10:28:53 Excellent. So our last speaker is Neil balmforth, if you can share your screen there, Neal.
10:29:03 Great, while he's doing that I'll just say I so Neil is a professor in the Department of Mathematics at university British Columbia.
10:29:09 He has quite broad interests in astrophysics chaos dynamical systems fluid mechanics, in particular, with applications into physical and non Newtonian fluid, we've already heard some speakers in the last few weeks talk about one of his ballsy theories.
10:29:25 And it looks like he's going to deliver another one today. Go ahead, Neil.
10:29:28 Okay, thank you. Bruce Good morning everybody. this morning. Yeah.
10:29:34 Yes, so I was asked to talk a little bit about this model so it's something of a retro talk.
10:29:42 It's either dated in some ways, and everybody can hear me right.
10:29:48 Okay.
10:29:49 Okay, so I can first of all point out that, you know, this model is mostly due to build young and calling it ballsy is the wrong way around maybe we should call it why slab.
10:30:04 Let me set the scene a little. I'll talk about a paper by man for a young that's on rotating share flow.
10:30:15 This is connected to what people have already talked about in this workshop about the formation and evolution of jets. So, the paper by Bill here is about a shift flow instability.
10:30:31 On The Beat to play and then he starts with Qg equation that that those details are not really important. What is important is that there's some bass flow that suffers and instability at some critical rentals number by carefully positioning yourself close
10:30:47 to the onset of that instability, you can perform a symbiotic expansion involving a small parameter thats related to how close you are to the onset.
10:30:57 And there's a multiple scale expansion or long space scales a long time scale.
10:31:06 And your objective is to build a solution for the stream function here and there is an amplitude no not amplitude in the stream function that depends upon the long space and timescales.
10:31:21 And eventually, whoops, will eventually by going through the esoteric expansion you arrive at an evolution equation for this.
10:31:28 A, and by solving it you can then learn something about the dynamics of the system, and in particular the slow evolution of zonal jets on the beater play.
10:31:40 And for this particular equation with certain choices of the parameters that are in it. What you find is that some initially random initial condition.
10:31:51 It grows and saturates into a series of jets what's shown in these series of pictures here.
10:32:16 A snapshots of this a it's actually the derivative of a eater is the long white coordinate here that's north south and the eater derivative of a since this is related to the string function that corresponds to zonal velocity.
10:32:21 So you seeing the zonal velocity snapshots of the zone of velocity, and a series of times, and you seeing the saturation, a two characteristic values one positive one negative, and the solutions look like spikes or layers and.
10:32:40 These correspond to jets. So that's, that's the essence organizers and what it gives you and there's a whole bunch of interpretation of that the main so the lesson that you learn is that you get this inverse cascade because the instability, it creates
10:32:55 a number of jets, which then merged together. And this merger continues in a cascade. And it's not finished by the end of this snapshot in this particular computational solution that.
10:33:06 But the point that I'd like to draw your attention to is that this evolution equation is really the current heavier equation.
10:33:14 So the current elite equation. That's the model that's been previously written down for face separation problems.
10:33:22 This is the more typical way that you write the current here you equation, the you here is related to the derivative of a, and what the con Hollywood likes to do is if you feed it, and some periodic space domain with some random initial condition or the
10:33:38 There's an instability grows from saturates and the saturation values are close to you being equal to plus or minus one.
10:33:57 However, if you started with a value, there was negative then you had towards minus one of you start with a positive value, you had towards plus one. And there's some competition between how those two different phases separate.
10:34:22 beyond the initial separation is that you get this network of layers. So you see it clearly here you go from plus one, two minus when you get the sequence of layers and on somewhat longer times what happens is that these layers, interact, the steps between
10:34:23 the legs that sort of drift together, and they collide and annihilate, and you get this course of being of the pattern, you can establish for the current helium equation that that must happen.
10:34:35 It's got some variation or structure. In fact, if you take the equation, and you define a new variable thigh. According to this.
10:34:45 And I can multiply this equation by fine, do a space integral, and I can establish this quantity.
10:34:52 v.
10:34:54 As a time derivative this negative.
10:34:57 So this integral here as to decrease in time.
10:35:02 And that's equivalent to that being illiac or not functional for this equation. So what this essentially says is that V has to continually decrease, decrease in time.
10:35:11 It's actually bounded from below, so it has to stop at some point. And when it stops that corresponds to a steady state. So this functional establishes that you must proceed to a steady state of the system, a minimum of this V.
10:35:25 And you can further establish that all the possible men available all the possible steady states, there's only one that's stable, and that's the one with the largest possible spatial scale.
10:35:38 So that means that you have to go through this uninterrupted coarsening of all of the interfaces until you basically get only to as at the end of the simulation here, and you stuck with that because you have to conserve the integral of you.
10:35:54 Okay, so you can do a little bit more with the current elite equation, you can look for a more detailed description of how the interfaces interact. So, the solution.
10:36:11 You can break down into a sub over all of the interfaces, and each interface and the current model is actually attach these tangents here are solutions individual solutions of the, of the steady current helium equation.
10:36:27 So what you can do is you can pose this kind of answers to say that I have a string of them, because this is a non linear equation a linear superposition doesn't quite work.
10:36:38 So there's some error, some remainder time.
10:36:41 And you can pose this as an asset for the current helium equation, because the steps have positions, those positions depend upon time what you're actually after are equations of motion for the interface positions, which you can extract by going through
10:37:03 the procedure of writing down in the equation for the remainder, which is given down here. There's this awkward nonlinear term, which comes from the fact that this is a linear superposition but it's a nonlinear equation.
10:37:16 So you get these nonlinear overlap terms. Right. Those correspond to the fact that the steps sort of overlap one another and where they overlap, that means it's this is in each individual step is not quite a solution.
10:37:28 So this isn't an interaction term. And then you've got the residual time dependence. So by making sure that you can solve this equation for Rie demanding certain solve ability conditions.
10:37:40 That's where you can extract the equations of motion for these steps. And if you go through some algebra.
10:37:48 It's a little bit complicated because there's some conservation law in the background that I don't really want to talk about, but you get this equation for the positions of the interface.
10:37:57 And up here.
10:37:58 We have a solution of the PDP next word solutions of these audience.
10:38:16 The ODS they work quite well at late times in early times, there's a bit of an ambiguity about what you mean by T is equal to zero, because there's this initial adjustment phase at the beginning where things are not really layered.
10:38:20 And then also when you set out at the beginning here the layers are rather close together. And this theory relies upon the layers, sorry the interfaces are close, close together, and this theory relies upon the interface is being widely separated.
10:38:35 So the theory is not particularly good at the beginning and so it doesn't agree to well there anyway so that's what you can do with the current helium equation.
10:38:44 And it's really that sort of setting that we were originally thinking this model for stratified turbulent interfaces.
10:38:54 In terms of. So this is this is that why slab model.
10:38:58 And as I said, the sort of physical setting is a stratified fluid in which you're staring backwards and forwards with some rod or series rods. It's stratified, and what's observed experimentally is that the initially uniform stratification breaks up into
10:39:16 a staircase of well mix layers and sharp police gratified interfaces.
10:39:23 OK, so the model is as follows you just pose evolution equations. The two main variables. There is the boy so your equivalent be the buoyancy gradient and a turbulent kinetic energy density.
10:39:38 And you could argue that what we're indulging in here is sort of full upper body waving it's not just hand-waving. We go through some very simple dimensional analysis, it's mixing land theory of the crudest kind to write down these two evolution equations.
10:39:55 So, in order to do that we need we need some constructive relations in order to understand how B or D and E very, so it will assume that they are defensively transported.
10:40:07 So we need some different activities, and if there's a mixing length, we can then formulate the facilities, from the mixing length and the square root of this turbulent kinetic energy density, and you just need to pay attention to the dimensions of these
10:40:20 two main variables in order to know how to use them. So I'll say that the the buoyancy field that's got a different activity of L times the square root of e and the turbulent kinetic energy is similarly defensively transported.
10:40:36 And I'll just include another scaling constant meter there, the Boise flux here is going to be L route eg g is the buoyancy gradient.
10:40:45 That's an important quantity, the mixing lands. Well, in the physical setting that we're talking about.
10:40:52 There's a stir with a characteristic size D.
10:40:56 But I can also come up with a Glen scale from these two objects, and it's basically just the square root of EOVG. So that's an alternative lens scale, and that lens scale you sort of expect to be more important in the limit of strong stratification where
10:41:11 the stratification is just suppressing vertical emotion.
10:41:15 So, is a simple bridge between D and this alternative land scale, we invented that mixing lengthier there's another scaling parameter in there gamma.
10:41:27 So that's what all you need for the Boise equation is simply transport a buoyancy. In this diffuse of fashion for the turbulent kinetic energy density, then you need a little bit more because there's going to be some production and dissipation.
10:41:45 energy. And also there's going to be conversion to kinetic energy from potential energy that's fairly simple that's just related to the buoyancy flex if I say I forgot to make that upper case.
10:41:57 So the production and dissipation of the dissipation right that's sort of typical and in kind of simple turbulence models you just formulate the decay rate based upon your mixing lengthen your energy scale.
10:42:13 And there is a sort of subtlety involving what you've put in for the production rate for this stirred stratified fluid and and that produces a bunch of arguments that were made in this particular paper for this energy production right, it's sort of saying
10:42:30 that the editor kinetic energy that adjusts to the energy of the stir on a particular Eddie turn over time.
10:42:38 l mixing plant that's a typical Eddie's eyes right.
10:42:41 Okay, so that's sort of the crude physical basis, the dimension was equations that come out, I'll just skip straight to those and their quotas at the bottom.
10:42:53 So here's the buoyancy flux.
10:42:56 This is the mixing length that's this thing. So the formula look a bit clearer because we've got rid of a number of the dimensional continents. So I've got the evolution of the buoyancy.
10:43:23 For the Boise gradient as it's written here, and this is the turbulent kinetic energy equation. And once you scale things there are only two dimension less parameters left over.
10:43:13 One is this beta, which is just the difference in the 70s. I'm not going to attach any significance to that in fact I'll set it to one.
10:43:27 And then there's another one which is related to two of these other scaling constants. And that one. I'll call it siloed, it will be small.
10:43:37 And you'll see why in a second.
10:43:39 Okay, so that's the model. Let's see what it can do.
10:43:44 Well, the first thing I can do is, is I can look at spatially homogeneous states, those where I have G and E being constants, and they're connected by the fact that in the energy equation.
10:43:59 Yeah, then the generation and destruction of kinetic energy must be in balance.
10:44:07 And, and so everything's parameters in terms of each one of these. So, if we choose GG zero is it spatially homogeneous state that we're referring to easier, is dependent upon that and so it's the flux, and this is the flux gradient relation that you
10:44:23 get from that, and it's got this very characteristic and shape.
10:44:29 I've plotted this for smallest value of epsilon.
10:44:36 And if you look at the equation then you find that the end shape here it's got extreme or that roughly aligned with two thirds, and to
10:44:49 the range where you have a negative gradient in this flux gradient relation, you can establish by a simple linear stability theory that the state is spatially homogeneous state is expected to be unstable.
10:45:05 Now, what you typically observe in the physical scenario in the, in the experiments, is that you start with this initial stratification you start mixing it, and it breaks up into a staircase and the steps the interfaces are strongly stratified their high
10:45:25 buoyancy gradient, and then you have these well mixed layers where the Boise gradient is relatively small. So in this particular model that chorus whoops, that corresponds to a choice for a Cylon which is small, because on the left side, where you have
10:45:42 a stable gradient. That's this parameter this dimension was variable g being or the one that corresponds to a well mixed layer. And on the right where you have another stable gradient that's an interface.
10:45:58 That's a sharp interface, if epsilon is small, which means that the values for G over here become very large and this is silent to the minus one half.
10:46:11 Okay.
10:46:13 So you have this this typical end shape for the flux grading relation you expect instability from linear stability theory over this window.
10:46:22 Let's just connect everything with the con Hillier problem. So it turns out that if I make up silent a little bit larger. This window of unstable gradients, it becomes thinner and thinner and it eventually goes away at a specific value of a Cylon, and
10:46:37 if you do a little bit more, as some topic magic in the vicinity of that point, you can reduce the model to the equation. So we're sort of expecting something that similar to current period.
10:46:49 Yeah.
10:46:51 If I stick with the smallest value of silence and soul of the, the reduced model numerically. And what you find is that if I start with an initial gradient that somewhere over the unstable range, the instability kicks in.
10:47:06 And what happens as illustrated by this solution in the bottom right corner there which is taken from the original paper is that you generate spatially in homogeneous states and pictures a little bit small, but what happens is that you get these little
10:47:23 spikes around a relatively flat into spike state right so that that's the buoyancy gradient so that corresponds to well mixed legs with spikes which I gradients in between.
10:47:36 So, for the buoyancy that would be all the density that would be a staircase.
10:47:42 Okay.
10:47:43 And and because we've chosen up silence small and we emphasize the spikes a high so you get high gradients in the interface.
10:47:53 But okay, so if you choose certain boundary conditions which are basically no flux conditions on the buoyancy gradient and he then the boundaries themselves they don't do very much and you conserve the integral of g for all time, and the staircase can
10:48:11 be long lived there are interactions between the steps the interfaces that we'll talk about a little bit more.
10:48:20 There was also a distracting effect if you have no flux conditions on the density on be not G, because then what happens is that you get edge less that kind of erode into the staircase for the sides and then ultimately everything runs down, so I don't
10:48:34 want to talk about that I prefer to talk about the internal dynamics of the staircase. So I'm going to use these somewhat unrealistic boundary conditions where things are not going to be eroded from the size.
10:48:47 Okay, so here we go. So dynamics.
10:48:51 So this is looking in a little bit more detail about what happens.
10:48:55 So I've got a solution for G here. So you see the initial state random perturbations that grow up with some characteristic scale into the spiky nonlinear states, what then happens is that you're going to more complicated dynamics in which these spikes,
10:49:13 they slowly drift together and merge and then they grow in amplitude once they do that. So that's illustrated in this little panel here, where you see to spikes, the drift together and merge into a bigger spike.
10:49:26 That only happens when the initial spikes are somewhat small when they get bigger, what happens is that the peak is approaching the unstable gradient on the other side here.
10:49:38 And so the peak of the spy custom sort of bottom out that he might say.
10:49:46 So maximized and instead of growing the spike size went to bigger size spikes merge they actually get fatter, as illustrated in this particular case. So to begin with, spikes merging get larger later biggest banks emerging get fatter.
10:50:04 And you see that in this series of images here. He's a spacetime diagrams buoyancy gradient the kinetic energy density and the flux is a little, maybe a chick.
10:50:18 So you can read that while I'm just describing one other important feature. So, the longer time dynamics of the spikes is to come together and merge so you get something like the course name of the current Elliot equation.
10:50:34 And one interesting thing about the whole process is that after the initial transition, the buoyancy flux basically becomes constant it saturates in this deep red here there are these little intervals where there's a sudden sharp adjustment in flux that
10:50:53 corresponds to a merger of the spikes, but otherwise everything operates at roughly constant buoyancy blocks.
10:51:00 Okay, so you can kind of look at the zoo ology spikes, a little bit in a bit more detail because of that. Here are some simple spikes that you can come up with the buoyancy gradients down here the kinetic energy there, the flux at the top.
10:51:15 So what what happens is that you can get these quality steady states where they're low amplitude spikes to begin with and as they get higher than they saturate the larger amplitude, stable gradient.
10:51:29 All the while, you know they do this with roughly constant flux.
10:51:35 It's only these larger amplitude fattest states that participate in what you would call current Hillier like dynamics.
10:51:46 For the staircase, you get this merger dynamics of these smaller spikes, where they could grow in size, rather than become fatter. So these states you can construct them explicit very explicitly by looking at quality steady constant flux solutions in
10:52:02 the model equations. So, throwing away the time derivatives means you don't care about the bonus equation just tells you the flux is constant, so that they boil down to these relations, this is the prescription for the mixing lands.
10:52:18 This is the buoyancy blocks and that's the energy equation that kinetic energy equation, which takes form of some sort of odd looking nonlinear oscillator, so I can integrate this, and I can write the thing down in terms of some potential here, which
10:52:31 depends upon either he or G, they're related through this.
10:52:36 I've written it in terms of E here.
10:52:40 Flux appears as a parameter, and I just arranged the integration constant there, so that you have some overall constant shift, and in the plots here what I've done is I've set the overall constant, constant in you to be such that the extreme or that you're
10:53:04 getting this potential that larger kinetic energy or smaller Geez, that has zero, you and all of these solutions. Right. For larger, he, it's heading towards, especially homogeneous state where the ed is at zero.
10:53:24 So these states or correspond to this kind of a potential.
10:53:29 And these solutions they basically correspond to excursions from this extreme or to the left and back again in G it's the other way around.
10:53:39 Right, so that we've been extremely that corresponds to allow an extreme that corresponds to an interface. And the fact that we have to have in G a relatively small value in the layer and a relatively high value in the interface.
10:53:55 What that means is that this potential must be tuned by selecting the flux, so that the excursion from the extreme of the leg is almost to the origin of eight.
10:54:07 And that basically tells you the F has to be close to this particular value which indeed it is. So this is how the system selecting itself by creating the flux of the system is being selected in order to create the staircase.
10:54:23 Okay.
10:54:24 Alright, so I'll just finish.
10:54:28 So the end shape or s shaped curve that you get here. You know what, that's a way of describing instabilities and patterns and many many systems. It was the Lh mode that was talked about this morning.
10:54:40 There are many, many other examples of it reaction diffusion models neuron models cataclysmic variables and astrophysics soft matter, people have been talking about things set up this way.
10:54:52 Place your surges and geophysics.
10:54:55 There's a whole bunch of things that want to talk about Yeah.
10:55:00 The steps themselves are not diffuse if you need of some kind of a turbulent motion in the steps.
10:55:05 The model avoids and ultraviolet catastrophe as in the original Phillips has meant to model.
10:55:12 There's an initial spacing that's controlled by the most unstable linear mode, the latest beta spaces are controlled by how much time you wait because of this course me.
10:55:22 And you might worry about some of the mathematical details of that.
10:55:27 I really like to draw attention to the statement here though, which is that the con Hilliard equation tells you that these interactions, take place on exponentially long times and that's clearly true for this, why slab model.
10:55:55 And that's very delicate. It's dead easy for perturbations to the system to destroy those exponentially long interactions in the con Elliot equation. It's known that if you just add random perturbations and you stop the interface coarsening altogether
10:56:01 and the interfaces end up walking around randomly.
10:56:05 And okay, I should leave it there. Thank you.
10:56:09 Wonderful, thank you very much Neil agenda, I'm going to stop asking a question it's kind of half baked it, I'm thinking of staircases in the Arctic Ocean, for example, there are examples there of merger between the interfaces, based on what Mary Louise
10:56:35 has observed. I'm kind of wondering where the merger comes in in your model that might be MIT are missing or too much to add to say an ocean model.
10:56:40 Well I think part of that is this comment that I made at the end which is, you know, the way that you think about emerges in the con helium equation is that you've got these coherent objects, they're like particles, and they sort of talk to each other
10:56:54 by their exponentially small tales. That's how they know about each other and that's how they know to guide themselves together. And if you've got perturbations that just wash out the tails then that's not going to happen.
10:57:06 So it could be that there's just a lot of background noise that washes everything out. And I've also said at the end it is this conceptual model at all relevant.
10:57:19 I've also said at the end it is this conceptual model at all relevant. I, you know, it was a while ago that we did this model and at the time we were trying to sort of come up with a vision of the dynamic seen in the experiments is there sort of a reaction
10:57:31 diffusion type problem.
10:57:33 And and we came up with it is very crude mixing lens type model that that has some of the dynamics that you seem to see.
10:57:43 And now, You know, you might wonder where they this model can be, you know, tested against experiments and it's not clear whether it would survive, given that it's so crude, but you know the dynamics of layers.
10:57:58 That's one of the things that you could look at.
10:58:02 So I do have a couple of extra slides here.
10:58:05 Well, they were actually motivated by something that Paul Linden said they were having a meeting at the time, Bill gave a talk on this model, and other of these rhetoric talks, and it was criticized by Paul at the time because some features of the dynamics
10:58:20 and one was, was the typical spacing between the interfaces. Another one was the instantaneous relationship between the flux and the gradient, I don't think those are that serious but the way that interactions happen in the model is a little bit dodgy.
10:58:41 So, this is typical where you have a small one and a fat one, and they come together, you know, they gravitate towards one another and then merge, whereas if you look at the experiments, a lot of the time what you find is that there's something different
10:58:54 than happens, which is. So for example, in this experiment, this is by hopefully in London.
10:59:12 Similarly, see things like that in the original paper by Park at all. Actually, they do show some instances where layers do seem to come together and merge, as in this way slab model.
10:59:20 But, one does wonder whether there is another kind of interaction that characterizes interfaces of staircases in these contexts, that's a bit of a problem because that's a qualitative issue with the model actually Kimbo thought about this and pointed
10:59:39 out that there were two types of merger modes, or two merger scenarios that were possible in these sorts of models, and he actually solve this ballsy model.
10:59:54 And here is in fact the solutions the Boise model where this type of alternative dynamic happens.
10:59:59 So you don't see the layers coming together. It's just a one of them heroes at the expense of the other one.
11:00:05 So, there is actually a bit more to be understood from the mathematical perspective and the Interaction Theory for this, this model, it's not simply can't hear the equation dynamics.
11:00:17 You know, the layers themselves you know they could be more like spikes and they they sort of have variable mass and they can create mass. That's a bit more like solid songs, although the way that the objects interact with one another, a bit different.
11:00:31 But anyway, I think all of this, it's not completely clear whether this conceptual model is really going to tell you that much at the end of the day about the physical experiment is.
11:00:44 It's sort of one of the guys from frustrations that I have about this whole thing.
11:00:50 Anyway, that was probably a much more long winded answer to your question Bruce, that was guided by the fact that I actually had some exercise.
11:00:57 It's certainly thought provoking we do have a number of questions so let's see if we can get everyone. The four people on board meeting with Pascal, and Neil thanks for your talk, and I was wondering from.
11:01:11 If I understood correctly they're both in the chameleon model and in the y slab model. It appears that mergers or are almost inevitable with the exception of that last example you gave from Timor's modeling of that where the interface just disappears
11:01:29 right but so do you think there's any modification that can be made to get to a stable staircase I mean what's missing in these models to get to a stable staircase.
11:01:42 Well, what we have is numerical evidence that you get uninterrupted coarsening for current hell you know that has to happen, there's, you know there's a mathematical statement that establishes muster.
11:01:55 For this model, there is no obvious, the app enough functional, we at least we didn't find one.
11:02:01 And so there's no mathematical indication that it has to continue to an ultimate state of that kind.
11:02:10 I mean it could be that it's simply because of the parameter settings and you're in some limit.
11:02:15 I so I don't know, I, you know, we haven't really explored all of the parameter ranges and, and, you know, there are some constituent relations that have been hardwired into the model like the form to the mixing length.
11:02:31 You know, maybe somehow what the model is doing is contingent upon what those functional forms were.
11:02:38 And so have you played around with those maybe the staircase could survive.
11:02:49 Okay, so as we move on to David do the quick question.
11:02:53 I yeah hi Neil, and a couple of questions so when the thickness of the spikes.
11:03:03 They look sort of almost like boundary layer structures is that just what is that dependent on from your parameters just from your epsilon. That's right.
11:03:11 Yep silence small guarantees that those things is a narrow.
11:03:16 So can you can you solve it through some sort of. Yeah.
11:03:20 Yeah. Yes, I believe you can have you have you solved it.
11:03:26 I've been trying for the last couple of days.
11:03:29 So you can kind of think about these constant flux steady solutions and. And here, here is basically up against Jesus said, and these are different spikes.
11:03:41 And there's sort of a region.
11:03:44 In the tail down there, and a region in the spike here. And there's, there's some way you can come up with, you know, based upon the smallest of upside when you can come up with a description of that, then the next step if you want to do understand how
11:03:56 they're interacting and so forth is to try and feed them into this and sets for a train of them, and to try and come up with some Interaction Theory. Now, one of the interesting features is that you've got the solutions that have different attitudes,
11:04:10 they have different attitudes here until they start to saturate. So it's really a two variable theory, not only do you have a position but you've got an amplitude.
11:04:18 And so the kind of the same topic description involves differential equations for both position and amplitude.
11:04:26 And, yeah, so I haven't quite got any further than this.
11:04:41 But there is probably some, some awesome topic theory based upon the smallest of siloed that can guide you to build solutions and look at how they interact.
11:04:41 And these nonlinear steady solutions that you found through solving that equation Have you looked at the stability of that.
11:04:51 No, no.
11:04:54 These are just study solutions you build them and then you throw them in here and you assume that there is no faster instability.
11:05:00 So, so if you really wanted to establish that there was no other instabilities other than the ones that are captured by the slow drifting together or the slow adjustments.
11:05:15 And the amplitude, then you have to do a little bit more. Alright, so there are a number of their questions. And since we're near the end of the session we, I would be nice to have an opportunity to get have everyone asked them, but for those who don't
11:05:23 have to leave sooner than later, I just want to remind people that there are new to new posters which are in gathered town. One is on the gas giant zonal jets and laboratory experiments, as with def definitely the muskie W and become about.
11:05:39 And another one on horizontally averaged.
11:05:47 Double diffusion convection by data choose and simpler. So you can see them and gather tell. So I just want to make that announcement and also for those people who have to leave shortly reminder that on Tuesday the talks will begin at 8am PST an hour
11:05:58 earlier than they did today.
11:06:00 So with that, those announcements made let's continue to some questions. First from Pat.
11:06:11 Two very quick questions. Thanks for the talk. Have you ever considered the, the ball z problem with a net flux, say a boy of buoyancy or been put it more simply matter across the system.
11:06:27 Now, it might be interesting I mean we've done analogous things in the raw speed problem and of course you then you get global by four occasions and it would be interesting to compare notes.
11:06:36 Second question in these con Hillier now VA Stokes things which, you know, it seemed become very similar and in structure one thing that pops out at you, which we didn't hear about the other day was the hinge scale, right where you have a balance of the
11:06:52 surface tension of the droplets with the straining field of the fluid which the fluid being forced or noise imparted.
11:07:04 What role does that play in setting the droplet size. Do you think in, in, you know, something like since you're basically forcing the fluid. Is that what setting the basics size of the element in the balls the story.
11:07:23 Yeah, so the you know the field equations here and not they're not couple to any, any novice Stokes dynamics.
11:07:33 I mean, you do have a fluid UFTKE right and you have you have forcing in a cheap sense that's true.
11:07:42 Right, so you know you, you've got this GL code and somehow.
11:07:46 Well, I guess, in those descriptions of multi-phase fluids, what you do is, is that you you have a field, a color field that satisfies the con helium equation.
11:08:00 And that's that's a way of replacing shop interface right and then to surface tension because based upon what you looks like.
11:08:15 I guess that there is some analogy between, you know, g being a con Hillier like field and he being a novice folks equation. I guess I guess I it's, it's hard for me to sort of bridge the gap between those descriptions or multi-phase fluids and this model,
11:08:33 and the like I cannot, I can answer what sets the initial scale of the staircase here, that's just in linear instability. Right, okay.
11:08:46 I think you need, what would what we would need would be a some kind of feedback term in the fluid equation and then you'd end up with total kinetic energy.
11:08:58 Something like total surface tension energy or droplet energy and and the buoyancy and then, then you might connect the two, but that's just a speculation
11:09:12 about we move on to the next question from Alan, with regard a couple of questions, you, you raise.
11:09:21 As far as does the layer broadening continue do stochastic fluctuations.
11:09:30 Dissolve our road the interfaces. You also were discussing specific context of the Arctic regime that we've seen earlier in the program.
11:09:47 A coworker of mine, ran a one year simulation of the Arctic staircase of 15, you know under Arctic conditions, more or less, as we have seen, so, so it's a 50 metres staircase result five decades down to the bachelor scale of the slower diffuser.
11:10:03 and it's a stochastic models so it involves stochastic fluctuations.
11:10:08 The simulation was completed. So ran over one year, and it so it does give an impression that the, although the fluctuations, do cause some dissolving interfaces by and large the staircase is robust under the fluctuations and slowly but surely the general
11:10:31 tendency is for the layers to to broaden and I'd be happy if this opportunity any point to run that simulation, run the movie for anyone who's interested if there's a way to do that.
11:10:46 I think it's remarkable as Arctic observations, the, the resilience of the legs over kilometers in the horizontal is just incredible.
11:10:57 Anyway.
11:11:17 Yeah, I'm talking about a 1d simulation so it only says this is reduced modeling so we're only looking at vertical
11:11:09 or horizontal you know it's a separate point I just, I've always constantly amazed by Mary Louise his observations.
11:11:17 Yeah. So, anyway, I'd been the simulations just a week old so I'd be very interested in any point of someone who really has been looking over time at the Arctic data to to just render an opinion as to whether this.
11:11:35 The movie does it does not look much like the Arctic phenomenology
11:11:42 like the Arctic phenomenology maybe that's something you can bring into one of the working groups Ellen.
11:11:46 Yeah, in fact there was thinking of doing that.
11:11:48 Very good. So we'll take one, One more question from Edgar Please go ahead.
11:11:58 Edgar I think your microphone is off. Yeah, sure.
11:12:02 So Neil thanks very much for a very nice talk I just wanted to make a brief comment and that is that another indication of the sensitivity that you mentioned at the end is the possibility that the profiles of your spikes develop oscillations in their
11:12:20 tails.
11:12:20 And that would of course arrest.
11:12:23 You know, the, the continuous coarsening process that you described.
11:12:28 And that could be a function of the, of the parameters of, you know, an extended version of your model. That's right.
11:12:35 Yeah, if you enrich the model you might be able to get facility details and then everything would just get stuck in place.
11:12:46 Very good. Well, let's think Neil and all the speakers from today.
11:12:51 I'll just find my little reaction button here, giving you a thumbs up as well as a clapper ama.
11:13:00 I remind everyone that there's posters and gather town, and we meet again on Tuesday for another series of four talks, beginning at 8am PST there after the working groups will begin.
11:13:12 Take care. All right.