08:03:21 Okay, I think it's probably
08:03:27 time to start so welcome everyone to today's mechanisms discussion group.
08:03:34 So this is, this is the batting order for today.
08:03:39 So we're going to be discussing what you might call caps on models, blind models or beyond the blind model.
08:03:47 And we've got three talks, wasting grow.
08:03:52 Pat diamond and pushing that from needs.
08:03:56 So we're going to kick off with way Shin so we're very pleased, its way into the second day running is giving a talk after midnight time which is very good over, so thank you.
08:04:08 So I'll stop sharing this, and you can you can start whenever you're ready.
08:04:36 Okay.
08:04:37 Thanks.
08:04:38 Yeah, it's a great honor to be here again. Yeah.
08:04:44 Lack damage said, since I have gave this talk yesterday, so I will repeat some slides, but still transfer few differences. And so today I'm going to talk about which is the most important and how does the multiple sharing work together for the staircase
08:05:05 formation.
08:05:07 And the.
08:05:08 Sorry went moment on a to choose on the most falwell for by turn. Yeah. And for the background patch. I also to want to introduce a little bit about the selection and feedback loops for the formation, our layering and patterns like zone of law and staircase
08:05:30 in the mathematical confinement system. And then I will move to introduce our recent result of the model study of the staircase permission sustainment.
08:05:42 Finally it's a conclusion. So, this, I think all of you have heard a lot about the various Larian patterns, and for mathematical fields and zona flow it's quite important, as a God.
08:05:58 He also hard he because as regulator, they drift away with transport and him, improve the confinement, and also close the feedback loops, without introducing any additional transport, and the other is the e commerce space staircase in the system and the
08:06:18 In this system, and the equal space there's case, showing here by The Cosby Show Rachel, always Tam, and with the production. It's very similar to the previous staircase and optimistic jack.
08:06:32 Yes, sir.
08:06:33 But here, will want to stress that there's very clear skies lexing for the formation of this pattern, and to quiet different regions I exist to the relative relatively narrow, I mean the relatively narrow path a steep raging cards they lack of cut the
08:06:55 jumps in the staircase structures, where we see it every day, I think, and the other raging is the relatively stable batter batter wide relatively flight, but wider origin harder stamps and their ski.
08:07:16 I mean that's krY samples and the steps are quite different, actually. So, when it you have a better understand of this case lacks in process to understand the formation mechanism of the staircase.
08:07:32 And so we need to mention the mixing lounge, because it's Clark he has a feedback loops.
08:07:38 And the key point here is a dynamic skill, because it's Uranus the gradient of our systems function of our gradient of the systems.
08:08:01 Typically, Ryan skill is very.
08:07:56 It's a very typical mayor of the dynamical scale and because it involves inversely proportional to the PVA gradient.
08:08:03 If we take the right skill as a dynamic skill, and we have the mixed allowance to express the lack of days.
08:08:09 So we can clearly see that if that's the scale of the systems, and smaller than the rain scale and it will correspond to the short memory and the idea, it like turbulence and this dramatic scene, as was a correspondent chill low memory and the web lacquer
08:08:30 turbulence and wake mixing. So, that was those Background The last come to the radios the model your yos the hair. And the final model currencies to three equations the evolution, our mainstays and evolution of mean what a stay and evolution of turbulent
08:08:48 potential as topic.
08:08:50 And, besides the state of flux so for those equations that way have the conditional for the particle.
08:08:59 I mean for the density of losing and the radio of flax for the World History evolution and the traditional component.
08:09:10 But for the potential as turbulent potential as a way.
08:09:16 Besides, they diffuse a component. We also have some couple in terms, and some displays in terms and production terms.
08:09:25 So, I mean, the key point here is that all those transport coefficient depend on the mixing Lancer. So if we take the right scale as the example of the dynamic skills, where we have this kind of S curve figure.
08:09:46 So as the indicators as they buy stable mixing in this system. And if we have some net positive feedback loops in this region, lack if we're Steve who the names of a gradient, and it will drop some in the potential as target and the further job of the
08:10:06 mixing last and the further steeple of the PV gradient. So, if this positive feedback loops holds true in this region, and it will finally drive some pattern, and this to some nonlinear features in the main profile.
08:10:23 So let's see the result. So the density staircase, and what his take is has been observed from this model action here. So, it indicates that staircase is a consequence of a modulation or by stability feedback and existence as a as a staircase is very
08:10:53 to the software, because you can see that the shell curve is zero they formed due to some self deepening our lives. And for the further, applying of this radio star model on we have some studies of job is to
08:11:03 have this pet of this pattern. I mean, we scan some dimension is parameters like Reynolds number, some dimension is our fan of a number, which way to distinguish the aerobatic and hydrogen dynamic name each here we we chose the alphabetical image, and
08:11:23 number, and is the ratio between the drive and the discussions and have some figures in the, in, in the published papers and released a little bit of them, and lecture here, if we are enhanced the collision not to save it is a staircase pattern is weakened.
08:11:44 So that means they decrease our plant number, we awaken our temple the staircase. And also if we decrease the Reynolds number, we can observe this similar dumpy of the pattern.
08:12:01 Otherwise, if we enhance the drive, and the staircase pattern is also enhanced. So that's some parametric scans of this pattern.
08:12:15 And first of all, we want to make a to now what is the main feedback loop for the formation mechanism.
08:12:25 And one candidate is the, what is the alpha, because the feedback loops can work through by the theory of independent comics announcer.
08:12:35 And the other candidate candidates are the density gradient and what is the gradient. So, they have a three candidates here so we want to know which mechanism is the K mechanism.
08:12:48 So, way first today we kept the cars Vissarion in the system and surprised late, we found that that there's no staircase, per se, for a long time.
08:13:05 But if we take the right skill, like in the published the paper in the system, the right skill, liquids are revealed include the boss the density of engagement and the world has degraded and if we have rain skill, we have we will have the staircase, and
08:13:18 some three stages that will clearly show and by this fever. I mean the density gradient versus position and Tam.
08:13:36 So, but there's two candidates exists here so anyway further part of the artistic gradient, was to have the own own staircase. In this system, so that means it's the Penn State gradient job playing our payroll of the formation of the staircase here.
08:13:58 So that means they feedback through the nonlinear gradient to travel I means a great density gradient of mixing is the Caleb.
08:14:07 So this result is in some, in some sense, it's qualitatively currencies that with some external experiment observations in somewhat typical talking about Mercy is in China.
08:14:20 I mean, like it out to a.
08:14:23 And first of all, we also studied the sensitivity of this pattern to some initial gradient to trial lecture here, If we raise the initial gradient.
08:14:38 We can see that the number of the stamps that will increase.
08:14:42 And it's because they free energy is enhanced to the SS term, and the corresponding lays the height of the system will will decrease the with the, with the, with the increased our initial density gradient, but quite a, quite an interesting result is here,
08:15:03 if we further enhanced the initial desta gradient here, the number of the stamps that will decreased will be a decrease the and they had his cross boundary only increased.
08:15:16 And this result is because there exists some balance between the free energy, and the dissertation process though so that the Finally, we kind of that there's a staircase pattern as because they do these patients can play a role.
08:15:38 So, and also here as clear ratios. If we take some proper reach initial gradient tool we have some at minimum walls tab scale of this pattern, and this mean normal.
08:15:54 Step skill, physically is determined by the balance between these two process.
08:16:02 And another interesting result is here.
08:16:08 We also find some nice uniforms structures like, because in the luxury here in the different radio stations, the size and the height of the stamps will be different.
08:16:19 So, this net uniform structures is quite interesting but the reasons we can talk talk more later page.
08:16:28 In the later pages.
08:16:31 And last but not least, we also studied some interactions between zone on zonal share and Michelle's.
08:16:38 Yeah, Max you on here as a first just that way fixed zona sharp at Verizon Meteor way, I can see that.
08:16:48 If we enhance the measures further the staircase is wakened on the event destroyed.
08:17:01 So that means the patterns retains some memory of initial knowledge, initial validation.
08:17:12 And if. Otherwise, if we fix them easier but viruses on on Shall we actually have the same results. I mean, they are stronger so no shell will awaken or even destroys the patterns.
08:17:25 So, that means, both the zonal and the media will affect the formation of the staircase. So, way as. So when it to care more about the about the effects of this area.
08:17:49 But in this model. The in the consideration of the media is not so how does a consistent play with the model because if we do need if you want you to study further of the effects of media, actually this.
08:18:12 model so maybe that, that's our next step for the walker to have.
08:18:12 How to say, better extent of this model.
08:18:30 And so far the nine uniform structures, we find that some part of the reason is because we found that when they zone our enemies are comparable this now uniform structures is appeared.
08:18:33 So, this is part of the reasons but more reasons is still a need to be explored further.
08:18:44 So, they can close in our sample here we have some more studies about the existence of a set destiny staircase, and what is the aggregation in the catalog tiny system.
08:18:56 And as a principal feedback loop is through the nonlinear density dependence. And what what has surprised us is the sharing of feedback loop as effective, but the reasons is not so clear for us yet.
08:19:11 And the interaction between zonal amnesia affects the formation and generates some that uniform structures as the pattern can exist at some minimal skill and this minimum of skill and side by the free energy is a disease patients, they carried out some
08:19:28 share. So, one word for the conclusion is that we think the origin of the staircase is very simple, some characteristic existed in the history and this they are formed by the self 70 of the modulation.
08:19:47 So that's our fault. The talk for today. Thanks for your attention.
08:19:54 Okay, Thank you,
08:19:59 Brian.
08:20:02 If you have questions, please raise your hand in the chat.
08:20:08 Oh, that'd be a, please.
08:20:12 I waiting.
08:20:16 Just, just to know, have you tried to to add some noise in the equations to see whether the mechanisms is sensitive to fluctuations, or not at all.
08:20:28 Not yet, actually when I discussed this yesterday I haven't.
08:20:54 Although there's some consideration on with this patients in the model but the noise, I think is not considered Where is a present model would be interesting to know about it.
08:20:54 Just a remark. I'll talk more about that coming up. So, it will be, you will you will do as not as easy as it sounds, but you'll find out.
08:21:05 We shouldn't Can I ask you about that.
08:21:09 The production, the production production of energy to yes that's the one, because in the blind model that's very crucial isn't it, you know, they look at a few different sorts and it all.
08:21:23 That was absolutely crucial on whether they get the, the son shaped flux gradient curve. Can you can you say a little about what you did there.
08:21:33 What you chose, or whether you tried different ones.
08:21:41 The comprising between this and the blind model.
08:21:46 How to say, yeah, here the production term made a.
08:21:55 consider though, I think, for some.
08:22:01 How to say maybe some forcing lack through the, they free energy can act as some production term here, and I think some similar false, false, driven by some free energy it also exists, they may be in that blind model.
08:22:24 Yes, because in this they don't remember them all they had constant power constant force and something else.
08:22:32 One of one of them gave the right flux gradient relation.
08:22:36 And two of them didn't. So it's kind of it's kind of sensitive to that.
08:22:42 Yeah.
08:22:44 Can I can ask another blind comparison question.
08:22:47 Yes.
08:22:49 In the blind model their steak staircases sharper, it seems.
08:22:57 What do you ascribe that to
08:23:06 this.
08:23:12 Sorry, I have.
08:23:15 I don't have very clear mind. For this, not missing for how many of you later. Yeah, yeah, yeah, yeah.
08:23:28 Okay, good, please.
08:23:33 Um, I guess it's a related question to us.
08:23:36 When you come back to you know your your thoughts.
08:23:41 When you use can the regions of the Atlantic City, say that, as density increases, you, you lose the staircase pattern which you could probably read it the other way, meaning that you have less steps and higher heights.
08:23:59 It goes for a strengthened staircase, which in some sense is not, you know, really, as we see it with respect to proximity to marginality etc when you increase the drive you would lose the staircase that and, of course, you didn't have that near, but
08:24:21 that's maybe related to what you chose for the production. So, if you were to, to, to be forcing a different kind of in a different way. Maybe you will have a different behavior as you as can The, the, as your clothes or not to to emotional state and
08:24:44 as you increase the drive so have you investigated this.
08:24:52 Not as as deep as, as you said, but way actually way, compare how to say the balance between the energy and the dispatching here on, so we left, I introduced.
08:25:13 Yeah, I think we think the initial stages of the enhancement of the free energy is, is, as, as playing arcade oh so that's taps and as he has increased and the height is decreased, but later on they feel safe.
08:25:34 Discussions can become stronger, with the enhancement of the free energy. So,
08:25:42 decrease the opposite stamps and the hands of the height appears.
08:25:54 Well, on all of the types of staircases maybe a week one, with several steps so or large one it into a transport barrier with a with a big one. The big says that size and so I'm not sure I understand how dissipation really getting related to dissipation.
08:26:12 I can I make two comments here first of all on what was done to the forcing it was a mock up of the linear growth in the problem. Yes, we want to clarify that.
08:26:24 I mean, it was a little different what from what's done in the original the original Bly and it was to mock up the fact that you had these things are linearly unstable.
08:26:36 And on in terms of your dissipation question i mean you got to be careful right because you have, the more steps you I mean they're perhaps not as big, but there are a lot more of them right so it's a question of what fraction of the volume or the region
08:26:53 is dominated by dissipation would be, I think of something to include in your reasoning.
08:27:05 And if I may have a second really short one. When you said it's, it's a close to experimental observation I guess it's the previous slide, what exactly do you mean by that.
08:27:17 Oh that I think they observed as a modulation process as mating sort of a test a gradient rather than sort of the Sherry,
08:27:28 that they modulating on the turbulence.
08:27:37 Sorry, did you.
08:27:40 Thank you. Well, maybe I guess we need to discuss this. Thank you very much.
08:27:46 Yeah, I can show you some pictures are some result about this directory. Yeah.
08:27:56 Going Good, good.
08:27:59 Yeah, so I'm not an expert in this field by any means. So this question is probably very naive.
08:28:06 But my impression was that you were saying that the staircases are actually moving. That was one of the early slides where the you know the position of the staircase is at two different times.
08:28:22 So are the staircase is moving because they are drifting say from the interior to the outside.
08:28:29 Or is it because there is a kind of standing oscillation, for example.
08:29:01 Yeah, yes. Lucky, lucky you can observe here. They after the mergers, of the awesome nonlinear mass mass of skill structures. They pattern finally mitigated to, to the edge
08:28:57 of the systems.
08:29:00 And it's always regenerated near r equals zero and then propagates outwards like a traveling way, or.
08:29:20 Yeah, I mean it's also amazing, all of them. Yeah.
08:29:26 I think it's, it's, like, like our traveling vibe.
08:29:30 I'm sorry. I have to disagree. All right, we have both under different circumstances, okay Edgar, depending on depends on the boundary conditions and it depends we do cases which are fueled in cases which aren't.
08:29:47 So some cases exhibit. This kind of escalator structure as I like to call it some cases exhibit. This what she's showing here which was really from stuff of short on and me which is that was that's a driven case, I think it was a driven case.
08:30:08 And that makes a content that the the stuff spins to condense at one side and make a steep barrier. And, and a flatter region in the core so depending on how you set it up you can get different outcomes.
08:30:26 Okay.
08:30:27 Okay, thank you for the clarification. And by the way, I mean in some of these things it's of course we use what we like to call sand pile boundary conditions.
08:30:38 In other words, Norman on one side and dear ish lay on the other. Right, so you induce an asymmetry right there. Right.
08:30:50 Okay.
08:30:52 outsiders question. Yeah, please.
08:30:56 Is it at all useful to measure spatial temporal correlation functions and such systems so that's, of no use.
08:31:10 Where haven't Mayor the correlation.
08:31:19 But, how to say
08:31:31 maybe halfway.
08:31:38 Actually there are some compelling terms in the model, but
08:31:50 the How to say maybe by the first apply of considering the media further when you consider the correlations between
08:32:06 IPOs that they some correlation
08:32:11 factors.
08:32:13 Thank you.
08:32:18 Okay, I was gonna ask just a question about which I want to know really games nice picture, so maybe GM should answer really if and how the one we always see yes that's the one.
08:32:33 How state is that very long lived can, if you're still there.
08:32:40 Yeah, I'm sure, but maybe Wishing Well, it depends really on the, on the forcing. If you change the boundary conditions, and you.
08:32:50 In this case, you'd have
08:32:55 thermal best on both sides, so it makes for a very stable system, there are long lived and they remain mostly at the same location, if you're a floods driven.
08:33:07 And the more so if we increase the drive, the more they meander.
08:33:12 The move regional leash, and they may disappear and, but they always reform in the wake of, for instance, an avalanche and possibly at a different location.
08:33:23 So this whole thing has a dynamics.
08:33:27 And if you were also consults below the instability threshold and drive the system through a source central source, you would see the staircase.
08:33:41 First, start on the outer region, because that's where you get first instability, and then progressively move in words as turbulence moves also inwards.
08:33:54 So, it's really, it's very dependent on the, on the profiles that that you that you have and how you foresaw system.
08:34:09 Okay, thank you.
08:34:15 Right.
08:34:17 Anyone else have any questions for wishing.
08:34:23 Okay, well thank you again.
08:34:26 Yes, thank you. All
08:34:32 right, we should move on to our next speaker who is Pat.
08:34:38 All right, can everybody see this.
08:34:42 Yeah.
08:34:44 Alright so, more in the same, so to speak, and you can try some of the questions on me and.
08:34:55 In, instead of ballsy.
08:34:58 I've become quite fond of Bly has as a code word for the paper a bomb for us Llewellyn sniffing young.
08:35:06 It's a little more elegant and opens, different possibilities for tons and so forth. So, anyway, keep that in mind. So, what am I going to say, a little bit of a prologue what really is and is not in the model.
08:35:27 I mean, we presented the basic results but we might talk about, since we're among friends, I hope, where the buried bodies are although in some cases the bodies are sticking up above ground to bed.
08:35:42 Then will you know issues in this.
08:35:45 I talk a little bit about flux driven cases because that's what we're interested in, particularly in the fusion side of things and where to next.
08:35:55 In other words, how do we go beyond the this model that seems to be getting so much attention. So some thoughts on models.
08:36:05 I mean, lie, and it's, you know, knocked up knock ons and all is a model right and we all know the old saying of George box, may be a little bit tired at this point that all models are wrong but some, those are useful Bly is definitely useful, but also
08:36:26 I would say there's another old saying, which I don't know who came up with it I got it second hand to the effect some models are too good to be true.
08:36:37 Other models are too true to be good. And that of course is a cautionary word, it's easy to say all the things left out and Bly but that that's a cautionary word Are you you know about trying to jam too much stuff into this simple model and we'll come
08:37:02 to it, at least on our side there are four papers there's two of our Ashish sure Yvonne and me whenever ash was a postdoc with me.
08:37:08 One with Misha Malka of and me and, which was really the first word that came out a bit later because of disorganization on my part and then the paper with ration and David and I are going to talk about the first and the third more because that's more
08:37:26 what I call the hashtag our walk a tiny drift wave turbulence as opposed to the pure.
08:37:33 You know Qg or beta plane thing it's more interesting and you.
08:37:39 So, first of all, what's in the model I think we you know we were supposed to be communicating with people from other realms here so just so people know this is the basic model before you do anything right i think there's a mistaken impression in this
08:37:56 program that all of plasma is in China, he has to go on MIMO which is a misnomer. Okay, and is. And so the idea is you have separate equations for density and vortices it with parallel and perpendicular diffusion right and you'll notice if you add these
08:38:17 or subtract these two equations up to the perpendicular diffusion, you get a conserved quantity in that conserved quantity is the potential vortices D and minus del squared phi.
08:38:33 Alright so this stuff cancels. and you have a mean and a fluctuation in n and you could have like wise and vortices it and the derivatives or invectives.
08:38:42 This system supports linear instabilities, which by the way has to go Amina does not a bad been some false advertising is.
08:38:53 And these linear instabilities are the infamous drift waves going back to sag day of an all, and they are negative dissipation and they require you know this parallel friction to shift the phase of VNN and they also require the frequency below omega star.
08:39:16 The Zone all modes are the K parallel and Casey day equals zero.
08:39:21 And by the way, you'll know the zonal modes, obey a slightly different equation and the me than the drift wage to another point that's been glossed over the zonal force is the rent is the rentals for us I should say is the vortices at flux, and the zonal
08:39:41 component of the particle flux makes congregation. Interesting question is the cross correlation of the zonal density in the zonal vortices at which you can see your recent paper by Rambis are seeing in me.
08:39:58 So then the model, then, is one of the kind of by stable mixing and maybe the way to see it is the density is a defeat evolved the mean density evolves by a diffuse of relaxation of the density gradient so it's just kind of freaky in, and the vortices
08:40:19 it is a little trickier and of course the point on the vortices it is there's viscous and non viscous stresses and the non viscous stress is what we call the residual stress and that of course basically covers how the density gradient drives the flow.
08:40:38 then the vn stuffy. Then we find it more convenient to work with potential and stuffy than energy and I'll say thing about that is given here and this looks like a conventional mixing length model right so you have the chi times the gradient squared.
08:40:59 And you have some diffusion and then you have the damping due to cascade and here by the way is the production so you know the form right it's a gamma times of potential entropy, the mixing length is what it is, right, is what was described.
08:41:17 And it is a hybrid of an expectation scale and the emerging terrain scale which is a consequence of frequency mismatch versus, in some sense, align with due to finite amplitude and by the way you can introduce some additional factors here and there is
08:41:36 a scale crossover which is the transport bifurcation and basically for exploitation scale small compared to the Rhine scale you have strong mixing the state is mainly Eddie's for exploitation large compared to the ryan scale you have week mixing in the
08:41:55 state is mainly waves and then you get sharpening feedback.
08:42:00 And by the way, and trying to think about this, I've been reading the ancient classics on mixing length theory going back to Ed's beagles reckless youth as it were.
08:42:14 And I mean there were ideas floating around in the 60s and early 70s about a two fluid mixing late model and in some sense, one could look at this as a realization.
08:42:26 So, a bit more on this I think it's important to understand where the equation comes from not just that it makes nice color pictures right.
08:42:37 So why are we doing this so YYPV. Well, the point is PV is mixed so it's the natural quantity for a mixing length model and if you were like me a plasma physicist PV is a conserve Face Face density.
08:42:53 So the natural formulation is in terms of potential and stuffy right i mean it's just perfect. And that also leads you to connection to other things, what's different.
08:43:05 So that's the first point its potential and stuffy and not kinetic energy is in the original Bly.
08:43:13 Secondly, of course there's two main fields that you have to keep track of right and that's the fact that the density gradient ultimately relaxes drives the fluctuations the fluctuations then produce a stress and drive this year flow.
08:43:29 So you get three way interactions, which is a bit beyond the original blinds who have exchange and couplings right and all of this means that the rentals work and particle flux couple the mean and fluctuations.
08:43:44 Couple of the mean and fluctuations. Working with n strappy makes the nonlinear damping easy because you have a kind of a forward transfer if you want to call it a cascade or forward cascade.
08:43:55 And the course the critical components are the diffusion and the viscosity right these are the turbulent transport coefficients that govern everything on PVM version we take the mixing length to be bigger than the gyro radius so that simplifies it that
08:44:15 sort of in in GFD speak means a small.
08:44:22 Yes, a small well whatever the row is that kind of limit.
08:44:26 We're also working in for in the idiomatic regime for dissipated drift waves and that buries a body legitimately, I might add it says the phase between the density and the velocity becomes very simple it's simply a coefficient that the alpha right it's
08:44:46 a dissipated density response and by the way that you get an instability because even that week density that dissipate of response produces a phase shift between density and velocity.
08:45:01 So that's a major simplification which is solid within the range of applicability, the chi is similar to D and this is an outcome of the structure of non resident diffusion.
08:45:14 Here's the vortices at flux okay and you see the diagonal the usual diffuse of bit and the pie residual the which is the non diffuse of flux and that's driven by the density gradient, the softest part of the whole story is the flux of potential and stuffy
08:45:34 fluctuations, which is what we call the turbulence spreading or the entrainment term and we'll come back to that.
08:45:42 So, the critical things in this model or the D in the chi, they regulate the exchange between the mean and fluctuations.
08:45:51 whole thing in the denominator should be in brackets, we had a typo there.
08:46:07 So what do I mean by that I mean that as the as the mean PV Grady and Stevens you go from a diffusion coefficient sort of like this to a diffusion coefficient sort of like this right where you have greatly increased memory because the way frequency is
08:46:28 going up and of course the wave frequency is due to the mean PV gradient which is dominated by the density gradient. So the waves enhance the memory so I think this is a realization of what in McIntyre speakers is Rossby wave elasticity and keep in mind
08:46:47 the real frequency scales is this.
08:47:00 And this kind of complicated function will lead you to an S curve.
08:46:55 Another very body is the exponent here, and use one would like Kappa to be too so the exponent is one as in blind but I have to say there are cases where the exponent isn't does that simply won't work to get an S and that's an issue go with for the plasma
08:47:15 people going back to the original Fred Hinton stuff on this. It's another another sore point and I would love to know of a way to get a rigorous bound from the fundamental equations.
08:47:28 So going beyond.
08:47:31 So remember I said that entrainment was soft.
08:47:53 So, gamma has been, you know, saying from the beginning, I'll start looking at how resilient It is so one very simple thing we did, and Xavi a raise this refrain again, was we in the potential and stuffy things since we knew it was soft and we wanted
08:48:08 play around with it we put a multiplier in front of the D few 70 so in other words it's beta L squared times epsilon to the one half. And this is a measure of how strongly you spread out things right that you've the fluctuations and what's interesting
08:48:15 there's a scan of beta from very small up to 10.
08:48:22 And the story is bait you know you see that there's quite a different staircase structure all things, all other things being equal, right, you can add, add you know if you run up to five you basically wash out the staircase completely if you go low, you
08:48:40 get a lot of steps and there's a decline in between so it says entrainment or turbulence spreading. And it's sort of like avalanche chain has a significant effect on the staircase structure and we think this is an important point.
08:48:56 And one of course could go further and we'll we'll talk about that in the future. Mergers do happen, and here I didn't understand it at the time we got it but due to Neil's nice little talk last Friday and interfaces you see his.
08:49:13 I call it a tight to me he had a different buzz word for it but the idea of a merger, where the two guys don't move. They just eat the guy in between it and here it is it's a figure from our trash in my paper and.
08:49:29 So clearly, both times of course earnings processes are observed. So it begs the question for future study room keeping the previous view graph in mind of looking at the interplay or competition of spreading and mergers, and you might want to scan the
08:49:45 course in time or the merger rate more precisely versus increments in the coefficient of the spreading term I think this would be an interesting exercise to do.
08:49:58 Back to Edgar's question, there is global dynamics so people tend to think that, you know, people know about mergers, well, going back to con Hillier but particularly from Bly 98.
08:50:11 But the point is we have a symmetric boundary conditions kind of to mimic the same pile so we have no women on the top side and directly at the bottom.
08:50:23 This introduces in a symmetry and we get escalator modes as I like to call it.
08:50:29 And she are migration.
08:50:32 And in plasma physics we're obsessed with nonlocality Please don't ask what that is today that's next week's discussion, at least as far as I'm concerned, but you know these global effects I hadn't seen before in anything, you know, in any of the, the
08:50:50 Bly work, and the message here a credible model must address the staircase dynamics I mean a static staircase, doesn't mean much. and the dynamics can be both local and global.
08:51:05 Okay, and again the This includes both an escalator in density and a migration in the sheer lattice.
08:51:13 You also can get these kind of condensation affected. And here we have the density gradient versus time and you know you've seen these type of pictures with the small steps, and you get the mergers the mergers don't happen at the same rate and they keep
08:51:32 merging and coarsening that at the same time the whole thing slides. So you have a steep barrier on the outside and sort of more shallow gradient on the inside, so.
08:51:47 Seems you can get a collapse of the staircase into macroscopic barriers so that this of course is very crude but one wonders Is this a realization of a way to connect the staircases to the Lh transition, I mean I'm not pretending that this is like an
08:52:05 Lh transition but it shows very clearly a collapse, to a steep gradient region on one side and it happens to be the outside. So I need to quantify that there's a lot more.
08:52:19 So that brings us to flux driven studies so this is something member I asked Neil this in his talk on on blind and NDA, and they hadn't looked into this probably it's, it doesn't, not so crucial to them.
08:52:37 So, we added a drive and the point is is the conservative drive so the point is that being. We, the the drive is a divergence of the gamma drive which has a coefficient with the strength and the deposition profile.
08:52:52 The point is this makes sure that you can serve things properly, right, you have the input just to the boundary. And what happens.
08:53:03 Well, you know, you can first thing you can do now is look at the global, and I emphasize global means this is the average gamma versus the average density radiate to not the local, and the global gamma versus Brad and plot shows by stability bifurcation
08:53:23 and history says here's Mr s. One more time with feeling. So you can see their global confinement by four occasions and the staircase state the physics beyond that behind that, as you'll see, is the weighting of the L zero n dl rhymes and forcing good
08:53:41 confinement the Rhines dominates. And this allows one to start thinking of the merits of the staircase stayed relative to various other things.
08:53:52 And to show it a little more clearly. These are extreme cases this is not meant to model the 3d or anything like that so relax.
08:54:03 You've got you guys but I mean you see you here is a case with a steeper outside gradient and a flatter inside.
08:54:24 And you can see the intensity profile here so you notice the intensity is dropping in time from the outside and that's where the gradient steepening. And here's the shearing profile also evolving and the whole thing comes to a kind of quasi steady state
08:54:32 in in in that with an S curve structure.
08:54:35 Now if you wanted to look at this a little further you can go back to our old idea of the flux landscape, and I mean the flux landscape was in some of my own papers with Vladimir Vladimir was another great student who went to a one off the wall street
08:55:04 he survived all these years and is doing great so you know definitely getting that grant proposal ready for him. And you can see from the years here, and it's simply the family of s curves at different positions in time right so here we have gamma versus
08:55:16 x and here, and then the other landscape is versus grad end so the landscape is, I shouldn't have said time really it's a, it's a, versus position. So in some sense it's an array of slices and it's that make them a surface and the slices are the original
08:55:34 S curve.
08:55:37 And, you know, the red or the good confinement the gray or the normal confined and then you can you know you see here this thing building up and appearing to march inward right that is that the signal of where the transition to the good confinement is
08:55:52 occurring right and here but that's as you progress in space, and the flux landscape is extremely useful when you're trying to figure out how far the merrier penetrates that and you know you when used in concert with something like a Maxwell rule.
08:56:11 So, all that's well and good, where do we go next. so I remind you of the. Some models are too good to be true other models are too true to be good, because this of course is where that that kicks in.
08:56:27 So, you might say, you know at this point you say this Bly approach has already been flagged to the fleet. Right. It's yielded a great bounty of results but what else can.
08:56:42 What else can be said here.
08:56:44 While there are a few things I think one can do the stochastic field effects that Samantha Chen talked about is one.
08:56:54 I mean, I'm almost done David and and the
08:57:01 the other one might be that might be interesting is the thermal Rossby wave and the reason I bring that up is PV conservation is broken by buoyancy so that'll bring a new twist and the other feature is you're going to have to face the phase between the
08:57:19 empty right so that's going to be a new twist we got off easy in the dissipated drift wave.
08:57:27 And then there's of course multiples, you know, multiple scale games.
08:57:32 Now there was some noise about noise, well I mean it seems to me if you're going to have in homogeneous mixing, to be fair, you ought to have in homogeneous noise and that of course forces in the consideration of incoherent mode coupling and that's an
08:57:49 interesting question of how you represented and mixing length theory you can of course represented do in a closure theory which is a great torture Hi, can you can look at this paper and it forces you to deal with things like there's no cross correlation.
08:58:10 And you see things of course going back to Rosenbluth Hinton that you can excite the zonal modes without modulation instability and but we calculated it in some detail and I remember Brian Pharaoh mentioned this critical obsolescence thing in his talk
08:58:28 I mean you, you know, an interesting question would be kind of a multiple scale or in, you know, mixing length model that would incorporate this effect, one knows the structure of the equations, but the details I think still need to be worked out.
08:58:50 So there's an old classic on dressed parcels the two component model that I found very interesting by Spiegel and go off in a paper entitled on taking mixing length theory seriously which basically says you have slugs and a wave dressing and this is akin
08:59:13 to address test particle and plasma. But I wonder if the lie has already been there and done that, because in a sense that's implicit in L zero l Ryan says your base scales right in the sense, the reason transport is lower in the Rhines ranges, is because
08:59:28 it's more waves. So you have more memory so will you really gain with that, I don't know, you could exploit the relation to wave kinetics which is nothing more than a glass of equation for a parcel but they are the parcel is a wave pack and it's a parcel
08:59:47 of wave exploitation and remember for zonal symmetry.
08:59:52 You guys may have not got the message. This in in previous talks right there's a degeneracy between the wave action density and the potential and stuffy and this was noted by the boule and NASA Franco but of course has been in all this work on wave kinetics
09:00:10 we've been doing since. well since 1998.
09:00:14 So I wonder in the end with all this are you going to end up with a parcel picture with some mean field coupling that's not, you know, how much better Will you really do than the by one wonders so it's easy to propose extensions but the extensions may
09:00:34 jeopardize the simplicity and clarity of by 98, we'll, we'll keep trying.
09:00:41 So, you know, this problem, I think, is will be at it will probably have some meetings and 10 years from now and talking about this issue it had, you know, the S curve even fascinated Salvador Dali right that it's going to be a long and winding road,
09:01:02 and just like the career of the original Bly and after all the puns we should show the original Bly, who has quite an original quite an interesting story which I read blade one night making these view graphs, but that's for another discussion so thank
09:01:19 you.
09:01:24 Thank you, Pat was a fine performance, very interesting. At least we had some, some questions for you.
09:01:34 Questions for packed
09:01:38 and semi a high but I will not make noise, noise this third wave. I know a savvy ay was commanded by Lord Nelson himself for clobbering see battle that just a little added bonus for you.
09:02:01 So thanks for noticing.
09:02:06 Anyway, it was interesting by another set of chaos you showed showing the other edge flex versus other edge density we see the density gradient. Yeah, brilliant and then.
09:02:20 Yes, I was not clear to me what was the meaning of the A B C D But I remember at some point you were arguing with Marco, that the transition in some cases would take place at the Maxwell flux that is some way and I was wondering.
09:02:34 Yeah.
09:02:37 Well, I mean, first of all if you just have a to d curve like this, you're going to get the Maxwell right but I mean, I think, really, in the landscape.
09:02:55 What happens is the transition is triggered.
09:03:01 At the intersection. Basically, of the surface of constant flux, with the flux landscape, which then becomes that's what our ancient work was about and which becomes a
09:03:22 how to say it become identifies at what position, the Maxwell criterion kicks in, or if you kind of see what I mean I can, I can draw it here.
09:03:36 Okay, interesting anyway.
09:03:39 But I mean, they there, there are some differences. Just to be clear, because here I mean in simple terms. In this system, you know, the prototype with the Maxwell it always seems to me is the Fitzhugh the boomer which is a boring reaction diffusion equation
09:04:01 it has the potential, it has all the action in the potential, and the diffusion is a constant. As you know, in these systems which are more like the con Hillier, the way I like to say it is the reaction is in the diffusion, right, as it were, because
09:04:23 of the suppression effect.
09:04:26 So then of course you do get a Maxwell rule but there are some extra pieces from surface terms in it, you know upon integrating and also it's not quite simple minded Maxwell but close.
09:04:40 Okay.
09:05:00 Okay. I did not prepare a slide on that. Sorry, but I thought
09:04:55 again.
09:04:54 On that last remote, I think you Jen each might be sure discussing something tomorrow. So on that and Misha, I guess we'll be talking about it in the in the plasma.
09:05:11 I think Michelle's gonna talk, basically about our 2010 paper which is going to be back to more or less to Maxwell, so that much I know,
09:05:22 but I had a different question there was interested in the world in your work with also Misha, and the cap over to your cat our the exponent, and how to connect it to first principles because that's indeed is key you want to have Kappa over two equals
09:05:44 otherwise you don't get the the the staircase. And something that we don't understand yet, but might interest you is that, depending on whether the shear leaders are converging while receive converging or diverging spreading.
09:06:02 They seem to behave differently. So in that respect Kappa is probably a function of spreading itself.
09:06:12 And I don't know how, what to make of that but that is the, the Hinton rule that you you have a shear to the square, meaning that whatever the, The, the shearing rate or whatever the shields layer, it behaves in the same way that doesn't seem to be extremely
09:06:37 It behaves in the same way that doesn't seem to be extremely well I mean, the Hinton thing I mean no it's an interesting paper is complete at some level becomes complete rubbish to speak bluntly, because now I mean they just put, they just put the, the
09:06:49 fluctuations or the turbulent transport as you know as a constant and they just apply the one plus alpha v prime squared thing in the denominator, right and that's it right that that that's really, that's really cheating.
09:07:07 Right, so but I mean I think you were making a violent advertisement for.
09:07:16 I gotta keep track of my propaganda here you're making a nice advertisement for this plot, right, that things, things go south as you crank up the effect of the scattering of the fluctuations that zero surprise to me because the, the sheer or the the
09:07:34 PV gradient ultimately is slaved through the fluctuations that's where it comes from. Yeah, but maybe there's an additional direction, meaning that constant in turbulence intensity and you would say constant spreading depending whether or not converge
09:07:52 towards this take a step or the shoulder year.
09:08:00 You seem to have a different behavior, on, on the same. So in some sense you're saying it in this model it would be something like a variation in D of epsilon.
09:08:08 Yeah, I think so. Well, I mean, but I would say of course remember everything's coupled here and I, this may capture that.
09:08:16 So I think the, even the simplest minded model had captures the vestige of that effect.
09:08:26 And, but I mean the what you say seems very plausible right I mean, but where I think we need maybe to do a little better in the epsilon equation is to come up with a mock, send up of conservative noise so we have a, shall we say, some model of incoherent
09:08:48 mode coupling. Okay, but it can be in homogeneous but conservative and I think that's doable, like many other things. So, yeah, and then the Epson will feed back on your kappa, kappa well i think it feeds back already but it'll feed back more.
09:09:12 Yes.
09:09:12 Echo.
09:09:14 Oh, yeah. I have a naive question is, as usual.
09:09:19 So, the house ago buckets on the model has the flavor of, you know, cross diffusion.
09:09:28 Yes. Was the the key term, you know, in the mean flow equation is, you know, is the diffusion term that involves the mean density.
09:09:40 And of course in systems like that, if you impose a gradient of the intensity you get separation.
09:09:47 Spontaneous separation in you. And I was wondering, I mean, is this a useful way of thinking about what's going on in this system, well I mean it the certainly the separation.
09:10:03 I mean this picture. well that's not the best.
09:10:09 Just find the right picture this something like this may smell of separation right in the end right you can, you know, this is a extreme case but I mean this looks like a separation from normal profile, look at where we well this is driven of course but
09:10:42 it separates into two different zones of enhanced confinement, the steep radiant region, and sort of normal confinement the flat gradient region. You could think of that.
09:10:44 You could think of that. I don't know how good an analogy, but that smells of a decomposition, or maybe a separation so I think all.
09:10:55 I think it is with cautionary words, it may be useful. Yes.
09:11:01 And actually I have a since we have this picture up.
09:11:05 So you talked about the Maxwell construction But that requires that you have a conserve quantity. So is the PV conservation. Well, to use the Maxwell construction in this way, you can use the Maxwell construction on density, I mean the whole PV, going
09:11:23 to the basic equations is PV is conserved right i mean if you look at, of course, up to the you know the scar cities, and by the way we we live in a world of rather large parental number if you take these things seriously Peavy is conserved right you
09:11:47 subtract these two equations and apart from these you have a conserve PV. Now, in fact most of the PV is in the density, right at the point you actually build the thing up so that's kind of when I mentioned Maxwell.
09:12:03 That's really based on a one field model those old papers were based just on a one field. Okay, now if if you had a significant energy and the flow, then you would be right you would you would have you would have more on your hands and you can cope with.
09:12:22 Thank you.
09:12:26 Your question from Allah is when your friend to entrainment and trying to understand if it's a synonym for spreading or something different because usually you would think of spreading is more of a micro mixing it, and it reversible mixing entrainment
09:12:42 is kind of more of a macro mixing without any irreversible mixing all right i mean i'm not going to debate semantic search that's your semantics and I'm talking about spreading.
09:12:54 but I've seen plenty of fluids.
09:12:59 My observation in, you know, things like pounds and and all I've seen entrainment used for what I've referring to so sorry.
09:13:10 Okay, I just wanted to make sure that you weren't referring to something distinct for now I'm in Townsend makes and I mean I love one of his papers that speaks of nibbling and engulf, which is I think the more I love the nibbling and engagement, and some
09:13:26 more accurate way of describing the, the two, the two processes and I would say I'm referring to nibbling, if that helps you.
09:13:38 Oh yeah, that's all I just wanted to clarification.
09:13:44 Right. Okay.
09:13:47 Any final questions for pants.
09:13:52 No Okay thank you very much.
09:13:58 Let's move on to
09:14:03 our third speaker who is born parishioner, who is a PhD second year PhD student at Leeds.
09:14:15 So whenever you're ready poor, please share your screen with us.
09:14:22 So can you see this.
09:14:24 Yeah, they look good. Yeah.
09:14:28 So
09:14:31 I'm going to talk about this smaller, we've been developing for diffuse connection, which is a lifestyle model.
09:14:43 Basically the motivation is that there's been a lot of work into DC layering by DNS and some experiments, but we don't have that much.
09:14:55 We don't have that much insight into the, into the development, beyond initials instabilities politically.
09:15:04 We've got work by route going is down condition, which tells when we expect layer into a car, but we don't have anything after that, to look at the subsequent development of the layers.
09:15:21 So, there's a lifetime all the blind model for target Strathclyde flew all that in 2013 there was another model by Pearl Harbor, which treated the reissue of the fluxes so gala be constant, which basically mapped double diffusion and supply system.
09:15:42 And we're trying to come up with an alternative to doing that with a full three equation model for temperature salt and energy.
09:15:51 Starting from the original business equations.
09:15:57 So these are the equations and, which should be non dimensional life so the really number is equal to one.
09:16:05 We've got key points. These are the number, the 5070 ratio.
09:16:19 The density ratio Arnold, then this small ass is a sign of a background brilliance. So this determines which the two double defusing for James we're in, so Ephesus, one.
09:16:25 And we've got salt friend Greg on the assets minus one to go to face connection.
09:16:34 The goal is to come up with a lifestyle horizontally operational. So, for temperature Symbian energy, all those functions of time and said, I'm going to walk walk through this averaging process from the center of the equation.
09:16:53 And so we split the fulfilled into a mean and participation part.
09:16:59 We take the average of the original equation, then we got an equation for the average in terms of this prohibition flux which we can then subtract from full equation, got an equation partition on then.
09:17:16 If we make the approximation that we scale time derivatives with some, some time scale to build and scale.
09:17:28 The bodies, squared, with minus one over x squared for some length scale.
09:17:33 Then we can prompt rise this flux in terms of the average quantities.
09:17:43 Then if we write the vertical velocity squared, the average vertical velocity squared as the energy that this girl told her, be a maximum length which we're going to prompt dries up some point.
09:17:58 And then to get an article timescale was this characteristic length over characteristic speed.
09:18:05 Then we can formulate this model equation for the average salt.
09:18:14 Then we've got an identical process from temperature to the energy we formulate the energy equation and standard way by doubting it delving into random equation with you on do you see an averaging process.
09:18:29 With this system.
09:18:33 So we've got these terms in flux terms and molecular deficit lease
09:18:41 on them the energy equation we got to say in turbulent flux diffusion, these two terms, or transfer from potential energy which come from the winds the term in the, in the momentum equation.
09:18:56 And then we've got a dissertation.
09:18:58 So all of these first three terms. So all the terms of the energy equation except for the dissertation are coming directly from the averaging process, the dissipation.
09:19:10 And the way we prioritize that is, we're free to chase up.
09:19:17 Then, difference from lie system is the blind system how to a production term, which we're leaving out here.
09:19:42 Because thought was a kind of thing for for energy through story.
09:19:42 But we want this to model so that all the instability comes from comes from the double diffusion alone.
09:19:41 So, what do you want it to just come from the interaction of the temperature itself.
09:19:54 So the parameter ization of the length scale is one of the really important parts of the model.
09:20:04 We expect it. And when the plan seared into small, like in layers, we should have a large length. When the gradient is large, we should have small length.
09:20:18 For example, this is the form of the length us by Bly which interplay between, starring unskilled D, or the ultimate of length, and proper Alan fall hard and burn used quite a different form, which doesn't depend on the boys reading the towel but just
09:20:37 in the length.
09:20:38 Sorry, just an energy and interprets between some large length sell go on some small length scale, which are just parameter and good.
09:20:53 And we expect that neither of these is really perfect for the double diffusion case.
09:21:02 First of all because we don't have this, we don't have this starting, let's go to fall backbone.
09:21:09 So, we need to think harder about water long length is going to be.
09:21:16 But also, we expect there to be areas of convection in the layers, which means that will have places where the blinds ready and as negative.
09:21:31 As the buoyancy rating gets to negative, then the denominator in this length is going to go to zero so we're going to have blow up.
09:21:34 So we've been thinking about possibly having a switch. So, when the gradient is negative, have one formulation for the length.
09:21:43 But it's positive, go with something like this up but not actually fully decided on anything yet.
09:21:58 So even as I coming up with the length yet, we can still do fly a lot of analysis into the system.
09:22:06 And we can learn a more general form of the system. And
09:22:12 look at it stability.
09:22:14 Social right of temperature ranges GSN Gritten does D.
09:22:19 Then we can write the mall in this form here.
09:22:23 So we've got our diffuse the flux is here. And then a term with nozette derivative the energy and the energy equation as well.
09:22:36 So if we look first at uniform steady states, then that means that those are fine when this P equals zero, we just assume that these exist for no perturbed with perturbations proportional to exponential violence, plus lambda t.
09:22:57 And we gotta roll through integration, which is cute I can launder, the grocer it quarter six in the way of number and
09:23:12 then if we let a small and limit.
09:23:16 Then we've got one for events stability here with lambda, like dp buddy.
09:23:26 And so FTP buddy is positive, then we're going to have an instability.
09:23:33 On this is effectively, just from instability in the energy equation on it so. So, if we considered D buddy to use equal pay.
09:23:43 And then we've asked.
09:23:47 We've also got two other two other modes which come from the Order, order and forth terms in this, which gives instability yet.
09:24:04 This script fg CD, minus FDCG is negative, where these derivatives are finally about.
09:24:22 This is actually equivalent Toronto is gamma condition.
09:24:22 And it's so it's exactly equivalent.
09:24:25 You can reduce the rappers condition.
09:24:27 If gamma is a function of only the ratio of D amp D.
09:24:39 So, neither would have looked at where we can have stability in general system, we can look at some steady states in this exact system.
09:24:48 So if we start p equals zero, then we have this expression.
09:24:54 And we can see that
09:24:57 we can see that this is only possible.
09:25:03 In
09:25:05 this second term has to be positive, to balance this first term, which means that if we're in the salt fingering regime if x is greater than zero, then one of our are not has to be greater than this, which is going to be less than one.
09:25:23 Which puts us in a part of the temperature really against the salt really number of this yellow star sauce in the sort of traditional salt fingering regime.
09:25:37 But if we're in SS negative. So, in this quadrant here.
09:25:45 Then, we find that aren't all has to be greater than one. So, this is saying that we can have a steady state, even when the background gradients suggest that the system should be stuck around still
09:26:18 We've got some interaction between these fields which means that a solidly unstable state, and can actually study.
09:26:19 So this.
09:26:22 This study states, you can actually get an exact solution for for them.
09:26:27 Is that alpha, which was a discussion parameter equal to zero.
09:26:34 Under yet to study sets, one with Elliot, is equal to zero.
09:26:41 Another lead is one minus time or or or minus one.
09:26:45 So this, the length Times Square to the energy needs to be positive.
09:26:53 So, this exists between our as our as one over to log
09:27:01 On these sets are independent of the choice of maximum length.
09:27:04 So, anything we can say about them here. It doesn't matter how we how we prioritize this L.
09:27:13 And if you are conditions for instability, we find that this look, this large one.
09:27:20 One line is Tyler over our minds one is still to both instability in the energy equation, the route code type instability
09:27:33 expound on it today Sarah solution in small alpha. So, we say he is alpha to the End Times. Some he sub nl, which is going to be a function of a is alpha to the M times felons functionally.
09:27:51 Then, Bhutan one solution which is zero.
09:27:54 This is exact, we obtain another solution, where each behalf goes like.
09:28:01 It was like, I'll cube, your relation between these experts of alpha.
09:28:12 And if you substitute these NTRP condition.
09:28:18 We find that dl but he has to be less than one overall to the five.
09:28:27 So this gives us a condition on the form of L.
09:28:38 The form of lm as a function of en allies instability. So, for a general very general form is a nk.
09:28:46 Then, for all valid choices of K that satisfies this relation between Canada and we find that p is greater than zero. So, for this form of lm, we have this energy and stability.
09:29:08 So I was looking at uniform steady states.
09:29:10 But we can also consider non uniform states. So just setting the left hand side of our equations, at the time derivatives to zero.
09:29:21 satisfy this a bit. I'm going to write to capital D is led to the half.
09:29:27 And you see, the temperature and self equations, become total derivatives. So we can integrate those up.
09:29:35 And then substitute those expressions for t said and I said, into that energy equation, which will then give us single od for deep.
09:29:48 So, we're going to assume that part of ingredients are zero on top and bottom boundaries, which means that we can set constants of integration to zero, and substituting these in gives us this equation for ded is a function of a.
09:30:09 So this can be viewed as a as an equation just free.
09:30:16 And my parameter rising wireless.
09:30:21 But we've been thinking that actually cause a way to get some insight into what the light scale should be.
09:30:29 We can give a profile for the energy. So, kick, the typical energy in a, in this darkness, or even just a typical data energy. Energy profile across one step.
09:30:42 And then if we substitute that into this, then that gives us the first order already for D, which we can solve, and then find out from that.
09:30:56 And hopefully this will give us some insight into what form this mixing length can take.
09:31:05 So in summary, and we've made a lifestyle for double diffusion.
09:31:12 The choice of the length scale it is critical to how the model works.
09:31:19 And we don't have a lot of scale, we've settled on yet.
09:31:23 We find a couple of different possible instabilities, we find one that's in the energy equation on its own, without input from the temperature himself.
09:31:35 And then we find another one which is equivalent to the gang condition.
09:31:40 We also find that welfare small.
09:31:44 There's a stable steady state for certain range of Arnold.
09:31:50 There's also an only stable state.
09:31:56 And if we wanted a larger alpha we need to prioritize the night and skill.
09:32:19 And we are using these non uniform study steps to, to help try to do that.
09:32:11 Okay. So I think that's me.
09:32:19 Thank you.
09:32:18 Very nice, clear talk
09:32:25 if people have questions, raise your hand raise this one, or if it is please go ahead.
09:32:38 Or am I am I still up. Well yes, I don't know, that's what I was saying I didn't know if you were asking a new question or if you are still.
09:32:46 Sorry. The answer was no. No. Okay.
09:32:48 Okay.
09:32:52 Right. Pascal.
09:32:55 Hello, can you hear me. Yep. Okay, good.
09:33:01 I am sorry I just keep out to the crucial moment and maybe I'm just asking a silly question but I don't quite understand why there can't be any steady states everywhere in your, in your, in your diagram at all parameters because basically you would expect
09:33:18 that it's possible. it's a turbulent statistically steady state in some sense right.
09:33:23 You mean here.
09:33:25 Why can't it be all. I mean, from the perspective of turbulence you would expect whatever you choose for your parameters that will be a turbulent state right.
09:33:37 So why don't they exist everywhere.
09:33:52 So it comes down to.
09:33:57 These, these two terms balancing the discussion balancing transfer from the potential energy.
09:34:04 So, for some.
09:34:11 For some, for some gradients.
09:34:15 Then, the only way that that's possible is with the energy being zero. So there is no there's no non static studies that's, sorry.
09:34:26 I should possibly have clarified out there is the that equals zero se it will exist.
09:34:33 I mean, I get that.
09:34:35 Statistically stable stable parameter space that sort of makes sense, right. Yeah, why not up there in the statistically statistically completely unstable case cannot not happen.
09:34:54 So, top left, top left square in here.
09:35:01 But in not part of the regime diagram, you've got both the temperature field. The salt field are unstable. So you've got both of them would lead to convection on their own.
09:35:13 Yeah.
09:35:17 So I think that's why we're, we're not finding them being able to balance each other and counseling.
09:35:25 Okay.
09:35:39 Right.
09:35:42 Yeah, I mean I think what books have you tried i mean i don't think you'd expect them down and you said you might expect them everywhere Pascal, I don't think you'd expect them in the bottom right.
09:35:52 No, and that's true. I mean, but what about in the DC region as well.
09:35:57 Because that becomes somewhat turbulent right and yeah but yeah I mean you've got to balance that you've got Dr.
09:36:12 You've got destabilizing gradients and you've got dissipation so you expect them to exist somewhere but not everywhere.
09:36:13 That's what I would have thought.
09:36:19 I want us to have to think about this boils down to the model. Yeah.
09:36:25 I need to think about it more deeply sorry for the silly question.
09:36:30 It's not.
09:36:34 Okay, Pat.
09:36:38 I have a couple of just very simple questions. I need to wrap my mind around this more.
09:36:47 Of course rad co pointed out to the need to regularize the gamma effect right he had a problem. And that was an interesting aspect in his talk. How do you do that here.
09:37:01 So, I having this having this third equation, or for.
09:37:09 So rod.
09:37:22 We think this can produce a
09:37:27 producer.
09:37:30 Okay, so, I mean, one question is which term in the energy court yeah I should have realized that it is doing the regularization is the turbulent diffusion term, or is it something else
09:37:53 or is are your basic or are you saying that there was basically in a kind of an energy non conservation before which you've now rectified by having the energy coupling to the, the equation, I'm just curious what's, what's the key player.
09:38:11 That's not possible.
09:38:15 After these are, these conserve energy, oh I believe no I can see that but I'm saying I, in other words what you were saying his previous thing in a sense didn't conserve energy is that the punch line.
09:38:34 I'm not entirely sure.
09:38:35 When his previous thing is a linear instability pass I'm not sure why. Not all right that's that the answer that is yes, it didn't.
09:38:44 Linear instabilities don't conserve energy right i mean i mean i would say is is a linear instability. With that goes fastest shortest wave. Right. Right now you know that largest wave numbers and, and this is this is a horribly nonlinear system.
09:39:03 Know that I get it but interesting. The second question is, is kind of you know layering in general evokes in my mind some kind of fighting, or transport process that resists relaxation and I mean, One way to do that is to suppress the turbulence which
09:39:30 is kind of the game we play in plasmas.
09:39:35 One way is to convert it to something else, which is you know like dry heat gradient parental stress or something and one way is upgrading and transfer were like a pinch, or chemo Texas is the other way there was same idea stated where relaxation say
09:39:57 in one of trs drive some upgrade in transport in the other. And that may not be relevant but which one is it right what's going on here.
09:40:12 In simple physical terms.
09:40:17 I one thing about this system right you get nothing if you just have heat right you need he can solved. Correct.
09:40:25 Yeah, you need you need to sell. Right.
09:40:29 Otherwise it, otherwise it turns into Bly, with no source. Yeah, exactly. Well that wonders is is one gradient driving a flow in the other somehow and that's what I'm trying to understand in simple terms here.
09:40:49 Yeah, I mean I think that's right in the right path if you if you subtract those equations, then you get to be equation, don't you but you get an, and then you get the extra bits, which is the non blind bit.
09:41:01 Right.
09:41:11 What is the extra bit to look like I mean, what.
09:41:19 Yeah, something like this on then, something like a man with an LED out plus one times x plus tire on the bottom right so that it smells like that's kind of an additional flux, in some sense, yeah so you end up with bt is some, Some buoyancy flux, plus
09:41:35 some temperature flux. Yeah, well that to me smells of what you know of the in our classification that's the pinch variety but I mean that let's not worry about nomenclature but that that helps me some.
09:41:50 Okay, so then there ought to be some on this system, given that there ought to be some kind of constraint of entropy production, ultimately, right and then had the books balancing and all that have you have you worked that through.
09:42:12 No.
09:42:12 Okay, interesting though. Thank you.
09:42:20 You can go to question, please.
09:42:23 Yeah.
09:42:24 So, can you hear me. Yeah, that's good. Okay, so I have like two questions and I guess there are simple ones.
09:42:33 For the first one, I think I'm a familiar with radicals common instability and I can see that in this work there's a similar type of linear stability analysis.
09:42:47 But I just want to understand for this linear stability analysis which is described here, what is the what is the stable state that is subjected to the instability, is this the same with the work of radical which is like the homogeneous salting bring
09:43:06 food everywhere or it's like some something different.
09:43:10 So in this in this stability analysis we've assumed specially uniform ingredients and energy and dance value analysis about these uniform sets.
09:43:24 And then later on here.
09:43:27 I was look for where are these uniforms exist.
09:43:32 Okay, so for the uniform state. There are like physically there are salting going everywhere, right, like in equilibrium against.
09:43:45 Yes, sir, especially in the homogeneous turn builds.
09:43:49 Okay. Okay, thank you. And the second question is like, I just in the beginning of the talk, like, I guess you mentioned that this work will try to contribute to the to our understanding in what will happen after the development of the original common
09:44:17 because we don't have enough understanding on what's happening after the linear, the GM is stability. So, next kind of summarize. Now, like, what can we learn from this analysis on what the system will behave, after a developed the initial combined stability.
09:44:31 So, After after the instability.
09:44:52 We see it emerged into their system, which the government stability predicts but doesn't. But, Rocco doesn't want the government's ability doesn't say hi that happens exactly. So the goal with this is to be able to actually sort of follow the system throughout
09:44:58 its entire lifetime.
09:45:00 So, see how, how these how the initial instability develops in layers, how these layers. Behave afterwards, like if they have long last, to emerge to they just disappeared.
09:45:16 Okay, but like, I, I haven't seen any like the time evolution in this theory is this heated.
09:45:26 No, it's not been done yet.
09:45:27 Okay, okay. Okay, thank you. Awesome. That's the goal.
09:45:32 Yeah, I mean, you absolutely got it you can't I mean, the goal of course is to solve those three equations those three DVDs, as initial value problems.
09:45:49 I see, okay, and then to, you know, to work out when and if these, we get layering, and if so to say how it, how it's happening. Yeah.
09:45:53 But, but that also has not been done yet.
09:45:56 I see, okay, thank you for that clarification.
09:46:04 Okay, I'm assuming it goes just waving rather than going to ask a question.
09:46:16 So we have, if we wanted another 10 minutes or so.
09:46:23 So I wonder if anyone has any.
09:46:26 Oh general points about what we've been discussing today, you know, mean patch raised a lot of interesting.
09:46:31 Oh, philosophical questions about, you know, such models and that, well, how far you should push them.
09:46:42 And what you know what you get out of them.
09:46:44 Wonder if anyone else has anything they'd like to contribute down that in that vein.
09:46:59 Oh you know what we should be doing between, you know,
09:47:10 less, less demanding than then massive computational simulations are there other. Are there other approaches that are, that they useful in this is clearly, no one sort of approach.
09:47:30 And this is clearly, no one sort of approach.
09:47:36 Is it too early in the morning, too late at night, philosophical questions.
09:47:41 Don't well gone Adrian, you haven't said anything today, off you go. I guess was just asking for clarification. You're, you're asking.
09:47:50 Like other methods like outside of the blind model is that what you meant well I was just I was just asking. Yes, I mean what what people's, I mean back, Pat put forward you know what the model is a good for maybe what they're not good for.
09:48:06 And I certainly think they've got some validity.
09:48:09 But, you know, whether you push them too far that's that's always a danger.
09:48:16 I just wondered what people's views were on how far you should push them, you know, how far you should resort to solving solving the full equations on the computer.
09:48:28 Gotcha.
09:48:30 And I elaborate briefly, sir.
09:48:34 Yeah Yeah, please. I mean I think you know what struck me looking is, it's a trivial comment I mean obviously be reduced models. I think have to pass some test.
09:48:48 The model captures phenomena and can handle things outside the original construction, otherwise it's just a curve fit. All right, that's one comment. And the other thing is that it in some sense be simple, whereas simplicity like beauty is in the eye
09:49:07 the beholder. But I mean I was struck listening to the story of Bly 98 you know they had the weekly nonlinear theory for man for a young and there's been no shortage of simulations and you had all three elements brought to bear at once, on the problem.
09:49:31 You know, and I kind to motherhood statement at some level but I mean that's the way to go right and to try and coordinate things.
09:49:45 Yep. You know,
09:49:49 can I suggest something go.
09:49:54 Yeah, go and Boris, yeah.
09:49:57 I'm sorry, is, you know, RNG technique, which can be used for this. And in early days of our engineers and developed a model which was mentioned that several times.
09:50:11 It had the kind of continuous transition from alarming nervous quality to turbulence and just saying can be done is the facilities as well. So, I don't think anybody has done this in terms of applying our NGO, or now we have a different model called Q
09:50:37 and A sequence a normal skeleton relation to develop models continuous transition between alignment and total volume to score the disability facilities and try to see what will happen is it will eliminate problems with dissipation for example because
09:50:50 dissipation will become part of the problem.
09:50:56 External parameters and
09:51:06 Okay.
09:51:10 Thank you.
09:51:12 scandal you still want to save me. Yes, please.
09:51:18 Yeah, I mean it's more of a reflection on your original question.
09:51:22 Is it useful obviously yes and I really want to make a big pitch for astrophysical stellar interior fluid dynamics were astrophysicists all the all they want is very simple little formula that they can plug into their stellar evolution code and the simpler
09:51:37 the better right so absolutely these reduced models are super useful in two weeks. I'll show a little bit on how well the government's ability as proposed by right, cool.
09:51:50 So without any modification works to explain layer formation fingering or lack thereof stars in in a military double defensive convection so the DC regime that Paul talked about today a little bit.
09:52:06 And and how well it works so at the basic level when you take constant gradients of temperature and solidity, you start with that the gamma instability works extremely well except in the case where the structures become vertically elongated on the scale
09:52:21 of the layers and then it doesn't work right and Timor touched on that the missing link is something that transitions to, because the government's ability fails wants the background states and are layered.
09:52:33 So the missing link is something that would be a reduced model that goes smoothly from a basically gamma type stuff to a layered type C. And I think what you were attempting to do David with Paul.
09:52:44 Paul with David is really useful because if it works then you can have something that will work in both regimes right. But then, just segue back to what Pat said it really has to be tested against DNS, and the difficulty there is going to be to get the
09:53:00 interface will transport right and so I'm plugging in for my session on Friday about interracial transports really want to know what isn't even about international transport.
09:53:11 And what kind of reduce models they use just for that part of the bigger problem.
09:53:18 Yeah.
09:53:21 Good point.
09:53:36 dm want to say something.
09:53:29 Yeah, going back to your original point and to what Beth was mentioning, also in his talk maybe an intermediate plan could be, well, that knows it. Well, it has not been shown here but he also has worked with another student Heinonen where they tried
09:53:49 to look at the the basic, the weight of the basic terms in the in the relevant equations about the city equation cetera through regression through machine learning, and what turns out is that the gradient of water city there from the kind of full fledged
09:54:12 stood simulation was important.
09:54:16 With what vision was showing the water city regions, didn't seem to be that big of a while that relevant that is to staircase sustainment, so an intermediate step could be to to force one to the value of the full fledged model and see how what what constraint
09:54:40 you have on the other fields, and are you able to match them. And that will help you, you know disentangling, the relevant loops, also because in those simple models you have sometimes no constraints.
09:55:05 are to some relevant operating points.
09:55:06 Yes, yes. Thank you, guys, that are make a follow up there, since I was addressed I mean first of all that's coming in week for the lad by the way is not a student he's dr Heyman and he just defended on Monday and he hasn't been in the program because
09:55:24 it was last minute heroics with the formalities of the thesis as I think you all can appreciate.
09:55:33 It is a machine learning approach a glorified regression is you say, I would you know it's very interesting but well call me out and tour, and I mean I you know will say more when when when he shows up he's on for week four in the transport barriers group.
09:55:53 So, yeah, watch for the announcement, that's the week after next.
09:55:58 It sure helps too so sort of know the answer and understand the physics and that by the way was has a gala waka Tani again in the ad about it, regime, you know to get that thing to work out is all I'm going to say it would be interesting if you didn't
09:56:16 know the answer and the physics what might happen, you know.
09:56:22 Okay, so that's something to look forward to in week four and transport.
09:56:29 You can you still have your hand up I don't know if you want, if you're on to say something else.
09:56:37 Oh no, sorry, I just didn't put it down. Well, that's fine.
09:56:43 Okay everyone, I think it's it's nearly six o'clock suitably normalized.
09:56:49 So thank you everyone for participating today.
09:56:54 And we'll see you next time and probably see some of you tomorrow. If it's Thursday, it must be something else, it must be plasma it's I guess. Okay.
09:57:03 Bye everyone.