09:03:39 Good morning, good afternoon.
09:03:50 So this is the session on the plasma session. so it's dedicated to talking about the incidence of boundaries forces forcing and dissertation on the things the year on transport barriers stuff like that so we have three regular talks and that is going
09:04:11 to, to show a few a few slides.
09:04:15 So, let's, let's first start with a Lutheran daughter will give us an experimental take on detect of boundary layer on transport formation in plasma so probably be talking about plasma interaction and what happens in different channels, calling coalfields
09:04:36 will give us a fluids talk on how leering happens. Instruct fight fluids under the influence of forcing shear and what changes with boundaries that will give us a short talk on some early work he did when he introduced the s, the same as him so the plugs
09:04:59 region landscape and has become a very important framework for transport verification and Misha, plus will elaborate on that and probably talked a lot about Maxwell rules ketosis protocols and what flux ingredient can be expected.
09:05:18 So in order to stir a bit if the, The discussion here a few points that could be could help fulfill the discussion. So, going back to what was discussed.
09:05:39 On Tuesday, the there was this interesting comment by your bed that active versus passive forcing leads to different states.
09:05:51 So the plasma at least is definitely Fox driven so certainly active forcing, is it play. So compared with a fleet or other system can we have.
09:06:01 Can we have an idea of general principles that could be going on the spot is bifurcation that you'll see in Tesla as it is the common many flavors who can have direct transition during limit cycles, etc.
09:06:16 They act differently on different channels. So, should be conduction conviction.
09:06:25 And they come to be upon what seems sometimes that apparently minor modification of conditions at the boundary so begs the question to better understand the propagation of the information from the boundaries.
09:06:39 to.
09:06:41 role boundary condition focusing on the sustainment of the layering and the feedback loops and the forward transition, at least in plasmas has attracted a lot of scrutiny, because there is a famed power threshold,, but that transition.
09:07:03 I think that's, I'm sure it will be mentioned, is also a very important thing to to look at to distinguish between models and the histories, some sure will be discussed as shown here.
09:07:29 So, this is a plot on elaborating on the, the model that Pat and Misha probably going to talk, and we shall probably come back to this as to how to transition occurs and how robust, the mechanical room is interesting questions is, what
09:07:38 does that mean play on model reduction, possibly president or other types of model deductions.
09:45:45 From the kind of invitation I got I really want to talk about how important boundaries can be how important the forcing mechanism can be, and also a kind of prevalence
09:46:05 of intermittency which is also quite interesting. And, and I still don't claim to understand what I'm going to be telling you about but. And I think it's the may be very look forward to hearing some ideas from the audience about what on earth is going
09:46:22 on and I should of course thank the people who did all the work the students, people who were students at the time when they worked on this or in red, magenta, or the postdocs and the low line and the sort of key is because this was a lot related to the
09:46:38 mathematical underpinning of stratified turbulence project that we worked on for five years and 2013 to 2018. So, what's the basic.
09:46:50 The basic idea that I mentioned in my first in my last talk. In January, really in stratified foods you can have these two kinds of dynamics I think it's very important to distinguish between them.
09:47:07 There's kind of overturning dynamics, where you are defusing out sharp interfaces, but you can also have this kind of scaring dynamics were interfaces managed to survive.
09:47:20 And really the question that I want to think about is, does it matter how it's forced. Does it matter what, whether there are boundaries or not in the flow.
09:47:27 And how does what's the interplay between instability of the flow, and the phenomenons that you observe and this also resonates with some of the really nice ideas that Pharrell was talking about, and other people have been looking at about the kind of
09:47:45 stochastic aspect of flows and linear type theories can still be useful even when the mathematician and me kind of things that they might not apply. And as we'll see.
09:47:57 I'll show you some data were looking at the linear stability of a turbulent mean profile is very useful for explaining the phenomenon of whether you have a kind of intermittency on off and nature in the particular interfaces that you see.
09:48:14 So, the first. So, executive summary, you know, When can you have overturning in a stratified flow. When can you have scaring, and you've always got to remember there's a competition between if you're lifting dense parcels up that's got an energetic costs
09:48:28 so as you get more strongly stratified, you kind of don't expect that phenomenon to happen. However, you can have the very idea of a staircase of course is you can have regions where you are weekly stratified near region way or strongly stratified the
09:48:44 the interface. So therefore, can you see, when can you see those kinds of dynamics.
09:48:49 And of course you can see those kinds of dynamics when you have vertical share flows. But then, that begs the question of course when will you what gives you this particular forcing this particular setup where you have a layer of the velocity I should
09:49:04 have shown in these blue lines when you have the fluid going at one speed, and then a share layer and then fluid going it's another speed.
09:49:12 In the presence of a density stratification and if you have relatively slowly varying density so that the scale of which the density is varying is the same as the scalar which the velocities varying, you need to overturn that interface that that structure.
09:49:30 And the only way you can, the main way you could imagine being able to do that is the sort of classical Kelvin Helmholtz overturning.
09:49:38 But the problem with the classical Calvin and Hobbes overturning is it, it needs the stratification in some sense to be weak. And also, it's very energetic and then flares is the sort of idea you know, it suddenly burst into life but then overturns and
09:49:51 then rapidly crunches afterwards, though not in the sense that I think that you mean in in plasma physics, but it's a short term phenomena.
09:50:00 But if on the other hand you have a sharp interface in density, with a more broad velocity distribution, well then you can have a different dynamics because you must remember then in the regions, away from the sharp density interface the stratification
09:50:16 is very weak. And so you can roll up the vortices at above the interface density interface and below the density interface, and you get a different kind of instability called the home bow wave instability where you have propagating 40s above and below
09:50:30 the density interface which can still be vigorously turbulent. But the density interface survives, and you can have a longer lived and mixing event and that that stratification can be robust.
09:50:45 But, though I spent a long time thinking about these sort of situations. This is still really quite artificial Where do you get this background, share flow that magically you look at the stability or.
09:50:56 And so then we've been more recently, very much interested in thinking about situations where you get to share flow through a forcing, either through an artificial body force in the code modeling something a numerical code modeling something going on
09:51:12 from larger scales, passing internal wave or something. Or alternatively through forcing it boundaries and so very nice work by ci su looked at this problem of if you fall if you have plane correct flow one layer one wall moving at a given speed, another
09:51:30 world going into another speed, you have gravity going in the direction in the wall normal direction. So you have less dense food near the top more dense food in the bottom, you actually can get this combo type dynamics, long term lasting turbulence above
09:51:50 and below a sharp interface provided, your strategy overall stratification is sufficiently high and your parental number is sufficiently high so that the density profile doesn't diffuse away on you, if, if you have very weak stratification you get a complete
09:52:08 overturning this sort of turbulent picture on the left. And if you don't have such a sharp interface so you don't have your parental number is too low, then it will diffuse away and laminate arise.
09:52:22 And so you can see the sort of long lived interfaces driven by walls. But in this case, remember that the vortices it is horizontal. And so it's kind of fighting against the stratification, but you do manage to see the sort of phenomena, but that that
09:52:41 Hi, Franklin on that so you're not really diffuse so that the density field is not really being diffused away. So we then thought, well okay this is one way of forcing to just force it a war.
09:52:53 What happens if instead you suppose that you've got some you you have a relaxation back towards a mean share that could be because you're actually in let's say an exchange flow in a channel or something like that or you have some larger scale flow going
09:53:07 on in your system. And so then we looked at that, Katzman just been published in Jefferson. And you see, the idea here is that you have a stratified share layer, but you forced the, the flow to relax back towards the stratified share layer.
09:53:24 Even if the flow becomes turbulent. And if you have an overall stratification that's very weak, you get which is shown on the top right, you can get really vigorous bursts of turbulence and it mixes everything, everything up.
09:53:39 Now, if you keep a sharp interface you can see this homeboy type dynamics that I mentioned propagating Battisti above and below the density interface, running along, and you see a very clear signal of these waves, and you can have turbulence above and
09:53:55 below the interface. But you, the interface survives.
09:53:59 But then. Curiously, if you go to an intermediate Richardson them but you can see at the intermediate Richardson number you can see intermittent dynamics.
09:54:09 Sometimes in the flow you see flows that look kinda like this you know very vigorous turbulent regions you know it's the it's the buoyancy it's the dissipation and mixing is what I'm showing in these three pictures here but it's really showing it can
09:54:24 be turbulent at an intermediate stratification, but also it can have a sharp interface. So, which is another mystery that I don't fully understand there appears to be situations where you can have overturning some of the time, you can have scouring some
09:54:38 of the time, but the interfaces, the staircase structure comes back and goes and comes back, it's not just that they can be the staircase structure can be robust, it can break down and then reform again, and break down, and then reform again, because
09:54:54 if I trace the time evolution that this particular flow that I'm showing here at this intermediate Richardson number. This is the main velocity profile.
09:55:04 This is the main buoyancy or density profile. And this is a measure, relative to this Richardson and the, the critical value of course for the Richardson number is a quarter.
09:55:14 And so, it's how strong the density gradient is compared to how that should be the sheer not the type. It's the sheer it's the two it's the numerator and the denominator of the Richardson number, and the top picture here is when it's strongly Calvin Helmholtz
09:55:30 kind of overturning, and that is the dotted line incident here, you'll see it's during the turbulent phase. So, the Richardson number has dropped has got above the quarter it's breaking down, and yet you have a diffuse density interface it's spread out
09:55:46 it's doing this sort of overturning dynamic and the velocity profile is spreading out which is why the Richardson number is going, getting large, but you've got this very strong mixing event.
09:55:55 But at this time and it just the way the data worked out for us we have these different moments we have this kind of on off behavior of the density profile spreading out or then forming a very sharp interface.
09:56:09 And this picture here on the bottom left is from that very sharp interface stage, and it's characterized by in the immediate vicinity of the interface.
09:56:27 The stratification is very strong, the local Richardson number is very much greater than a quarter, because n squared is much more than the sheer squared.
09:56:27 But I the side of it. You see this, this quantity here has negative values, which means that it's weekly stratified either side so you can have this scouring dynamic with vortices overturning.
09:56:39 So you see that if we force, either make the flow relax back to me and share or force at the boundaries, we can have interfaces that survived in its own even in this statistical sense intermittently they reappear and disappear and reappear and disappear
09:56:54 because also I guess when you have a mech mixing event in a stratified fluid there is a tendency for collapse and for it to get want to relax back towards being more uniformly stratified.
09:57:11 But, as I said, Of course you don't necessarily remember my talk.
09:57:22 A month ago, this is all, there's all the energetics there's a, there's a battle going on in these sorts of problems, because if we have the velocity share in the this sense vertical which means that you have horizontal volatility.
09:57:31 When the instabilities or the turbulence grows up it's going to be dominated at least initially by horizontal volatility and being dominated by horizontal volatility is has an energetic cost to lifting dense parcels up putting like parcels down.
09:57:44 So it's also been the interest of putting out, introducing share in a horizontal sense and the other sense so that you have vertical Battisti where there isn't.
09:57:53 At the beginning, and energetic cost to having to mix things up. So that, of course, leads us to these experiments we've been doing with the Taylor correct apparatus.
09:58:04 So we have vertical stratification in this tailor correct operators and the cylinder rotates, so it's very very vigorously and unstable very it ends up being really quite turbulent.
09:58:18 And if you look at, take a pixel line here of the salty water, so take a line straight down here, and then stack that line one after another. You managed to make these, these video where there's that video first.
09:58:32 So you see interfaces appear and disappear and interfaces appear and disappear. You see the because of the refractive index changes you can see whether interfaces.
09:58:47 And then at a given height, they appear and disappear very periodically. And if you do this trick this stacking up of the vertical lines that pixel line there and then from the next Friday next Friday next frame.
09:58:56 You see these beautiful on off intermittency phenomenon of the interfaces. You can see at the beginning, there's a clear signal of spirals right and then that is associated with the initial instability because remember you're starting from rest here,
09:59:13 so you have an initial instability of the sort of strata rotational instability. So you see these spirals forming. But then, after some time, these start to become horizontal and you see these ones near the middle of really really horizontal, so you have
09:59:29 an interface, it goes on stable, it disappears, you have an interface, it disappears interface disappears interface. And it's really curious to understand why that's happening because if you look at the vertical flux of density in these kinds of experiment,
09:59:44 you see the kind of non monotonic curve flux curve, like the for the Phillips mechanism which means that you can indeed reinforce these interfaces. And so we were trying to understand how this phenomenon happened.
09:59:59 And so what we did what we did what Kanwar did for his PhD was do simultaneous piv and
10:00:09 Cliff Lazar and juice for essence experiments in the vicinity of one of these interfaces. So we look at the radio that sorry the, the azimuth low velocity along the radio line and simultaneously look at the laser in juice fluorescence dying the fluid
10:00:33 below the below an interface, and then seeing how it goes when it comes. Where do we observe it as it becomes about the interface. and this dynamic is determine allows us to tell we do refractive index matching allows us to identify a really interesting
10:00:45 structure in the systems that we don't at all fully understand yet. But here's the picture. So, this is as a moveable velocity relative to its mean. So if it is yellow.
10:00:58 It is fast as a new publicity and it's blue, it's below the mean over many, many rotation periods. And this is the radio.
10:01:15 It's a radial line, once again stacked but it's a radial line measured just above one of these interfaces. And so this is near the fast in the cylinder, and this is near the stationary artist cylinder.
10:01:21 And so, this is a picture that shows you this very very well defined periodic fast, slow, slow, slow, fast, slow, slow flow that is in the interior of the fluid.
10:01:34 And in the interior of the rotating cylinder and where that is the angular region, and where that's come from is bursting at a completely different time scale from you see this very very fine structure here from the inner boundary line.
10:01:50 And then if we zoom up you see this is 350 of these rotational time units. And here's 100 of these rotational time units so it kind of zoom up this data to be the data here.
10:02:02 And now we look at the current at the structure of these fast and slow bursts of of as a neutral flow, compared to what's happening in the density field.
10:02:12 And remember the fluid that was below the interfaces died fluid, and we measuring just above the interface. So, observing die means that you're seeing mixing, that the interface been smeared out.
10:02:25 Whereas if you don't see die that means the interface is very sharp, this was only a couple of millimeters about the interface so it show when you see dark that means the interface is really sharp, because you're only got a fluid.
10:02:36 And so you can see the correlate the structure of what happens this, you know, this being here is like that beam. Sorry I'm pointing the wrong one is this one here right.
10:02:45 It's this, this here is that 60 units are you see that it's just scaled up this first hundred and 10.
10:02:51 And you see you see the bursting outwards.
10:02:55 The when you, when fluid is ejected out with this high frequency The interface is extremely sharp.
10:03:02 It's extremely sharp and there's a bit goes on and then it bangs into the far wall. This is the stationary well and there's also extremely sharp there and it's gone fast.
10:03:10 So what you have to understand fast fluid, of course, is fluid that's come from the near the inner cylinder, and it's been ejected outwards, and it scours and sharpens the interface.
10:03:20 And then when it bangs onto the far wall, it causes a mixing event that slows everything down. That's the signal you see here slows everything down and mixes from underneath.
10:03:30 And so you have a process that you have a very very high frequency ejection from a boundary layer is somehow that triggering a very organized, much lower frequency signal.
10:03:44 And it's really quite mysterious why that's happening. And it is also mysterious that as you make as the interface gets weaker, the signal gets less significant.
10:03:54 So this is a more strongly stratified interfaces when the, the overall density jump is about six, and this is down to 4.5 and you see this banding signal disappears, as you as you decrease the Richardson number.
10:04:08 And then this comes to the when I said the connection with the Brian Pharaoh was talking about, because you can do a stability analysis of a turbulent flow.
10:04:18 So you can look at you can, you can do an empirically measured velocity profile that this is the very fast near the inner cylinder, very close to constant angular velocity like a one overall square, I want to angular momentum rather one over are in the
10:04:34 interior much slower, and then a boundary layer near the inner wall.
10:04:39 And that is the azimuth of velocity structure. But then in the vertical, there is a density structure that looks like this you know sharp interface. So it's a difficult to be words non trivial stability calculation students non parallel flow but as I
10:05:07 And so the dashed lines are the experimental measurements and the theoretical lines are either. Our empirical boundary function what we actually measured, that's shown with the magenta line.
10:05:23 And then we also used data from Detlev loss' group Heisman the towel wrote a paper where they also mentioned in the stratified case, the velocity profile which is these other these other scales, and you see we really well predict, two things, the turbulent
10:05:38 profiles is determined by this, we get their, their period their frequency very accurately.
10:05:46 And we get that the dominant mode is indeed m equals one, which is what is observed as an M equals one signal going round this cylinder. That is, so the turbulent flow that is being triggered by boundary layer rejections forms a linear instability of
10:06:04 the turbulent flow that is in some way stable, robust that drives a mix an intimate and mixing event that periodically scours and overturns the interface quasi period, well very periodically scans and overturns the interface.
10:06:22 So you see this interplay between the instability in the boundary forcing appears to be perfectly designed to keep an intermittently appearing interface, but you may think this is a peculiarity just off the very particular case of the, the flow we considered
10:06:40 Taylor Colette's got, you know, curvature effects and so on. So we also looked at horizontal plane correct flow, Dan Lucas did. So this is kind of playing correct flow with the, not with the density gradient in the wall normal direction, but in the span
10:06:58 wise direction, so we have two walls shared going like this in my usual idea, and you have a vertical density gradient in that flow, and you have a very interesting structure that per week stratification the flow can be turbulent for stronger stratification,
10:07:14 this is a Reynolds number stratification boundaries. You have laminar flow or realignment arises for sufficiently strong stratification, but then a beautiful study by Virginia towel and mass a group who were low bar and LIGO came up found an instability,
10:07:35 strong stratification which is shown in this green region.
10:07:38 And so it's linearly unstable, when the flow is sufficiently strongly stratified.
10:07:51 Unlike playing quit playing.
10:07:51 So first we do a sequence of simulations and layers spontaneously for the layers that spontaneously form scale like you over n. And the switch off that occurs that they're at a given stratification to switch off as you increase the stratification for
10:08:03 a given Reynolds number is because the scale gets thinner and thinner you may end bigger the scale of these layers gets thinner and thinner. And eventually, it hits the scale that of stuff being ejected from the boundary layers you know like we saw in
10:08:19 the other case, the length scale of the boundary layer ejections hits the length scale of the structure of the layering that separation between interfaces, and it switches it off.
10:08:32 That's the dynamic we observe, but then if you think about. So the way to move my signal the circles case here, these are the pictures of the density, the stream is velocity, and so on.
10:08:47 At that particular parameter choice, but now if we look what happens when the flow is linearly unstable, we see some very interesting dynamics.
10:08:56 I think it's very interesting dynamics. So let's think about this case where the flow is linearly unstable. And yet, below the sort of trend line of we would think of this turbulences stabilization.
10:09:10 And then you see the onset of something that really looks like a linear instability got this angled lines, really quite reminiscent of what I showed you the tailor correct flow, and the flow onset and form sharp players, and then it becomes a turbulent
10:09:25 on a nice, long time ago. Okay, and it's quite turbulent. And that's that's the looping round. But now if we go to the region, up to this one up here which is, it is Lindley unstable, but he's above what we predict for this real laminar ization of the
10:09:42 turbulence.
10:09:47 This is what we get.
10:09:53 Yeah, we have this intermittent bursting layers for form.
10:09:58 And we have an intermittent on off turbulent behavior.
10:10:03 We get layering.
10:10:04 We get turbulent bursting because that, and it is associated with the linear instability, and then it dies back again, and it happens.
10:10:11 So, I've talked for more than 15 minutes, but really what I wanted to say is in the stratified flows, because you have two different ways the flow can mix, you can strongly overturn, and you can have this scouring because you can have vortices at Eddie's
10:10:30 impinging on the sharp interface in certain circumstances like cheese who and john Taylor worked out. It can state that can be robust, you can have a situation where some of the time you ever turn some of the time you scour, and it keeps coming back and
10:10:46 linear instabilities can tell you that that really might happen.
10:10:50 And it seems to happen, if you have vertical share if you have horizontal share if you artificially force the flow, or you force it from a boundary, you, it really does seem that you get this layering phenomenon, all the time.
10:11:07 Okay. So, thank you very much for your attention.
10:11:11 Thank you very much.
10:11:22 An interesting visa. Do we have questions. Well, Adrian freezer you will first.
10:11:26 Please go ahead.
10:11:29 x.
10:11:29 This is really exciting stuff thanks for sharing this, I was. I'm sorry this is not a good question because I just was wondering if you could clarify a plot.
10:11:39 Before you got into the tailor what flow, the last slide before killer quote flow.
10:11:45 This one.
10:11:48 You know, maybe it was the next one you were it was it was one.
10:11:51 Maybe it was it, it was the last part of the tiller quite slow I'm sorry.
10:11:55 You started talking about. Yeah, this one. Could you say again what this bottom right plot is it seemed okay.
10:12:04 So, this is a trying to be a representation of what the flow we looked at the, the instability which right so this is near, so the red line the orange curve is the main philosophy profile which is all in the azimuth of direction in this telegraph.
10:12:19 And then we have a vertical density stratification.
10:12:23 And I have it, and then we looked at the stability analysis of such profiles, but this is a mean turbulent profile measured from the experiment, it's not the base flow that you would have heard Lamanna Taylor correct its turbulent.
10:12:39 Taylor career at several thousand rentals on the appropriate data number.
10:12:45 You do a stability analysis of that to the flow, you find that the most unstable Matt mode is an n equals one mode in the musical direction. It has a given eigen structure in the radio, and the vertical direction which is localized at the interface.
10:13:01 And the experimental observations that you have are twofold one you have the that m equals one is the dominant mode, and you also have the period, because it is a periodic structure that appeared you know said like the real part of the growth rate that
10:13:15 the imaginary part of the growth rate that those are phase speed associated with it.
10:13:19 Right. that is a period.
10:13:22 And then compare that theoretical stability calculations, which are the solid lines with the experimental observations which are the dashed lines for different relative gap quits that's what this is, because we were able to do different cylinders as well.
10:13:39 as well. So we were able to, so we saw that it wasn't said that the the n equals oneness, is, is independent of the stratification coral can walk in many different density stratification is many different rounds numbers, always found m equals one, but
10:13:54 found that the gap wit was the parameter that changed the period with three different sizes of cylinders.
10:14:03 That was that he then created an empirical curve for that which is these dashed lines I extrapolated from the three different gap quits to get a function that went through those three lines, and then did stability stability calculations at all these points
10:14:17 and and it's, it looks right to me.
10:14:20 I think it's, it seems to have the right trend, and it was, you know, without any trouble parameter. It has the right magnitude.
10:14:29 That makes sense. Thank you,
10:14:32 David, then that's it please think you might just have been addressing this anyway column so you but you have this linear instability and off it goes.
10:14:42 And then what, how, what determines
10:14:47 how long it goes on for.
10:15:03 That was the, that was the biggest what determines how long it goes on for it goes on until the density and, because it until the density interface gets too weak, and then it overturned.
10:15:04 And can you, can, can you capture that all within the same theory.
10:15:09 I mean, you're doing so you would have an evolving background state and what what would the theory look like for that. Oh, so, so we it's essentially a frozen in time theory we tried, we did a calculation we did the calculation for all different jumps
10:15:28 in the density field that you, you could observe you know Richard to numb from Richard to number 22 Richard so number five in this parameter which is basically the strength of the density john, because we would be driving a constant speed.
10:15:44 And we found that it didn't depend on on the growth, the growth rate and the wave number did not depend on the Richardson number This was the poor Kanwar right you were convinced it was something to do with coupled gravity waves or Sri, but it doesn't
10:16:00 know about the stratification provided there is an interface. Effectively what the interface does is it localized as the Eigen function in the vicinity of the interface, it causes a Zed vertical structure that is localized, but it doesn't it doesn't care
10:16:16 about anything other than localization for the within this as a musical velocity, because there has to be something that means the Eigen function has a given structure in the, in the radial direction in the axial direction rather.
10:16:32 And it's just that localization and the axial direction which is what stratification does to it.
10:16:38 The growth rates are quite significant.
10:16:41 So did so that. So then we were confident that it seemed where we think it seems to be a linear instability, because then the question is what it Why does the Richardson number change with time.
10:16:52 Well, it changes with time because there's a constant flux through the system so the jump in the density and across the density interface can draw.
10:17:00 And so that is rushed through what I'm showing in this picture here, the signal starts to disappear. As soon as this parameter gets too small. And you switch off.
10:17:12 But if you look at these numbers here six two through five it's when it switches off and go back over here.
10:17:20 Six through five is precisely the stage where the
10:17:27 Phillips curve starts to kick up.
10:17:28 So really the heart of this kind of small level of flux across an interface is associated with being in this periodic regime that can be described by a saturation of the linear instability.
10:17:42 But when that breaks down and you start to overturn, that's when your flux jumps up because it though I show on this figure that time goes this way, it's not uniform.
10:17:52 The experiment to go along here talk, a week.
10:17:59 And, but then this bit is an hour, it really rapidly goes through the last bit.
10:18:05 So, while you have a strong enough interface to cause the Eigen function to be localized. It doesn't depend on the stratification, and it grows rapidly.
10:18:15 But once the interface gets too weak a different dynamic occurs and you just rapidly overturn the interface, which is why flux is so spiky here for example.
10:18:26 Okay, thank you.
10:18:28 You bet your next. So, very interesting so sort of curious might this be thought of in the within the ballsy or as we like. I like to call it the lie framework with, with the, with some professional teasing of my good friend Neil balmforth yes the blow
10:18:57 framework as much with the, with the.
10:18:57 Well, it opens the door for other kinds of teasing by the way as we had yesterday, but anyway with it with the idea that the, the, the stuff coming off the inner boundary layer provides the stirring that's necessary in that framework and then you have
10:19:15 of course you're off and running with the, with the, with the mixing and then all the asthma de physics and then you'd have to add the additional greedy ingredient of the evolution of the flow, and the possible instability of the layered structure, which,
10:19:32 you know, would say you may be, and by the way we have analogs in infusion what's called and you make the zonal flow and then you have the instability of the zonal flow which is called the tertiary instability right which is in other words it's the instability
10:19:50 of the secondary state here. put it crudely.
10:19:56 But then the question was it a is that a reasonable way to look at it and be, wouldn't you have to calculate the instability in the presence of the turbulence in the layered structure because in, in, in the ball z Story Of course you're going to have
10:20:15 turbulence in the layers, or you have very vigorous turbulence in the in the layers, all the time. Absolutely, and, and, you know, this is.
10:20:27 Yes. You're quite right. It is mysterious that we were eight. The thing about this experiment that was most mysterious is, you know, the spontaneous forming of layers was not was seen by the Grenoble group, guys, as I said that my interest in this was
10:20:45 triggered by being on the PhD exam that particular PhD, and the fact you see the layers, the interfaces and the layers, they're very turbulent on the inside, and so it isn't at all clear to me, and kind of offends my sensibilities that this linear stability
10:21:03 theory appears to work.
10:21:06 But the fundamental point is you see a really really robust m equals one mode in this system. This on the on off of the interfaces. And so one way to interpret it is to say the mean flow triggers or perturbation that has a period of a given value and
10:21:29 an M equals one structure, but the way you are described what the way I hear you describing it is, I think, or an equally valid description, it is effectively, that this is this high speed ejections from the boundary layer are in some way, forcing the
10:21:47 flow, and that that forcing in the flow, it can think of no other word than resonates it resonates precisely to give you this m equals one phenomenon right so it's like a, it's a response in the system, but But, then, is the way these high speed objections
10:22:09 are going, are the things that are leading to the main profile that we do the linear stability of, so I kind of wonder whether it's you know, it's like you say it's the tertiary instability it's that it's that the, the state, that would be formed purely
10:22:23 by these high speed ejections is a very sharp interface, and that very sharp interface in the presence of this particular type of profile is is excitable to give this overturning the structure that has n equals one, and has the period that is entirely
10:22:45 consistent with a linear stability analysis of the turbulent main profile.
10:22:50 So, you know, I, for a long time we wanted to kind of talk about it like turbulent puffs in a in a in a pipe right you know that you were that you excited, you, you, you then have to dissipate you speed up you dissipate the energy you slow down and you
10:23:08 go backwards and forwards, but it then this, the linear stability argument seems to work startlingly well in terms of, you know, directly predicted the independent the independence, it was dependent on the right parameters, and it gave numerically correct
10:23:24 values for preference for wave number and the period, the growth rate is not particularly useful apart from the way answer to David it's it's fast, and so that the fact that the background flow is evolving slowly doesn't really matter.
10:23:39 All right, thank you. Interesting.
10:23:42 Well, if you're if you can make it relatively short. Yes, very short calm, very nice talk I really enjoyed this and it seems to me like the circular geometry is is key in this in this problem.
10:23:58 Yeah, you can you go around and you try again, try again right that's really the nature of the scouring. And I was wondering if you thought about what happens if you don't have circular geometry.
10:24:10 In other words, you just have a you know a planar interface where you know the instability is not necessarily, you know, an absolute instability.
10:24:21 No, we haven't. We didn't look at that but that you know I did, I kind of said we you know we looked at the, the horizontal plane correct which is not which removed the curvature but it's not the same thing.
10:24:30 And I think that would be that would be a very interesting problem and yeah that would be a very nice problem and for our camera would have found it easier because, as you can imagine, setting up the non parallel stability problem was a bit of a pain
10:24:44 in the annual region.
10:24:46 Yeah, I think that would be very interesting. And the way you just described it about whether it is absolute or not, is a very interesting question that I don't know the answer to.
10:24:58 Yeah, that's very curious.
10:24:59 Okay, thanks. Okay.
10:25:03 Well, discussions going on well which we should probably move to bed now.
10:25:13 slides and then to Misha. And now I there's just this is a commentary.
10:25:18 So,
10:25:20 this. So, this, this, you know, to continue in the Python vein this talk is dedicated to his MBA and he's not even dead yet. Right.
10:25:32 But if you recall that great movie, but it's it's a response to the question of yesterday, you know, the Maxwell rule and is it there isn't it for barriers, and it gets back to some very old work of mine with Vladimir Lebedev not yet another by the way
10:25:54 student who went off to financial business and I gather is doing extremely well over the years. So the question is, you know, how do you describe the, the state of the barrier.
10:26:10 And is it, is it a Maxwell rule.
10:26:13 And well the answer is almost so we did a very simple this, we did a very simple one field model, save for density now some of you may not like density.
10:26:26 So please, plug in your mind, t if it makes you come here because we're going to treat it as flux driven but then it's just a symbol.
10:26:35 And the point is you have to regard the flux sky as some function of positions a radius and the gradient, then, or more generally the end profile.
10:26:47 And there's a possibility of some regularization to control any, any funny behavior which you can associate say with some reason, you know, smaller scale stuff, or, or residual banana.
10:27:03 Banana orbits or something but that's not going to matter in the end, the point is the flux is spatially in homogeneous and quite general.
10:27:12 So if you have a source, and you are away from the source region the source creates a flux of particles.
10:27:21 So the position, the steady state is going to be described by where they basically the source driven flux is equal to the turbulent flux, plus the regularization and by the way this the beta is about three I think to satisfy various rules.
10:27:42 So, very simply, then what you do if you want to, to figure out the stationary state is you multiply this thing by the second derivative of and integrate, and you get a jump condition, right so you get gamma not times the jump condition in the density
10:28:03 gradient equals the gradient the integral across the jump of the flux.
10:28:10 Plus what's left of the regularization and you notice you get smoothly the regular ization and you can then take epsilon go to zero and forget it so there's no ambiguities in the regularization so this to answer the question of yesterday, is that is the
10:28:29 stationary condition. Okay.
10:28:31 And the message here is the barriers are like ice hockey, the action is in the corner it says we were discussing in Pascal's group. Some weeks ago and the court the corners the you know the changes in slow but court occur at the barrier transition.
10:28:49 And so the point is this sort of formal expression determines the barrier location in terms of the flux landscape and the source and I'll just show you a flux landscape in a minute so to answer his question.
10:29:07 This is not quite Maxwell but it's close I mean I'm good question of what Maxwell might be but the usual thing that people like to draw might be something like the right hand side of this equal to zero.
10:29:22 And I would say it's not zero it's equal to gamma and odd times the jump in the derivative okay so it's it's close to the Maxwell little bit not exactly and trying to having trouble advancing the slide here.
10:29:43 There we go.
10:29:44 So in the source region, you would not have the gamma not, you would have an explicit representation of the source and the balance would be basically between the, the, the, the integral over the flux there again equal Simon integration over the source.
10:30:05 And that would be a good deal harder to extract anything useful from this approach, it's not so easy to generalize to multiple fields except in a very formal sense.
10:30:19 And I might add, although one can have great fun writing it down and drawing equal area rules you, it is not not so easy to actually solve and extract useful information from and isn't we were saying yesterday, of course the information you get is only
10:30:38 as good as the model you have for the flux and there are several key questions about the model of the flux.
10:30:46 So let me though if you forgot what a flux landscape is let me show one other little picture just to remind you, and then I will conclude, too many things here.
10:31:04 Sorry, a little bit.
10:31:10 There we go, share screen. Bingo. Yes.
10:31:18 So, this was of course a picture of the flux landscape that I showed yesterday from a kind of an analog for drift wave turbulence of the blind model right and this just shows the picture as I was describing before, have a two dimensional function in terms
10:31:38 of the gradient and position.
10:31:41 And you can see of course a different position, as you progress in gradient, you can see the onset of the barrier and the growth of the good confinement region.
10:31:54 And that's basically all folks. Thank you.
10:31:58 Thank you. Thank you very much. Is there a pressing questions, a pressing question.
10:32:10 Hello, Judy. Just the way sense but just one shop question.
10:32:16 And I think you can get some information on the tangent to write that if you start from an arbitrary initial state. This of the old you get it would give a final state that is location variable, give it a try.
10:32:30 Remember when we can get some information also on the dynamics right yeah I mean can get some information particularly what you can do is you can try in the flux landscape, you can trace out the trajectory of the transient dynamics right you can you know
10:32:46 it's it's back to the.
10:32:48 It's a bit reminiscent almost Python esque of our nose geniuses and maniacs plot for you know in his catastrophe theory book right you can you can trace out the evolution, as long as the evolution is in some sense slow relative to you know to the time
10:33:06 scales of the underlying processes that you have coarse grained over, yes.
10:33:12 of the underlying processes that you have coarse grained over, yes. But I mean I think that the thing one is one is interested in, which is, how do you not only predict the pedestal but what does the pedestal mean you have to predict I would say at the
10:33:26 very least, the temperature and density pedestal with and in any practical case also, the rotation pedestal with. So then you're going to have three of these things coupled and very dependent on both of those suppression, and also the course the flux
10:33:44 and I mean, for example how you would the sensitivity of the story to the residual stress in the rotation for example is one major one major thing.
10:33:58 When you can write down the rule but get get a meaningful answer is another matter.
10:34:07 Maybe a short, short one doesn't strike me that you get histories is with this right back condition Can you can you actually.
10:34:19 Well,
10:34:19 let's see, I mean sorry I got too many slots open here.
10:34:30 I mean, that looks historic to me is, you know, you can see the S curve. Yeah, right. you do get history.
10:34:39 I mean you know this is a running you only gave me three slides. So I didn't prepare a slide on history, but if you have if you have a fan, if you have a family of s curves I think it's not much of a jump of intuition to see you do get history says,
10:34:59 maybe save other questions after Michelle.
10:35:07 Michelle talk right. So, thank you very much, but since we're running late and, Misha needs some time to Michelle Can you please start your presentations.
10:35:26 They see my window.
10:35:28 Perfect, thanks, okay it's okay full screen. Because I speak to Simeon. Yeah.
10:35:41 Thank you for giving me this opportunity to discuss is quite interesting topic and, but unfortunately Iris quote a little bit of surprise by surprise because that suggested need to return to the paper that we was here and I wrote, more than 10 years ago
10:36:03 and I sort of underestimated how long would it take to turn it off.
10:36:21 But
10:36:14 the basic ideas,
10:36:18 already been introduced and discuss by the kind of shirt on explaining details and here is my outline the beaches looks wrong but in, in, in reality it's just a list of the slides essentially maybe multiplied by two.
10:36:46 Come up this amount of material here. So, let me start with some difference maybe what do you have when you are interested in transparent barrier and talk about specifically so yeah I'm mostly interested in, in the edge area because ah it's somewhere.
10:37:14 Point A say you. It's the place where, where you get those losses of particles and heat,
10:37:17 and also your, get some source of freshly created plasma by ization so you get some material that impurities coming from the wall and then Josh exchange and all this complicated, your normal Korean plasma by radiation, etc.
10:37:37 If you look at this a little bit from a murder perspective and if you have to assume that there is a nonlinear relation between gradient of whatever gradient density of temperature and the flex informal, sort of, escrow or, if you look at this from the
10:38:15 If you look at this from the other side you will get the advantage of being single really functions is one has to do firms values of gradient for the same flux.
10:38:30 And it's classical instability situation, and two branches to upper and lower. But perfectly stable and intermediate it's unstable.
10:38:41 And if your concealer. This is a opposes from say driving.
10:38:50 Really Jubilee flocks and you get the, what is called better more words, you get, Nisha the low gradient.
10:38:59 And then you come into the situation that there is another state, or even two states and the system by jump to the upper brought brunch.
10:39:11 Any, any brace.
10:39:14 Again, if you're if it doesn't your dry rot your gradient, your flex excuse me good and it's pretty high and you start to learn.
10:39:26 But what do you observe, but observation, you can take from here is that there is a overarching spatial or a disparate there is no space in this picture, simply from a functional relation between gradient and flux.
10:39:52 But if you introduce spatial relations and there is ripen region, and the question is, where exactly is this jump may occur.
10:40:06 And actually
10:40:10 read as a criterion.
10:40:12 And what will be the beats of the group confinement the regime so this regime is of course it what what do you want to have, because it's a large gradient, and just say in flux.
10:40:29 Now that a little bit of history.
10:40:33 One of the first step system has been considered but by Brad Hinton and 91 he considered one one field to model for temperature so it's it's no classical component temperature gradient, and children and component.
10:40:56 But this this is big compared to know classical but it is addressable so
10:41:03 where you can strongly suppresses.
10:41:07 And again, you get this two branches and as I described before.
10:41:19 On the other hand, at least to
10:41:23 resist suppression factor density and temperature, pressure, so you cannot possibly do the accurate theory, by considering one field, one field bifurcation problem.
10:41:39 So renewed the Second. Second.
10:41:44 And again it was Fred Hinton who interviews at the distances of course and introduce it to feel more density and pressure.
10:42:03 And as I said that they are coupled through these, we wanted to share, which contains the density and pressure and you can have a product of gradients.
10:42:07 Primary term but you can also have a second derivative of the brushes of curvature, which, which can play also significant drop in.
10:42:22 In, determine what, what, what is a pretty important transition Barrett exactly sure that your maximum muscle.
10:42:30 In, determine what, what, what is a pretty important transition Barrett exactly sure that your maximum muscle. But discussed, minutes ago or something different.
10:42:36 So, now,
10:42:47 actually.
10:42:43 actually. Hinton and Sadler very limited treatment through special case where diffuse images of those hidden particles in some relation so the ratio of this turban and then, and no classical CLL more granted this is true and then you can
10:43:11 eliminate one of the gradients, you can eliminate in general case but if you eliminated visitation so this term disappears into have just the same gradient for for a boss granted this just different by a factor as a person, as a situation where if you
10:43:51 animation is different systems zeros a linear relationship between gradients and this of course changes the whole story significantly.
10:43:50 So we need to consider general case and the realization is quite different quite complicated than as Brad said it's not always easy to reserve so it's basically doing some sort of algebra, Britain Britain.
10:44:12 Yes.
10:44:14 Some, some general remarks regarding the necessity of considering the journey is.
10:44:27 And I probably may skip it, everyone understand that it's quite constrained in condition, it's not necessarily, not necessarily have a practical interest because of.
10:44:37 But then you can see the specific type of turbulences doesn't really hold water I mean this type of suggestions.
10:44:48 So, but anyway, we can still come up with some ideas, what is the current existence.
10:44:57 Can you tell and and for this simple case you have basically are talking about the product of two Fluxus particles.
10:45:20 And she didn't they should be between certain radius of CS lambda is the ratio of the, the one to the zero that should be large enough larger than 1609.
10:45:23 And it's physically graphic do.
10:45:27 It makes sense because in order to craft this visibility situation you, you need to practice, turn turbulent component of your transport in the first place.
10:45:40 Otherwise, that was messing to, to reduce essentially by by the shooting of the turbulence denominator, which is here of course, when CDs, known as zero so it says there is no this portion relation between different types of
10:46:03 patients Sermo and particles.
10:46:07 Then situation becomes much more complicated, and your craft better service or above horrible. It's interesting, even though it's definitely not one dimensional bifurcation program and if that, because I've known so if you read our know.
10:46:30 You know that there is no general bifurcation theory. Other than that one dimensional parameters.
10:46:38 Here you can pack all the transport coefficients in his his two parameters. A and B by forming this combination and when your, your current existence conditions.
10:46:53 Eg and in parameter space where you can class that is Aaron h stayed in the same breath.
10:47:03 And here's what you need to get this knowledge transition is possible. This is for the region.
10:47:12 So, so make sense.
10:47:17 So you if you can really show you
10:47:27 how you constrain your parameters, also aware of the bag the strength so that's offense standards the transitions it's not easy to in general case that there is no simple, you know, parameter.
10:47:40 So, you have to introduce this sort of awkward function but again the message here is that you need to consider a pterodactyl masquerading symmetrically in your final results and to introduce this function major game content contains inspirations of transport
10:48:05 coefficients tourbillon, over, over new classical background transfer, and its strength its maximum relative minimum value, shown here for different as a function of D zero or do you run at about four different guys zero over guy wants a bunch of course,
10:48:30 it's basically.
10:48:32 So, this, This bifurcation problem.
10:48:36 There is also analytic experience expression for this but this one will be reproduced here.
10:48:44 So now the curricula, where exactly this maximal road, etc. So how he approaches.
10:48:54 We actually managed to be they can be one simple case.
10:49:02 And in the, in the case of quasi stationary situation when you assume that around gradient and slave to the other and then the system will slowly, you can again.
10:49:17 Express one to the other and then consider revolution creation and gate.
10:49:23 And then you can add some fiber diffuse if you will notice. Chairman, by the way this is not out of the blue view
10:49:45 derives its diffusion pigeon from some underlying data manipulation or, or project ranking creation barrier for some additional degree of freedom, such as LA City and get that either with that is what do you do essentially Chapman, and scope approach
10:49:54 you eliminate faster.
10:49:58 eliminate faster. Step by step and it's right nationals at the Obama is a fiber defuses term.
10:50:08 Then you can serve. Already this problem and and come up is absolutely certain resolve that that should be muscle maximal type of transition.
10:50:24 But if if the fields are still independence if you don't assume that one is slave to the other and look at them independently.
10:50:35 That was think that Marshall, Marshall was a quote quote us said on the show that they are different than the transition is not necessarily Muscle, Muscle condition, it can depend on the ratio between different doofuses quite a few DVDs for them, and
10:51:05 density.
10:51:08 Okay then be, which we try to sort of clean it up and suggested a different approach to do this determination of transitional.
10:51:26 This better gear and again we get rid of this Okay, forget about this quiet for diffusion.
10:51:33 Let's look at this problem, as it is, and formulated familiar variation principle so that introduced some functionalities quite dopers integration in gradient and functional is, basically, you can show that it decreases as time is decrease and the dealer
10:51:58 but its breath while you see is the minimum obviously with medium will be reached, only if it has actually two meanings.
10:52:16 And the absolute meaning of this function will get each only this minima coincide and transition occurred in here. And if your contract emergency so back to the flux gradient, our usual escrow here is the Emperor, by the way, it's the fleet the labels
10:52:33 gradient.
10:52:35 and the flux.
10:52:38 Come back to the initial picture that they show the S curve.
10:52:42 Seven transition is exactly here. So these areas should be equal. So again, maximum muscle.
10:52:52 But it's get more complicated.
10:52:57 If me include the second derivative Basha into the expression for sure.
10:53:04 Then, very fast algebra relation between
10:53:25 the flex your source essentially differential differential equations and deviant music shows you this transition. Curiosity occurs to me. So basically it's quite variable, maybe situations, in a sense that when you increase your flux.
10:53:38 Initially, remember it and since it's a gradient. This is flax, and you can use it as a system jumps at the first occasions of the say to a more tolerable confinement state.
10:54:05 Basically, as long as they care. So, keep that on design it must jump to the age mode because it is no other choice for the system. It turns out that if it is significant then it jumps as soon as it can.
10:54:33 So, my conclusion I don't need to read them so I basically stopped here.
10:54:40 Thank you very much.
10:54:44 Thank you very much, Misha, so we have time for a few questions.
10:54:50 I'm sure there are going to be some.
10:55:04 This is yours.
10:55:09 Hi. Can you hear me. Yeah, I can hear you.
10:55:13 Hi, Michelle thanks a lot for the nice tool, are several questions actually.
10:55:21 One of those is that I'm not sure that I properly understood at some point you mentioned that the regularization term matters to find the Maxwell flex, or the equilibrium condition that Marshall mentioned that, having something which is not iPad diffusing
10:55:43 could eat some another rule. Am I correct when I say this or I misunderstood point. Yeah,
10:55:52 quite clear on that. So what Marshall said that very moment but I'm sure.
10:55:57 Also remember this was tragically not not so long before, before he died of course but what he suggested is, if you take either ization steering.
10:56:28 heat and density are different, not equal. When you, you can sort of shift back and forth that transition point by changing the direction that was his demonstration, essentially, our approach was without thorough be great to hear so if you introduce this
10:56:48 absolutely or increase your equation by two, and then we'll do matching as in 30 analysis so if you just step.
10:57:22 Your inner solution inside this see layer of the square root of epsilon.
10:57:12 And then your outer solution which where you can drop this by the diffusion correction altogether and then you can pretty much match real solutions.
10:57:25 And the you realize that the only way you can mesh is two different solutions out there and inner solutions, if the transition point is chosen to be exactly a massive.
10:57:42 However, if you have two different epsilon and two different fields.
10:57:49 Then you get another freedom of choice of this point, which will depend on on the ratio between these two parameters.
10:57:58 So, but I, unfortunately, this has not been published and if you I just did that because that would be quite beneficial is a reference to Marshall I think it's quite simple exercise I don't have maybe a practice, his notes.
10:58:20 He gave us his notes.
10:58:33 As usual, he just you know you discuss with him something again the next morning he brings you some calculation which are quite interesting to look at.
10:58:38 So that was my collection and yes it didn't
10:58:46 does matter if it's not that simple as be assumed to be just for one.
10:58:54 But, on the other hand the different approach, which is iterated on time dependent situation, clearly shows you that Maxwell is still more or less robust rule it survives the user is married equalization that just it depends.
10:59:14 Okay by
10:59:27 any fans. Didn't you mentioned you had the other questions or show others but maybe just one because it's quite late already. so if I may ask, last one.
10:59:30 All part has a question where, what is the chairman say when you ask a question and I just want to make a comment, go ahead. Okay, great.
10:59:44 on your slide, nine, actually, Misha you we show the quite complex relationship between the values, DQZVT.
10:59:52 That should be satisfied to to allow for an alleged foundation. So although it was quite quite complex and maybe not.
11:00:02 Does it
11:00:05 Excuse my interaction after bad give you such a workout and then you repeat what you said.
11:00:16 Okay.
11:00:17 Oh, I was referring to your slide number nine, where you want to show it, where you showed a quite complex relationship that should be satisfied by the values diffuse etc.
11:00:34 to to have the chance to observe and at age transition. And I was wondering, although it's not quite intuitive.
11:00:40 I think the way it's formulated yes exactly this one.
11:00:44 Just the.
11:00:44 On slide nine, actually.
11:00:52 Yeah, this is nine. Nine. Exactly, yes this one. I was wondering whether it's my brings some insight in the way of looking at the experimental observation on the transition and whether it could give some ideas on the way to look for an H transition.
11:01:10 And have you looked into this more more deeply.
11:01:15 We are. Yeah, and opportunities, which I'm not crazy again. But I don't know how as Pat said that the conclusion is as good as rulers and if if suppression factor is correct and if the abundance of bliss gradients is more or less accurate then, then this,
11:01:50 this is basically an exact exact sequence of integrations. It's there is no no approximation here.
11:01:56 Unless of course we made some mistake but I don't think so rejected.
11:02:01 Yes.
11:02:03 You can probably if you can measure
11:02:07 the transplant patients.
11:02:10 Then we can constrain the region and to this two dimensional parameters space, which is so that, what matters is that ratios, which, which I think it's quite natural.
11:02:22 And you can concern the boundary where you couldn't possibly have any relations, you shouldn't then, where you can write specific result. Yes. Yeah, I think it's possible.
11:02:40 Okay, but we didn't come up is any recommendation as I remember it was a last minute extension of what we did for this paper.
11:02:57 And it was a little bit clumsy I mean in terms of algebra
11:03:03 mourn for expression.
11:03:14 The duration that we sort of distilled from that
11:03:21 relation is is here, but it's still quite in love with a normal
11:03:24 No.
11:03:27 Well, thank you very much. I'm sure we'll have the last words as comments from that.
11:03:34 Yeah, I mean, I just wanted to make a few comments.
11:03:39 If you really want to reopen this antique.
11:03:43 This is aimed a bit at Yannick, I mean you want to junk the model. All right, and start over.
11:03:52 In particular, there are many outrageous things in the model that go back to issues of track stability but one of the more serious ones is the constant D and constant Chi right which you know to be complete rubbish right i mean both have scaling the gradients
11:04:09 and scaling is on the threshold and it would be interesting to reformulate the model with say a D as a gyro bone time some power of the gradient minus the threshold and things like that, and likewise for the chi, and maybe some off diagonals I'm not so
11:04:32 sure of that second point is I was saying yesterday when you were revisiting lie I mean the suppression exponent is of paramount. And they mean.
11:04:44 Hinton chose was able by the constant Chi and constant D to get by with the suppression exponent of one right he had enough powers in the denominator to win.
11:04:59 Right, but that of course if you start putting the proper dependence in the chi and be that isn't going to work.
11:05:09 Third, face the, the nasty details of the particle source right the particle source is going to evolve as you go to the condition for the transition.
11:05:22 And that, you know, that's sort of what I was saying that the Maxwell rule then is linked to the structure, the detail structure of the source and you know fueling to be to be important.