got raspy waves propagating
10:58:35 up from the, from the troposphere, and they grow in amplitude is one over the exponential route density so they get huge. And as they propagate up into that low density stratosphere, they, they just take the, the background for TriCity and mix it over
10:58:53 macroscopic thousand kilometer regions and of course, by the inversion principle that's going to produce a jet.
10:59:00 But that's not the mechanism by which the Jets form.
10:59:05 In, in your, in your usual kind of idea that you have an instability that gives rise to a small jet that grows exponentially into a bigger jet, and then equilibrate says a finite amplitude.
10:59:17 That's just going in and hammer in the flow is if with a body force.
10:59:28 Well, as I say there are there are many different ways of forcing these, these experiments and and crudely crudely staring with a, with an array of spoons, is, is the simplest possible way that one way that we could, we can we can do.
10:59:46 We certainly can also do experiments which start from merely forcing very, very slowly varying flows on a, on a very large scale and watching the Jets form spontaneously on scales that had nothing to do with the scale of the forcing.
11:00:10 So, you know, the there.
11:00:13 There are many different, different kinds of investigation that you can you can do with the, with these experiments or experimental systems just in the same way as you can do with with it with it with any any sort of sort of numerical model.
11:00:25 I mean the only other thing to mention is that you mentioned the invincibility principle that has an implied balance assumption.
11:00:35 Now, when you start steering real fluids that balance assumption is not always fulfilled.
11:00:42 And although it wasn't obvious with it with the staring experiment that I showed her the last year, as well as exciting Rossby waves, you will also excite gravity waves.
11:00:54 And those gravity waves are not about not part of the dance solution and so you can generally, you won't recover those components from from the convertibility argument.
11:01:06 So, with real fluids, you have to be a little bit careful that that you don't sort of throw the baby out with the bathwater, as it were.
11:01:16 But that's probably enough.
11:01:19 You know, some round off this discussion.
11:01:24 Any other PV mixing fans.
11:01:31 All right, but let's thank Peter again for very interesting talk and looking forward to more on constraints on wave breaking models in the future, and make for a lively discussions.
11:01:47 So, as they say in the Bugs Bunny cartoon I think that's all folks right for this group.
11:01:57 And I forget was it bugs who said that or was it was it one of the other characters.
11:02:05 And I guess tomorrow we have the grand finale where we attempt to summarize all this stuff and Yannick will summarize this group, and I'll summarize barriers and David who's here will summarize mechanisms.
11:02:21 So it'll be, it'll be great fun.