08:02:55 Yeah, I think it's time to get rolling So good morning everybody, or good evening or the error as the case may be. Good afternoon.
08:03:08 And this is the transport barriers group.
08:03:13 And we're off and running to our second little gathering and let me just briefly show a few things so if you didn't know it was barriers now you would.
08:03:33 And the theme of this week is barrier mechanisms which in some senses that continuation of last week, barrier formation next week, there's a clear departure to the topic of symmetry breaking, and maybe I'll talk about that the end.
08:03:51 So this week we have sort of.
08:03:55 We have three short presentations that probably won't be all that short right by wasting glow from us on a freshman from Technion and Yannick Sarah's and from ca.
08:04:10 And we have a general discussion.
08:04:13 Steve Tobias will be weighed need to K is still she has technical difficulties and yours truly.
08:04:22 But of course, we have many, many impromptu discussion leaders here ICDMAC Grecia, I mean we all have no troubles starting a riot, I think with this group.
08:04:38 And so let me say just to keep in mind as let the bees buzzing your mind during the talks.
08:04:46 So a few questions for discussion all the questions from last week are still in play.
08:04:55 One for plasma, or YZGS traffic turbulence, how do you compare mechanisms the interplay of shearing Rossby way the plasticity some uncertainty exactly what that is, I'll talk about that tomorrow and means hearing how to represent all this and models.
08:05:17 balmforth well and Smithsonian which we abbreviate is by someone more elegant than ballsy, and then the question which will hear some today from Yannick what is controlling Reynolds stress and flux phases and what role does that play in Dynamics.
08:05:38 And then can we understand the interplay of noise and incoherent mode coupling with negative modulation or processes specifically what's going on near marginal, and I think, I think, on was not seemed interested in that point and then there's the question
08:05:57 of barrier development in space and time.
08:06:00 So those are some additional questions to keep in mind.
08:06:05 So with that, let me stop the share.
08:06:14 And all three of these speakers are in some sense new of I think many of you know them. So, we should introduce them. So, our first speaker is where you can grow from harvest and hustlers watch on University of Science and Technology and Mohan, she was
08:06:35 a both a graduate student and a postdoc there and is now on the faculty in the whatever they call the, the J text lab which has a fancy new title that I have in the upper right, and it's in the School of Electrical Engineering.
08:06:56 She has spent some time at UCSD in the past.
08:07:02 And way Asian is known for work on plasma kinetic theory on impurity transport and now has ventured into the world of staircases with a think familiar gang of criminals and she's.
08:07:15 Let's see, she's won a young scientist award from an Asian Physical Society and she's one special national postdoctoral award in China. And today she's going to tell us about interaction between staircases and means here.
08:07:38 So go ahead, please.
08:07:40 Okay, okay, how many.
08:07:42 Yes. Yeah.
08:07:44 Thank you for Pat, father. Very good introductions and Hi everyone. Yeah, it's a great honor to be here to introduce our walk the interactions between staircase and the media with daddy the this world, as they, I think first of all, maybe two or two or
08:08:15 three years ago with pad damage, and my lady boss, Lou, and the way continue your use SD with also with some, some with errors.
08:08:25 So here is my outline.
08:08:30 First today I will mainly introduce the basics of some nonlinear patterns, and I will ever since they are some basic idea of schedule action, and feedback loops and wisdom background.
08:08:51 Then I will move on to introduce the recent result of a model study of the sustain and for missing mechanism of a staircase.
08:08:57 So, finally, as the conclusion.
08:09:04 I think it's already almost they started the month of the KITP program, so I think I may ask, are very familiar of the staircase pattern. Actually there's other similar patterns observed in different systems like the layer the stratification the transport
08:09:25 barrier, and the TV staircase and he Cosby staircase.
08:09:30 But today for me, I'm focusing on the mathematical fusing devices. So far this devices zona flow is also a staircase like structure. And from this ecology of zona flow and the drift away of the turbulence the right can see the importance of zona flow
08:09:50 to magnetic fields and because it will regulate the drift available turbulence and the corresponding turbulence, transport, so it will trigger some internal chat about barrier that improve the confinement.
08:10:08 Also, it closed the feedback loop, and without any additional transport. So, this staircase structure is very important for magnetic fields, it
08:10:23 Also, the ecology of feedback loops must enter the scaling of the spice of structures lacks a potential will also proportional to the mixing lands and related to the Icarus Berry.
08:10:39 Besides, they don't know flow.
08:10:43 The Cosby staircase is also a class they call high turn observed in the magnetic fields and devices, hair as the result of that is Allah code.
08:10:56 It found the CO existence of the e commerce space staircase, and they danced a car license.
08:11:03 So, this a cross the staircase is very similar to the previous staircase and automatic death, and we may learn more about the about this, maybe tomorrow.
08:11:17 Under the typical characteristic of all the crawl space staircase is that there exists a very, very clear case elections.
08:11:29 And there's a two regions for the staircase.
08:11:33 I mean, I'm here, the relative flight and a wide ranging may be called by the steps, and another region maybe narrow, but a steep cut jumps.
08:11:48 So we can clearly see that the ski between jumps and steps are quite different, that we want to learn our studies the physics of escape directions to have a relatively well better understanding of the formation mechanism for meeting mechanism of staircase.
08:12:13 And to achieve this purpose we need to pay attention on the mix the Lancer.
08:12:20 And if you're really our hybrid, our children love skills means a constant skill and our dynamic last skill.
08:12:29 Actually, the rise bill is a very classical dynamic alongside skill, it as the divine the eyes they were they the correlation rage across as they miss mismatch frequency and expressed as they are immersed immerse to the PVA gradient.
08:12:51 So, if we take the dynamic Lance to the definition of the mixing Lancer, we can see that makes you laugh at ourselves also related to the PV gradient.
08:13:03 So, I can see that. If the, the skill of the system as a smaller than Ryan skill and the system will have a short memory and the turbulence is right.
08:13:17 So it's corresponded to strong mixing.
08:13:21 Otherwise, is correspond to wake me see.
08:13:27 So, with those background, let's come to the radio, tomato from has gamma equation. And this model currencies tax rate. Finally, all, it causes the three equations, also the derivative in detail, it's not showing here.
08:13:48 It consists of three equations, also the terrorist in detail, it's not sure here. And for the first day equation for the ability of a mean density hKh include the they do feel safe, flax and the condition of blacks and the for the evolution equation.
08:14:15 is included some residual terms, and some typical diffuse a diffuse a reflects as a traditional term, also for the turbulent parties your ads choppy.
08:14:20 It has some to save high powered this positive and productive in terms.
08:14:28 I mean, the importance of a mixing long slow as the rip eyes is reflected by those coefficients, because those all those coefficients are dependent on the mixing Lance.
08:14:43 So, if we take the wind scale as an example of the dynamic skill, way, way kind of sees it as a flux of the potential of artistic versus the artistic gradient will heal the characteristic of as curb.
08:15:03 And that indicates, there, there, this is the by stable mixing.
08:15:11 Also, a voice deep, the PV gradient, we can see that there may exist as a positive feedback loop. And this positive feedback loop will cause our drabs, The formation of a pattern, and the leads to some nonlinear fixture features.
08:15:31 So, if we saw was those three equations together. Finally, we can observe the density staircase, and the world has take care of patients from the model modulation law process.
08:15:46 So, we can give one waterfall conclusion.
08:15:50 I mean the staircase pattern as a consequence of a modulation or by stable feedback existence is very similar to the shock absorber because they suck away from the deal to self steepening of our reps that.
08:16:06 Right. So, as we go further, I want to know about the feedback loop.
08:16:15 And as we introduce the mixing last May related to the cross species area. So, the first the candidate for the feedback loop as a care.
08:16:27 And if we observe would detail of the ryan skill, we can see that both the density gradient and the artistic radiant can also act as the feedback loops.
08:16:40 So, that exists the multiple candidates for the fall closing this loop. So, we are wondering which one is AK.
08:16:57 To find this answer with first only considers the equal speech area in the system.
08:17:07 Then, we find that the staircase pattern cannot proceed. For a long time, but fy recover. I mean, Cape the ryan skill in the system. We recovered staircase of structures.
08:17:19 We also see that the characteristic of the staircase show those three stages. I mean the first cell is a micro skill or instabilities will form our drive some nonlinear muscle skill structures.
08:17:44 I mean those massive scale structures, then merge, and the Mitch kt to the idea so educated to the edge. So finally, to form some staircase, or barrier profile in the system.
08:18:01 And since the skill included the density gradient underwater stadium. So way farther. Turn off the road test a gradient, we found that the staircase pattern is still there.
08:18:16 So, we concluded that the formation of the staircase is mainly through the nonlinear density gradient feedback loop.
08:18:26 So that's the Caleb.
08:18:29 And this theoretical result is cognitively consistent with some experimental observations on the mercy of HL to a also in China.
08:18:44 Secondly, way, that is the sensitivity of the staircase to the gradient to Dr. White sky and the initial density gradient chair.
08:18:56 And we found that the, the number of the steps will first delay increase, and then decreased with with was the initial driver.
08:19:19 Correspondingly, the height of the steps that will first delay, our job, and then increase the wisdom, initial gradient. So, on their master exists some minimal type of skill in the system and physically this minimum was type of skill will determine the
08:19:29 buy some balance between the initial free energy given in the system, and as a, as a further diffuse a but this patient processes in the later stage.
08:19:44 So animate the balance between the energy and the and the. Dis patients will define some minimum of skill.
08:19:54 And one more interesting result is that we also find this llama on that uniform structures of the pattern here it shows that is a different or is a different radio production so the size and the height of the steps may be different.
08:20:13 And this Nyoni formal structure is very interesting fast passes the raven will be explained later on the last but not least, is that way, also studied the interaction between zonal and amnesia.
08:20:31 For the for meeting and sustainment of the, of the pattern.
08:20:36 As a first step with fix the zone OCR patches gang of very nice year.
08:20:42 And we found that if we enhanced the meteor, the staircase pattern will wait we'll be vacant are even destroyed.
08:20:53 So that means that the staircase pattern remains memory of initialization as a second step away fix the minutia, but very zone OCR, and we find the similar results.
08:21:11 It shows the stronger zone OCR tend to waken, or even destroy the pattern.
08:21:20 So, if we combine these two steps together we always give the conclusion that both days Oh no, and me share will affect the formation of the staircase.
08:21:34 And the reason for the night only for more structures, is also explained, because this night uniform structures, I mean, in different regional productions, the height of the steps will be different.
08:21:52 And this now uniform structure can form, when they don't OSHA is comparable to the media. So we find the pattern of the reasons for this now uniformed structure.
08:22:06 So,
08:22:21 because I only briefly introduce the field results of our walk, a more work, and more research can be found in the papers so I mean I gave the closest is that we have studied the existence of a staircase, and the artistic crushes is a hardcover book tiny
08:22:30 systems, and we find that the nonlinear density gradient to dependence, as the main feedback loop. And what a surprise us is that the sharing feedback loop is effective, and also the interaction of zonal and the media will affect the formation and the
08:22:57 generator some not uniform structures and the minimum or skill of the pattern is the determine the buys a free energy that this patient, the period of the Xhosa and so on.
08:23:02 So one word for the conclusion and the reason for the staircase is very simple. I think it's very simple because both the characteristic in the participant, and then state are formed by self, Stephanie, or modulation.
08:23:21 So that's all for today's presentation, and tomorrow I may be share my results. If you have interested please come and the sense for your attention.
08:23:36 Right. Show.
08:23:46 Big clap for wasting outstanding final slide, of course, and questions oh we have some life already David.
08:23:52 Oh yeah, it's pretty interesting talk.
08:23:54 We. I just wanted to note that in the case of just for diversity.
08:24:09 I'm Richard Scott nine, some years ago and I think 2012 found that a big differences found when you do spectral narrowband forcing versus broadband physical forcing or any kind of forcing which is broadband.
08:24:16 So, in particular, we didn't do a broadband physical injection or diapers or quadruples in for volatility forcing in the bear tropic Qg system. So conflicts in terms but anyway very choppy who unfortunately equation with forcing and we found that you
08:24:33 got very unequal staircases how some large some small. So, only when the forcing was narrowband Did you get a beautiful classical staircase pattern but otherwise it's not generic.
08:24:49 Just a comment, not a question really
08:24:53 go ahead pet.
08:24:56 Just, just to comment here I mean, here the forcing, there is forcing because you have to somehow get through what would be the linear growth phase.
08:25:09 And the way it's put in its broadband, as in assets so there's even some, some correlation with what you say, although one would have to. I mean, one would have to do, it'd be interesting to ration to do further studies right to look at that.
08:25:27 Okay. Okay. Thanks.
08:25:39 All right, comments, All right yeah my think is next.
08:25:40 Thanks, we have two questions or clarifications. The first one is regarding. I think this slide just before this one.
08:25:49 Did I understand correctly that you would somehow invoke an evolution principle that would give you, you know, the system would pitch, the max number of steps, or the men in height, and you would say this is your evolution principle so that's where the
08:26:06 system would lie. So, if so, why I missed why you, you will have that evolution principle.
08:26:17 Why I have this evolution.
08:26:22 Yeah why amongst all the possibilities that you have scanned for instance credit, going from 4.5 to 5.7. Why would the system pick five, for instance, meaning it would take 20 or 18 number of steps and step size corresponding is that height of corresponding
08:26:47 altitude. And why not something else.
08:26:51 Well, maybe you listed what you said.
08:27:09 I, you are wondering about the range of the x axis right.
08:27:03 And what I understand is that you scanned
08:27:08 scanning read and you get the result number of steps and height of steps.
08:27:27 Amongst all the possibilities that the system can can try to accommodate, why would the system choose to be at grad n equals five 5.1, in your case, meaning you have 19 steps or 18 steps, and the corresponding step, height, or can you have other combinations.
08:27:39 certainly can have other.
08:27:42 I think what we indeed have other steps, by, by choosing, they are by changing the range of the initial gradient.
08:27:57 The answer is yes.
08:28:01 Are you sure you might be Chairman, I go, I think it's just a peek into curve. There's no statement that we will actually go to the peak.
08:28:12 All right.
08:28:13 Okay, I'm coming back to Tony I have a second one coming back to what you see on the ground and she dug versus the sheer feedback.
08:28:24 Well, David ritual showed last time.
08:28:28 That sheer in his system was quite, quite important or if he's gonna be the, there is a clear analogy with our Katanga to us here. So, why, in your case you understand better why sheer would not be that important, especially because you can think of.
08:28:49 You can think to translate your, your rule on on the, the, the, the length, the mixing length on the time scale. And there, it's not obvious that the time scale would be the time to kill of sheer but at least you can translate your length scale into a
08:29:08 time scale.
08:29:10 And then you have you would be with a probably shear in the mix. So, did you have any idea about that.
08:29:19 Oh, to be honest. Yeah, I didn't continue on too much says after this Walker, but we are trying to include the, I mean, the hot is a more self constant of the new share.
08:29:38 Because here.
08:29:51 Actually, the initial result is that if we include the Sharia in the system. I mean, in the this worker at the earlier stages, they staircase can confirm back to the source team and
08:29:58 it's not working. Only by keeping the sharing feedback loop. So, I mean, The plan is to study further as your side. I mean, they focus, it may be on the meteor effective.
08:30:16 I mean, we need to extend that model to include more FedEx
08:30:25 or May I have a supposition for you. Then, I think, you know, Ali I'll explain some of this tomorrow because I realized we're speaking in tongues for our colleagues outside the fusion business.
08:30:42 But this model is the simplest possible model and of course it's that is beyond pasa gala Mima. Yes. And, and that's that has to go a walk autonomy or collision or drift wave actually should be the second day of my save system, and the deal is in that
08:31:05 system the phase you know there's there's, there's velocity in particles the phase between velocity and particles is basically fixed by collisions right the publicity parameter.
08:31:21 Right.
08:31:23 I think if you were in another regime or a different system where there was more play in that phase, the sheer would get in that phase and you would see more sheer sensitivity.
08:31:37 We could talk more about that it may be a bit of the, of the strongly dissipated character of the model.
08:31:57 Thank you.
08:31:47 I guess Yannick didn't have his hand up but wanted to ask a question so Yannick yes hi wishin nice talk. Thanks.
08:31:58 I have a question regarding the coalition's are there, the staircase, if I understand correctly, what you said that you do not observe this merging on long time, provided you take into account the Rhine scale.
08:32:15 Am I correct, yes.
08:32:17 Yes. So, two questions regarding this.
08:32:22 Do you understand why first and the second one is whether you you have tested or you might think of a way to test the robustness and the strength of this absence of merging by for instance hiding some noise or
08:32:46 are not nice actually falls, I think.
08:32:54 I can't understand why they haven't included something like noise but way, way, way indeed scan some parameters lack the lack of the cloud, number, the ratio of the driver to the discussions, something like that.
08:33:17 I mean, kind of robust of the formation of this pattern, but as a noise. I think I haven't included.
08:33:32 it again if I may, Yannick Hold your horses until tomorrow. All right.
08:33:40 There are.
08:33:41 We have done. there are mergers okay and not only are there mergers, but their mergers of both types of the form that Neil bomb fourth was mentioning on Friday, right there's two different types of mergers there's that.
08:34:00 Both are seen. And in another form there have been some robustness studies done so that's sort of tomorrow's discussion. All right.
08:34:17 Anyone else for Wait,
08:34:25 let me ask you a question. I mean, we're, you know, just in your own mind way, where would you go next with this.
08:34:38 I'm like near tab for me to include the day, as I said, in course include the course be I mean the knee course be sharing to the model more consistently.
08:34:53 I mean it would be interesting to find in some sense of threshold and meanie Crosby, your to kill the staircase right.
08:35:04 That would be interesting.
08:35:10 And I'm also curious about the bloodstream and say to me, to extend to black Stephen case right well more coming on that tomorrow, so I mean there's, there's room for further studies there of course that's it.
08:35:30 That's an interesting question. and yes,
08:35:35 anyone else.
08:35:41 All right, let's thank you again for an interesting talk and I'm sure we'll be hearing more about this later in the discussion. So our next speaker today is on a freshman who I think many of you know but hasn't been introduced so we'll do so so Ana did
08:36:03 her PhD and Weitzman Institute actually with her advisor was Grecia.
08:36:10 And some of you like me may remember her from the 2014 wave flows program when she was a student and participating quite actively. She was a postdoc at the Princeton Princeton center of a theoretical science, and is now on the fact of the faculty in physics
08:36:31 in the Technion and she's going to tell us about quasi linear theory and steady state and without wave so please go ahead.
08:36:41 Thank you very much. And, um, yes, I'll talk about something that is very much simpler than, then I think most of the things that you've heard before so I will have no waves so no better playing and and I'm also going to talk about steady state so no
08:37:00 dynamics. And for this simple problem. I'd like to say, what are the things that I think are understood or could be derived analytically or hand-waving Lee and then what are the things that are, I think, are completely unclear.
08:37:16 So I start from from Debbie stokes equation in today, and you have a small scale forcing and you also have the friction and and if you look at what happens after you wait for a long time.
08:37:30 Then you'll see these. So this is a periodic box. Square purity box so you will see a vortex dipole in a sea of turbulence and the goal would be, then to describe this mean flow and the fluctuations around it and when I say I mean flow, I mean, let's
08:37:47 just track one of the vortices and average in time.
08:37:52 And let's, and obviously you have a mean flow and then there will be fluctuations. On top of it. And I do need to specify some things about the forcing.
08:38:03 So, one thing is that it's going to be small scale so for example for the simulations it's like 100, compared to like one over 100 compared to the boxes.
08:38:15 They, so both in the theory and although it might not be that necessary.
08:38:21 A white in time forcing is used and it has zero average, so it doesn't force mean flow by itself. And with of course convenient about one in time forcing is that, then you have a constant energy dissipation rate.
08:38:35 Okay so so these are the quantities that we're interested in. So we are zooming in, into one of the vortices.
08:38:41 It's quite as a traffic.
08:38:44 So we can describe it by a minister which is purely as a Moodle, but it has some radio dependence. And then we're going to denote by small, small v the velocity fluctuations around that to me so.
08:38:59 Okay, so, so if we want to, to make some analytical progress then we need equations for the main flow. And, of course, the range of stress.
08:39:11 So, obviously you know that one of the equations is just a momentum balance. So in this case it will be an angular momentum balancing will have flux of turbulent angular momentum, which is balanced by a transfer of angular momentum of mean angular momentum
08:39:25 to the bottom, because we have this friction so this is one equation that we have.
08:39:30 And then for the second equation will have to consider energetics. So, what happens in this system so of course you know there's no risk escape but so okay so in this system, we have forcing.
08:39:44 The first thing is injecting energy directly into the fluctuations.
08:39:48 And then the fluctuations are transferring that energy by nonlinear interactions to the main flow, and the quality linear approximation is going to enter because we're going to assume that right there's no really first cascade it's really these nonlinear
08:40:02 interactions between the fluctuations and then the flow, which has, which are responsible to this transfer of energy. And then the main dissipation is happening because of the friction, but basically the dissipation of the main flow, which means that
08:40:15 we have to build. So we're working in a limit of a very small friction, such that, in order to dissipate a finite amount of energy which has been rejected at each moment.
08:40:26 We have to have a very high altitude of the meaningful.
08:40:29 So this is the main energetic balance that we have.
08:40:33 And as you know, probably the the energy transfer is mediated by the reign of stress right by the way the work that the renewal stress does against the stream.
08:40:43 And in this case there are no stress is given by an angular momentum flux and the industry's is the is the shear thats related to right to to rotational motion.
08:40:55 So this is. So, okay, sorry.
08:40:59 So in principle we can ask about the energetic balance for the fluctuation so these are things that we had mentioned so far.
08:41:15 In his talk so we have a constant energy injection right so we're using a waiting time forcing so we kind of know everything about what's being input in terms of the energy.
08:41:18 And then there's the nonlinear transfer which is local so at each point the fluctuations transfer this energy to the main flow. But apart from that, and notice that here I'm completely neglecting the dissipation of the fluctuations because I'm assuming
08:41:32 that there is a finite fraction of energy that goes to larger skills. So in a sense, I'm kind of assuming this epsilon is finite.
08:41:41 It's close to.
08:41:42 Most of the energy that's being injected.
08:41:45 And then, of course, there, there could be spatial fluxes of energy. And that's the thing that's preventing the closure of this problem.
08:41:54 And since I'm using the quite a linear approximation which I'm going to justify by the fact that there is very strong sharing at the forcing skill. So that, and the sharing and also above, so that the shearing doesn't allow the fluctuations to interact
08:42:11 with each other. So we just dropped the cubic term, but you're still left with this V Peter.
08:42:19 So in principle this.
08:42:21 This is on this level this is, there is a close description to the system is the cause a linear system and it's basically what Brian feral also talked about, and stiff device.
08:42:40 Because of this pressure term which is not local and efficient said that, basically, what he what they did is they they threw this term away, and I'd like to argue by symmetries why this is possible and and there is a bit.
08:42:54 There is a bit more to do than just surgery but this is what I'd like to do here.
08:43:00 So first I'd like to argue that there's something special about the momentum flux, and that if you look at energy as a target with energy it doesn't really behave in the same way like a few squared and b squared.
08:43:12 So what's special about the momentum fox is that is the point. Okay, so there is this symmetry in the system which is the parity and time reversal symmetry, so basically if I, If I reflect.
08:43:27 I think that the angle five to minus five. And they take time to minus time. Then of course this is a symmetry of the Euler equation, and without forcing and dissipation and without an inflow This is a symmetry of the system.
08:43:38 We also have the main flow, but the main flow is also symmetric under these, so it reflects its, it changes its sign on the reflection of fire, and it changes its sign on the reflection of time.
08:43:50 So, it totally doesn't change and also all its derivatives are also invariant under the symmetry.
08:43:56 But on the other hand, the momentum flux is odd with respect to the symmetry.
08:44:02 And also this week Peter is also armed with respect to the symmetry. So this means that either forcing or dissipation must be really important in between the momentum sucks and if we didn't have forcing or DC patient, then we wouldn't we would have zero
08:44:17 momentum stocks, and I'm guessing this is maybe trivial things.
08:44:21 So, so the main thing that breaks the symmetry for the momentum flux is the fourth thing because the first thing is actually always there, and we do expect that we have transfer of energy so this makes sense.
08:44:35 But here with now important is that the force, we have assumed that it acts it's very small skills.
08:44:42 And this means that for most of the force mode so now we can think about the response.
08:44:49 The contribution to the momentum flux, that's coming from the modes that we force by the forcing. And for most of the modes that we force because they are the forcing correlation length is very very small compared to the box scale.
08:45:05 And it's very very small, if we assume that it's very very small compared to the, to the radio to the radiance of the main flow.
08:45:13 This means that the only thing that are basically sensitive to is the sheer rate. So they basically feel the main flow is if it's just a linear sheer.
08:45:23 So, we can kind of zoom in and use a Cartesian coordinates.
08:45:30 And if we did that, then we find that there is additional symmetry in these Cartesian coordinates, where if you take.
08:45:46 If you assume that this. The fourth thing is also statistically homogeneous. so it doesn't depend on the radius or, or, or any says a topic.
08:45:49 Then, if you take x two minus x and y to minus y and these local coordinates, then this the system would be symmetric to this.
08:45:57 So again, the lawyer equation is symmetric to this and also a linear shear inflow is also has the cemetery so it's it's your chance to itself under these couple of reflections.
08:46:08 But, while the momentum flexes invariant under the symmetry because you go so you start Yes, you go goes back to you and he goes back to be sorry you goes to minus you and he goes to minus v.
08:46:13 So, in total This doesn't change the VP term that's to sign so it's all under this reversal this cemetery.
08:46:28 So this means that it must be zero. And you can also do these linear, sort of, you can also do their xe, a calculation that shows that for most of the modes, this is really a good approximation.
08:46:42 And so most of the modes contribute all of their energy into the momentum flux. So this gives you a second equation.
08:46:51 And in total, you have two equations and these are already plus that WeChat shown as well so these are the analytical solutions that you get by solving these two couple of equations.
08:46:59 So you get an equation for a solution for the momentum flux and a solution for the main flow.
08:47:07 But, you might want to go a step further and you might want to ask about the energy profile so they use squared and the B squared, and then you can do, because now you know the mean so then you can write a two point of an equation for the two points correlation
08:47:22 function.
08:47:23 And it would look something like this.
08:47:27 And then you can well first say, Okay, I am assuming that the dissipation is really not important for the turban fluctuations because most of the dissipation happens in the middle.
08:47:39 So I'm just going to throw that away. And then another assumption which I think is less trivial is, let me first be interested in. In the long sort of large scale properties of the flow.
08:47:53 In that case, if I'm if I'm looking at correlations that are beyond the correlation length of the first thing, then I can take these. These correlation, these terms to zero these forcing terms there's zero because the forcing is correlated and they're
08:48:07 really, really small scales. So I'm just going to take it to zero, but then eventually what I'm going to do is I'm going to take the two radio, and I'm going to merge the points, and I'm going to argue that nothing dramatic happens because for the energy
08:48:20 it's really supposed to be determined on the large scale and it doesn't really care about the small scales, as opposed to, as trophy for example.
08:48:29 So, you can do this you can go through this and you get an equation you get a closed equation for the two points correlation function and if you assume.
08:48:37 In addition, scaling variance, because of the form of the equation, then you can get solutions. And you can compare them to numerical situations, of course you get a family of solutions and you don't really know how to choose that.
08:48:49 But you can so so you sort of by taking some features from the simulations, we then can predict that there is also part, which goes like one over r squared for the, for the square.
08:49:03 So these constants are just have to do with the fact that the condensate the vortex is moving.
08:49:10 But this second part is not something so trivial. And so just using the fact that there is a constant, and there's no contribution to your squared, you find a unique solution that should go like one of our r squared and this is what you see here that
08:49:23 matches very well with the simulations.
08:49:26 And I think the surprising thing about this, about this are there are two things. So one is that these are zero modes of the equations. So in principle, if you're thinking about Class A linear theory.
08:49:38 The Zoom Lens should not contribute right because somehow if they do, then what prevents them from growing and becoming dominant or just destroying the closet in your impersonation.
08:49:48 And, and actually also what we find in the numeric is that the scanning the way that they scale, identify what delta is but they, the way that the scale the scale if u squared so they scale like one over the friction, actually.
08:50:01 So in this sense they don't seem to be suppressed but they actually suppressed by some seem to be suppressed by some factors that are related to the ratio of skills between forcing, and the box scale and and that that probably comes about from from sort
08:50:17 And that probably comes about from from sort of a scalability, condition where you because they are zero modes, you need to have some balance between the dissipation which is of us squared b squared, and the forcing and the forcing probably contributes
08:50:33 in a very very small region.
08:50:34 Ok, and now I want to say, so please stop me if I'm running out of time, because maybe this is too low.
08:50:40 So I think I say, don't worry. Keep going.
08:50:46 Okay, so, so now I want to talk about something that we don't really understand. So here, before we had a periodic box and square box. And if we extend the box a little bit so we changed the aspect ratio.
08:51:02 What happens is that you get these vortices in addition to I mean you might expect you will just have jets. if you had just you could apply the same theory that are talked about, just by, you know, because what was really central there is that you have
08:51:16 But here, this is not the situation. So really it's not really clear what to do. And if you want to defined a mean flow here, so of course you can zone in the average, but then if you don't know the average then the fluctuations are certainly not going
08:51:36 to be small, and you somehow want to take into account these large scale structures which by the way diffuse and move, so they're not stationary.
08:51:46 So you do want to and they don't stay on the diagonal like this, they actually can move with respect to each other to larger or smaller degree.
08:51:55 So you'd want to do kind of device some kind of, it seems like it would it would seem like this is still a system where there is a lot, sort of strong mean slow and there's a small scale fluctuations.
08:52:08 But how how do you extract that, how do you write a theory or even a simulation, like, even in a miracle, sort of code that that treats this in this way.
08:52:20 And then there is an additional surprise so here this was a, so these plots are kind of short time averages they are averages over the time scale of the kind of thats related to the, to the main flow circulation.
08:52:35 And, and this is a snapshot where you see the two vortices and now if you decrease alpha with it. So, alpha is the friction, and that's the thing that controls the amplitude of the main flow.
08:52:45 So in a sense, if you increase the amplitude of the main flow, what you see is a bifurcation, where there instead of having to vortices, you have three.
08:52:56 And these, and so not only you have some cemetery breaking. And you, apparently, I mean, I'm guessing that these two solutions coexist in some regional parameters.
08:53:08 You also have these two verses they're actually lying on on jets and they're moving quite fast. So in this case you have a coherent structure which depends on time, but still somehow you would want to describe it as there is this coherent structure which
08:53:32 is very strong and there are small fluctuations fluctuations on top of it, but obviously if you do some temporal average that that doesn't really make sense. So, so it's not entirely clear, and even as a short time average like this wouldn't work because
08:53:36 they are moving on the timescale of the main flow.
08:53:51 So, yeah, so this is more or less all I had to say I just want to say that there's some I think some interesting qualitative features which are similar between this very simple system where you just to cast typically for small skills.
08:53:56 And this is from a simulation of really Bernard collection in also in an aspect ratio period periodic box with a non square aspect ratio. And what you see is you, again.
08:54:07 So here it's forced by 3d production.
08:54:10 And you see that there are these two verses as well and in red here you see the participant profile from these two these simulations and this is the participant profile from the 3d simulations and I think it's quite striking that you have this region,
08:54:26 which is flat, which you could construct a theory for but I'm not sure how relevant it is, since you do have these strong vortices which you don't know what to do with in a theory, but but I think there's definitely something that's going on here and
08:54:38 I don't know if it has anything to do with fourth density homogenization or something like that.
08:54:45 So, so I'll finish with just some questions that I had.
08:55:03 So, yeah. So one question is the importance of the zero modes and I think it's sort of related to do what that had mentioned but not necessarily so. If there is zero Mozart there, and also they scale.
08:55:06 So there was this issue of kind of justifying the cost saving your approximation in the better playing where, where you assume that everything skills like the momentum sucks, but actually what we see is that the momentum flux is very very small, so it's
08:55:20 scales like squared of alpha, whereas the, the energy, for example the fluctuations they scale, they do not scale, they are not suppressed, they're not smaller than you squared, in a sense, but it's only because of the forcing being very small scale.
08:55:33 So there's also this question of what is the role of small scale forces it's really important to have such a separation between skills or not.
08:55:42 And then the question that already raised is kind of how can we use a linear approximation of what can we do in the case where there are these complex flows and I guess also something I didn't mention is, I was wondering if these vortices if they appear
08:55:58 as some kind of instability of the jet solution or something else.
08:56:06 And then there is also the question okay we are we are putting by hand, forcing, and that's why we can basically solve everything because we can ask what is the response to some stochastic forcing.
08:56:18 But what happens if it's if the flow is driven by an instability or something like this. In that case, we also have to, I guess, do something with the fact that the system should be chaotic.
08:56:31 And what's the role of that.
08:56:32 And then of course they haven't said anything about waves and wondering how, how do we have changed the stuff that I talked to them.
08:56:41 So with that all thank you for your attention.
08:56:45 All right, thank you very much for very interesting talk.
08:56:51 Questions, ladies and gentlemen.
08:57:07 If I my favorite question, but I like on it too slow. Yeah, good you saw a little bit back as the slides to this. Yeah, this one. Yeah, no, this is a different aspect ratio in vortices.
08:57:12 What I just want to make a short.
08:57:16 Yeah, this one. Yeah, yeah sense. So, in a sense, one way, a good guess if you would, in their, in their aspect ratio equal to you you would give in your cascade go into a Lewis the wave number which means is they will digits, which are parallel to the
08:57:35 the short side but it's not an IT, this shows is basic trivial difficulty of linear approximation namely, it cannot work. Wednesday, she wants him to zero, or an October now but.
08:57:50 So, if you cannot have biologists in parallel jets, you always give somewhere.
08:57:58 Velocity derivative changing science of shift turn it into zero. So, Now, it means it zip flotation must be strong in this places, and it basically what happens is the result is that you this throw flotation organize themselves into this world this is
08:58:14 so this work this is in the sense in between kind of Jeff's where, where you would expect it. So, if you know average every single over very very long time this word This is of course wander around, you will live as all flow but the assumption which cannot
08:58:29 be described everywhere because in linear approximation, because this word This is assumption question you present a very strong for patients, but on the shorter time school every day.
08:58:42 this year, where me and flow is not straight lines is something different. It may be wins aspect ratio is very, very large, this could be done in, in terms of Kelvin vortices default kind of very very elongated Kevin Waterston so this is essentially what
08:59:04 we see here is very much like like given what this is organized political were so this way.
08:59:10 I find very tantalizing that I wasn't able to do it but maybe somebody someday.
08:59:16 Do it, because it was a remark know the question.
08:59:21 Alright, so the question order is Edgar Pawan Kumar, Adrian and then me. So, Edgar.
08:59:33 Hi Anna thank you very much for your talk, I just really had a comment about the last thing you, you mentioned in the work by David Hughes in selling Garvey, of course, they study a carnival system you know with with rotation.
08:59:48 And so there is a marked a symmetry between cyclonic anti psychotic voices in the system.
08:59:55 That is not presumably the case in your problem, which is not control. Yes. And so I'm wondering if you can comment as you can, you know, you can inject for diversity at small scales via the forcing somebody you're not doing that so can you comment on
09:00:12 this.
09:00:13 Yes, I think it's really interesting so you can see this, so actually you can see this on the spot right so this is why this orange vortex is here, and this red peak, but it's actually not here in the, in the case where the system rotates right because.
09:00:28 So, so this is actually something that I think is is is both interesting and also not clear to me because it seems like the fact that you're rotating is somehow regularize in what's happening here right here you are I don't know destroying the vortex
09:00:43 or it cannot be sustained or something like that right because, because I really do think it's it's because it's the same pattern but the only difference is that you don't have this vortex which has the anti cyclonic thing that that would destroy the
09:00:57 to the turbulence. And so instead you do have a just to kind of basically a constant mean flow, I mean sorry a linear shear here, right, because this is where to city so this is very very constant, so just have like a linear shear over here.
09:01:13 And this seems like what you would want to have over here as well but but you can, because you have the vortex, which maybe it's like a Calvin can get die or something like that but I think this is really interesting but I can't, I don't really know what
09:01:27 to do with the yeah how to see that the fact that you're rotating somehow or maybe it has to do with the 3d fluctuations I don't know that it changes that mean so in this way.
09:01:41 I think this doesn't answer your question but.
09:01:44 Ok, OK, I think it's still an open issue.
09:01:50 All right, whoa on your head a question. Yeah, yeah. hello Anna, thanks for the Nice, nice, nice No. My question is that, I think that this is a very natural.
09:02:15 Nonlinear energy transfer, you know, Linda phenomena at large scale at also sort the scale.
09:02:10 I think that the lead nurturing is applicable for the weekly nonlinear system. If somehow the system is astounding nonlinear you do think about this, if the system is this tongue in nonlinear II, I'm not entirely sure I understand what you mean.
09:02:32 It's totally nonlinear system because because the linear theory is applicable. I think that in practice, if perfectly nonlinear system, if the system is strongly nonlinear like.
09:02:47 Yes.
09:02:47 What do you think about this.
09:02:50 I mean I think so I think someone had mentioned that this, this is also nonlinear, just because you have the interactions between the main flow and the fluctuation So in some sense it is nonlinear.
09:03:00 It's true that the kind of the main, they're the most interesting on in you already we throw away but here at least the justification is the fact that really these, you don't have enough time to have the nonlinear interactions between the fluctuations,
09:03:15 because the sheer just catches you too fast. And so you just don't have time to do that. Of course if you if that's not the case, then, then it's right i think it's a question of what can you do because Because really, then it means you somehow have to
09:03:30 combine.
09:03:42 Okay, Thank you.
09:03:44 All right, next customer is Adrian.
09:03:49 Thanks for this is really cool talk it was getting really clearly. Thanks, I unfortunately have have less interesting questions.
09:03:58 I was wondering. Two quick ones, the same slide where you're showing the aspect ratio stuff from the first question.
09:04:05 I was having trouble following I was wondering if you could just briefly explain what these are plots of and then also the the main question that I had is.
09:04:16 If I understood correctly you assume that the fluctuations sort of see the background flow, like in a local sense that there, that the fluctuations that they don't see the global for flow profile but they just the rental stress depends on the sheer at
09:04:32 a point.
09:04:33 And I'm not having worked with beta plane so so maybe I'm just way off base I'm wondering how how strong assumption that is because certainly in, like, Calvin Helmholtz cheer flow stuff.
09:04:44 I think that that isn't a fair assumption so. Is that a fair assumption here.
09:04:51 So, the first, so maybe I'll answer your second question first.
09:04:56 So it is a strong assumption.
09:04:58 It's the, I think the reason, it can, or it does. I think work here is that, so it doesn't work for all the modes. Okay so, so basically what you do is you look at the forcing nodes and you look at their contribution to the rain no stress, and it won't
09:05:17 work for all the modes because there are some angles right it's k equals zero, right around cable zero for which this completely fails, but those modes, the sheer approximation anyway right and that's also true by the way for.
09:05:31 So basically what you need is you need the scale of the main flow right the kind of the scale on which the meaningful changes to be much larger than the scale of the, of the forcing.
09:05:44 And in that case, you, you, the kind of the angles or the modes for which you can assume this, that it's a linear sheer are a very small, they're kind of the ratio between the, the scale of the mean so and then scale of the forcing so you have like this
09:05:56 small angle and of course as you go into the core for example rights to smaller API. When the radio is comparable to the forcing scale, then that completely fails.
09:06:07 And by the way, it kind of this sheer approximation that everything is dominated by the sheer also fails when you go far away from the vortex and you don't have strong shear, which I think is also something that the patient was referring to.
09:06:19 And for your second question. Um, so yeah sorry I just kind of didn't really explain this. So, here what you see really what you see is you have this, this is the participant plots, like a density snapshots.
09:06:33 So you see these large vortices and you can see there's like a color difference between here and here to show you that there are jets. And then if you what you do here is you just average in time, but I'm relatively short time so it's shorter than the
09:06:48 time, like one over the friction. So, there's no better here by the way the best is zero. So there are no waves in there, it's not a bad thing.
09:06:56 It's just worth visiting.
09:06:58 So, when you yeah so you just average over short times. And once you see is these books so you see I mean flow.
09:07:07 And you see that, like you don't really see fluctuations and the, the red line is this apparatchiks.
09:07:15 And yeah and it's just streamlines that are plotted here. It's just meant to show that, as you change the aspect ratio, what happens is that the separate trick separates, and you get a jet in between, but you kind of see very weakly here but it's just
09:07:31 a fluctuation. And in the data that you're doing the short time average to get this, is that a fully nonlinear simulation or is that yes yes yes, this is DNS of the, of the novice stokes equations intuitively with forcing and friction.
09:07:47 Yes, thank you.
09:07:50 All right, so I'm next but I have several questions do you, do you have a long he or a shorty David.
09:07:59 Was that a more remarks than questions, I suppose, just just in response to what Edgar and Ms said okay.
09:08:10 That's what I thought. Go ahead and then I'll that I'll take that I'm next to everybody else go ahead sir, the cloud.
09:08:21 But that's a very nice comparison, isn't it, I mean if we only understood it.
09:08:28 So, so with the one in the collection one. We will never get your orange vortex so to speak, if we never always unstable, the anti Cyclone so always unstable.
09:08:39 So even if we seated seated it, we never get it
09:08:45 when we, when we did the non.
09:08:49 When we did the problem with the square box.
09:08:52 Then we did all sorts of things to try and work out, you know, what modes were important because it looks it looks like today, doesn't it it looks like everything's coming along to them, you know, inverse cascade into determinants but when we looked at
09:09:07 when we tried to keep those modes in the simulation then that's not what we got. Interestingly, so we concluded that if it was a different beast.
09:09:18 though.
09:09:20 It looks as if there's something similar in what you have, which is kind of intriguing to them really understand.
09:09:28 Just a question when when we did this one that you're showing them we found that this, the symmetry breaking between jets and last season needed only a very slight deviation from squares, D.
09:09:42 Is that what you find also. Yeah, so I think these areas also go back to to Freddie cliche to say, like, yeah.
09:09:52 Yes, I think it's okay.
09:09:57 No, but I think I mean I kind of tried to dig into your paper understand and I saw that you did that study where it seems like it's not just kind of a 2d small scale forcing sort of thing, but yeah but i think i think there is something there that's.
09:10:18 Yeah, there is something there that's similar and again, there are obviously other things which are not. Yeah. Yeah. you know, that's really nice. Thank you.
09:10:26 So I had a few questions.
09:10:30 First of all, on the infamous VP term presumably you could just calculate it couldn't do by by in compress ability. And that's it. Simple enough and by the way that would be the form of an entrainment term, wouldn't it because p would be of order v squared.
09:10:48 So, it's got it. The the silly jargon we use in plasmas is is turbulence spreading or something so it would be qualitatively different. I mean, have you done that and just and kind of explored, when it falls apart.
09:11:09 So what I did is I.
09:11:11 So what I did is I did so okay so I should say what I actually did. So this is of course as always the cemetery arguments you come up with him after you do something.
09:11:32 And also I based, a lot of what I say on the work of other people like Khokhlova negative and who they are the ones that thought about this tree approximation and also some things of shady shade, even though I don't know I'm not sure that they would agree
09:11:36 with all they always said.
09:11:39 So what I did is okay so I don't have the fourth density equation here but in the participate equation for the fluctuations.
09:11:46 There are two terms right so there's the direction of the vertices by the mean so and then there is the addiction of the fluctuation of the rigidity by the writer you double prime sir.
09:12:00 I the second derivative of the meat so, so, so that term I dropped and said that, that's small let's assume that small, and then just assume, whatever profile you want for the, for the other term, so I don't assume it's a linear sheer, but I do assume
09:12:16 that the second derivative is much smaller than like the second derivative term is much smaller than that one. So if you do that, and you and you track the contribution of the different modes, from the forcing.
09:12:31 Then you can see exactly that, for the modes which are close to cake was to zero, you, you can use the sheer approximation the linear cheerful summation but that's a very small angle, and you can go through the calculation and there are some subtleties
09:12:44 there because you take the friction to zero from the start. So they're going to be integrals that were the limit of integrations matter, but I did do the VP so I did calculate the VP, and you write the equation for the VP, and what you see is exactly
09:13:02 that it's zero just by the symmetry I told you, I mean you see that it's basically, it's purely imaginary if you if you compute the zero. I mean, it appears in an appendix of paper that I wrote but.
09:13:12 But yeah, so So to the extent that I did the calculation correctly, you see that it's zero because of the symmetry of the x two minus x and y two minus one.
09:13:26 And you can show that the not the the the contribution from the, from the fact that the the means of isn't really a linear shear is suppressed, as long as you are looking at wave numbers which are, which have k that is far enough from zero, basically
09:13:45 because the stream function is related to the fourth density by other fashion. So, so once you have the case, they sort of localized things in why a sec okay thank you a second question, sort of on the symmetry issue is do you do you consider.
09:14:07 I mean, it always strikes me in these things what we're really doing is we're exploring the stability or growth of a system with a week seed here. So if you want to understand you know for these problems of jets to various sort and what will happen, what
09:14:28 one should really I mean there's often a market difference in the symmetry of the system when you introduce the seed field right and that Tara leaders change and things that were imaginary become real etc etc.
09:14:45 I mean that you're making a lot of uses of symmetry Have you ever compare the differences in the cemeteries in the to the Bay Area equations and the seeded equations.
09:14:59 So I'm not entirely sure what you mean by the same because here I'm just assuming a steady state. So this means that I don't have I don't remember anything about the initial conditions.
09:15:10 I'm just like leaving forever in the States. And I do see that the state mean that, at least the first symmetries so I mean these are I guess are like harder to swallow because they're much more hand wavy.
09:15:24 But, but at least this symmetry is really something that you see you see that the fourth is is very isotopic. And then, yeah, so this is just a symmetry of the of the system if you don't have foreseen usurpation So, so there isn't really a strong assumption
09:15:40 there.
09:15:42 All right, well I wanted to ask you about the chaos issue which is very interesting. I mean, because we've been corresponding a bit on that. I mean, What is your view on.
09:15:56 I mean where, where I think that discussion goes is exploitation processes by for lack of anything better.
09:16:06 By mechanisms of incoherent mode coupling sort of a nonlinear noise, which of course is part and parcel people love to talk about negative viscosity but where there's one there's the other right i mean there it's simple consequence of conservation and
09:16:25 homogeneous mixing necessitates in homogeneous, if you will, nonlinear noise right again the two have to come in packages, how, what are your thoughts on that.
09:16:40 Yeah, so I'm not I'm not really sure how to, how to think about this so I think also, I thought that was was interesting so okay i'm not sure this is I'm pretty sure this is not answering your question, but I thought for discussion here now.
09:16:57 So, so in terms of forcing so i think i think Brian Farrell had an interesting comment that's that. Right, he can. So one way to represent the the terms that you're throwing away in the closet linear approximation is that you introduce a forcing which
09:17:13 okay then you can make it not white and stuff like this, but I think what was interesting in his comment was that. Well, the thing that you're not capturing is the feedback, right, is the fact that the forcing can change and these are of course also related
09:17:33 to the work of its garden so so I think this is a very interesting concept and idea and this is why I think for me kind of what's not clear about the role of cast is for example if you don't want to use external forcing and instead you use an instability.
09:17:43 Then, somehow, you're kind of introducing an average, where I guess you need to assume something right here I could do all the averages because I can always say oh it's just averaging over the noise so that introducing a stochastic forcing is very useful
09:17:59 in the sense, which once you don't have that, then somehow. Yeah, I'm not entirely sure what you're supposed to do and are you supposed to just like as soon as something about the averages or.
09:18:11 Yeah. How do you do that. I mean I would say a better approximation and sticking in a forcing going back to launch of an equation one on one would be to stick in a forcing and some, some viscosity or drag effect to make sure you conserve things right.
09:18:28 So you're in a particularly if you're interested in the cause the steady state and that's what closure theory will take you to. So in other words, if you have interactions, you will have both incoherent, and coherent mode coupling if you want to put it
09:18:45 in such high falutin terms. So in the two will have will come together right because, ultimately, the the pairing of those is related to the fact that the model has to conserve things like energy and then stuffy.
09:19:01 So, yes, but somehow I mean I see what you're saying now I think you're saying that the feedback is through kind of a dissipation that you could you could put, put in, like, similarly to what you do for the forcing.
09:19:13 But yeah, it's gotten in a way that's consistent with the forcing right that is yes exactly because yeah exactly, and you don't really have detailed balance right like you don't.
09:19:26 So, and you don't want to be, yeah, you really want to have dissipation which is, which is not in the same place where you put the energy so but okay that's that's a really interesting idea.
09:19:39 All right, Edgar's back will give Edgar the last word here yeah yeah sorry folks.
09:19:45 This is kind of motivated by some of the things that we've heard from David, and just now from Anna.
09:19:53 So I think, you know a key distinction that one needs to make in these problems is between what I would call active forcing versus passive forcing, and by active forcing i mean you know when energy is injected say through an instability, like in this
09:20:10 really been a problem that David talked about, and passive forcing where the force is prescribed you know once and for all.
09:20:20 So that's, that's one one comment. The other distinction i think it's it's very very important I agree with that.
09:20:29 Thank you.
09:20:31 The other comment I would like to make is about the work of Boucher and when I.
09:20:38 So they have made a prediction using techniques from equilibrium statistical mechanics, you know, looking for lowest energy states, you know that the jet like state is the low energy state when the aspect ratio the domain
09:20:56 differs from square by 10%.
09:21:00 Yeah, and that's exactly, you know what David and selling girl really find in in their simulations of, you know, rotating convection in non square domains, and it's also what Keith Julian and Meredith Plumlee found using this low rosebud number model
09:21:23 that I talked about earlier, for the same problem, you always find that it's 10% difference. That's the transition. That's the aspect ratio at which you have a transition from essentially vortex like states to jet lag states.
09:21:39 And it's a complete mystery to me why methods from equilibrium statistical mechanics, you know, make the same prediction, as in these, you know strongly driven distributive systems that are certainly far from equilibrium.
09:21:57 And if anyone understands why these different systems, behave in a similar way I would love to know.
09:22:04 Do they have a project prediction for the value of where the transition is because the prediction is that you will get a jet and you don't actually get a jet because you also get the vortices, so they don't have vortices in in the statistical mechanics
09:22:15 prediction. Yeah, exactly. They do find that the aspect ratio, at which you know you make a transition from, let's say, a vortex Darko to a jet lag state occurs exactly a 10%.
09:22:35 You know, non square aspect ratio.
09:22:37 So I think I missed the fact that you can. I mean, okay, you're saying that this is really a prediction of the theory.
09:22:57 Yeah. Are you believe should have nothing to do with these strongly driven non equilibrium systems.
09:22:57 So that's a, it's a curious, maybe it's just a coincidence. Maybe there is some deep thing you know that I don't understand.
09:23:06 Okay, I mean I'm sorry David I think we're we're getting a little is it can you make it really fast. Yeah, yeah and I was it was 8% done.
09:23:17 All right. Thank you. Good. that's my kind of comment. Thank you very much.
09:23:23 I think interesting is this discussion is we probably, to be fair to Yannick have to move on so let's thank again for very interesting talk and triggering a great discussion.
09:23:42 And our last speaker is Yannick Sarazen, and he also is.
09:23:50 I think hasn't been introduced so we should. So I think I first met Yannick in 1997, he was helping me out, the meeting at carry the room a near Marcee.
09:24:09 And he,
09:24:08 he did his PhD from University of Grenoble but really, really at CAA, and he's like many it seems to be a trend with ca he's been at ca ever since we're now he is director of research and he's made you know many notable contributions, and particularly
09:24:31 in the area of the study of transport you know avalanche thing and so forth.
09:24:39 Some of his notable papers. He's also one of the leaders in the development of the GES Ella code, and has contributed to the, you know, to the work on Lh transitions and other things going on in the CA group.
09:24:57 And last but not least he's a mainstay of the festival the theory which has been running for over, well now it's about 20 years.
09:25:07 So with that, Yannick please.
09:25:12 Thanks for the kind introduction.
09:25:15 Can you see my screen.
09:25:18 Yes.
09:25:20 Yes. Okay.
09:25:24 So I'm going to speak about the key role on the face dynamics on the Reno stress on the momentum.
09:25:31 Flex, and especially regarding a huge and plus metabolism that preceded this can be extended to other domains.
09:25:39 It's my pleasure to acknowledge could eat some of them are here, em, of union gallery and a lot of PhD student and master students who help in this activity and I also acknowledge the spot mentioned the festival or to be where we had some many interesting
09:26:00 discussion on this issues.
09:26:02 So the context and motivations.
09:26:07 As you likely know from the start that we already had Dublin's rotates Polina took him like if you look at here at Polycom what we call it put a little cross section.
09:26:20 This is a map of the polio virus et, which is self generated by turbulence and the color map correspond to the sign of the velocity. So, it rotates in one direction here.
09:26:34 And in another direction here and you see that tablets develops.
09:26:47 For scales, and sometimes for some parameters and conditions it can develop staircases. So this comes from a five dimensional committee simulation.
09:26:53 And the main flow is the crispy flow is related to the radio, electric field so this is related to what we call the cross be drift velocity.
09:27:07 And it turns out that the radio sheer of this political velocity, which is a mean flow because it's constant over the flux of phases which are to rye turns turns out to decline a turbulent 40, years, and it's it's likely the dominant mechanism for evidence
09:27:27 control, at least the iron scale. And it is expected to be, again the main player in driving transport buyers and it's beneficial for fusion because in can allow us to, to reach optimize confinement and hence better performance.
09:27:48 The main ideas here if you are vortices immersed in a sheer flow going to the left on the at the bottom to the right on top. Then you get declination, but likely can limit the deflect the turban infects in this direction which would be the direction of
09:28:07 the confinement, this would be the rate of direction here.
09:28:12 I come back to this in the next slide. So, it turns out also that turbulence as you, you know generates mean flows, which are called zone foes, and via this momentum flex the reign of stress, and I will focus during this talk on this issue what is the
09:28:29 dominant mechanism, which govern the rain on stress dynamics.
09:28:35 I will use two models and spend most of the time on a reduced nonlinear model, and then show that we recover some features. The main features with the outcome of this turbulence simulations, more advanced for fusion plasma.
09:28:54 So let me introduce the model.
09:28:58 We're interested in magnetized plasma with Nina when she news magnetic field. The geometry would be this one. So this is a sketch of the kind of fluctuation that you can get from a kinetic simulation.
09:29:12 Saw cycles and I nt cycles if you wish at small scale, and the radial direction we're interested in is this one this is the direction of transport, the direction along which we try to confined the plasma.
09:29:26 This is the y direction or Polo direction, and the flow, the main flow would be along this line so that in this Cartesian coordinate it would be in the y direction, and the z direction and this reduced model, we average over the z direction which is the
09:29:44 pilot direction, essentially the pilot direction. So the region small is either thermal confusion call is, then we are left with two equations, capital equation one for particle transport affected by ve crispy video city here with the source, and an equation
09:30:04 for chart conservation involving debo TCT here that are clashing of the liquid potential, which is again affected by the crispy flow and person extra turn here, which accounts for.
09:30:18 Well Paki new to the
09:30:22 centrifugal force, which you can view as an effective gravity term.
09:30:31 Yes, which is written here so the equals b flow is assumed to be divergent plus and this. We do small, and to further as I said the system here is still two dimensional to further simplify the system we first distinct make this case separation between
09:30:54 mean outrage over the wire direction, the polarization. And the fluctuating quantity which are the difference between the total field and the the main quantity, and especially we will be focusing on the main polio flow, which is simply the radial derivative
09:31:06 of the mean of the yes the mean potential.
09:31:10 So the density is split into those two components the mean density and the fluctuations. the what gct splits into the, here are the crispy meaning trust be flow, plus the fact waiting Birthday Party City, and we further retain a single way vector in the
09:31:32 y direction. So the fluctuating fluctuations are written this way and nk and five k all the only quantity we deal with.
09:31:45 So the parameter the free parameter of the problem is ky.
09:31:50 Doing so we end up with four equations to equation for the main fields. The main entity and the main flow mean density is governed by the trouble and flex here.
09:32:09 So the turbulence effects is related to the fluctuations of density and radio component of equals be flow radio stressing this term so they can be interest expressed this way, with respect to the Fujimoto the fluctuations.
09:32:31 And the equations for the fluctuations of their fluctuations Cindy, the full year they are complex fields. So in the end, although you have four equations you have six fields, in a sense, and the potential here you have the potential Botticelli, and this
09:32:45 component here is the analogous of the BTR parameter and beat up playing physics and is called the Dominic DTVOCG which is related to the mean gradients of density.
09:33:00 Okay, if you look at the the system here and look at the dispersion relation of the Federation, then this is what you, you get the first term is the simple Doppler shift.
09:33:15 And what you see it because of the square root here and that you have an instability. So with respect to golf comment.
09:33:25 Previously, it's an active forcing in this case because we do have an instability there, which is not a good to have you been up with this effect, you've got to determine if you wish.
09:33:37 And, in the absence of any mean flow then, this is what you would get for the growth rate, it's, it's case like the square root of the gravity turn times these beta parameters here.
09:33:54 What you see that the second derivative of the main flow X has a stabilizing parameter. So this can lead. If the phone evil ops to an instability threshold.
09:34:07 So now, we're interested in the in the face.
09:34:12 And so we we modify the way we look at the fluctuations and we, we, spit we separate we distinguish three, the magnitude of configurations and their face.
09:34:24 So the density fluctuations is an aberration amplitude and a phase.
09:34:29 And the same for the electric potential. And by doing so you can rewrite the flex the main flex tourbillon facts and the rate of stress as follows.
09:34:39 So you see that the deflects is raised to the sign of the difference of the face shift between the density and electric potential fluctuations and the rain or stress scales like the square or the amplitude of the electric potential times the gradient
09:34:57 of the face.
09:34:58 or the face, and the gradient of the face how to understand it, it's here a product the two dimensional space the radial direction which we are considering and since we are interested in single ky waves so this all the way that we would look at, and having
09:35:16 a non vanishing gradient of the face simply means that the face are shifted apart, and that you get this kind of structure, or the face.
09:35:25 So, from this expression of the rainbow stress, it comes out that it can grow, you either to the growth of the aptitude or to a change of the face radiant and the face gradient can be viewed as an analogous of the radio wave number.
09:35:48 But actually this the role of the face has been identified already as a key player.
09:35:55 And so this is the sentence his face patterning can generate its own flow without turbulence in emerging he this mean that the rain was forced so that divergence of the momentum flex can be related to the curvature of the face.
09:36:12 Although the turbulence intensity can be homogeneous and constant spatially.
09:36:18 And actually this is the proposal the identification theoretical identification here in this paper and there are some hint that it actually can lead to experimental behavior, and to the generation of zone flow.
09:36:37 Due to the face curvature in linear devices in this paper.
09:36:42 So, this is kind of complimentary way of looking at the face which I'm looking at with this model.
09:36:52 And to show you the dynamics of the reign of stress.
09:36:55 Here is a sketch of the time evolution of the.
09:37:01 of the total number of particles for two different simulations of the previous model, and for two different magnitude of the, of the source for a small magnitude of the source of the particle source, you first turbulence first develop, and then because
09:37:19 turbulence the generates a strong flex. Then you reduce the total number of articles and then you reach a steady state here.
09:37:30 But if the source is large enough, then you first. Reach first steady state. And then after some time, you you reach a better confinement state here, where you have a larger number of particle stored in the system.
09:37:46 And actually if you look at the profiles of the density of those two times in this simulation here at magnitude 30.
09:37:55 This is what you get the density, for the low confinement regime here in New. This is what you get with the source is located the particle associate located in this region, this is the radial direction, and after some time, you have a better confinement
09:38:12 and so more political stored in the system here.
09:38:16 This can also be viewed in terms of the effective use of the tablet and transport. It is reduced strongly reduced here, and this regime as compared to the regime, where you you have a strong transport.
09:38:33 So, the question is what occurs.
09:38:36 To make this transition from this regional fluke and firemen to this one.
09:38:42 And actually, the answer is given by the fact that the reign of stress dr soul flows, and which packed react on turbulence and saturated at the low magnitude, the small magnitude of the source.
09:39:00 They said the time evolution of the rainbow stress here in Doc, and you see that the rainbow stress remains vanishing almost finishing.
09:39:27 At some points. When the magnitude of the turbulence has reached its first saturation stage, then the reign of stress develops an instability.
09:39:23 And with the growth rate which is well identified here. And actually the rate of stress is driven, because of the gradient of the face and you can see that, remember the rain on stress in the square of the amplitude of the electric potential time the
09:39:40 gradient of the face, the amplitude of the potential remains constant, that the gradient grows exponentially.
09:39:47 So you we can barely understand the, the drive of this instability although it's the equations are still quite complex because you have to take into account and this this a close one of the remark of our freshmen Previously, we really have to take into
09:40:08 account the, the dissipating terms which I have not written in on the slides here but which review, really important for the dynamics of the system, but basically we understand that phase instability is governed by the curvature of the mode amplitude.
09:40:28 So that was the observation on this reduced model for interchange turbulence already been out like Terminus but then what, what about interconnect plasma turbulence.
09:40:40 So to to explore this kind of dynamics we used results from a generic at code acuity code which was presented by gmD for a year, during the first week, which falls.
09:40:56 The last opportunity equation for the distribution function of the is the elections are given a simplified response, and the self consistency is insured by the equation neutrality which leads to a poison equation.
09:41:14 So you can see a snapshot of the tablet fluctuations
09:41:20 CUNY.
09:41:22 Obviously to look at the rate of stress.
09:41:27 In this complex system it's much, much more complex than the structure is much more compact done for the region small.
09:41:38 But still, if you project the so you, we really work on the three dimensional data from the code.
09:41:45 And you can express the rain or stress as follows. So if you project the electric potential in the four year four year space in the, in the truth.
09:42:00 period the direction of the Troy of the tourists, then this is what you get me dippolito web number, and five is the magnitude of the victory potential for remote.
09:42:13 The free mode. So you recover basically the same kind of structure. And we've tried to split this, so it's a similar all the most, and we tried to spit this in terms of turbulence intensity here, and something which is related to the face gradient here.
09:42:31 So I do not detail.
09:42:33 The way we perform this. But basically, the main result is presented here. When we look at the cross correlation between the TOEFL Rainer stress.
09:42:47 With the turbulence intensity. On one hand, and with the face gradient. On the other hand, this is what we get we get that the rental stress is well correlated with the face gradient.
09:43:00 And this can be seen here. The cross correlation is almost equal to. 211 is blue here.
09:43:07 So this is the timeline, the cross correlation versus the, the radius here. And if you look at to
09:43:16 catch here.
09:43:18 This is what you get the time lag and the cross relation for the Reno stress relief fake to the face gradient. It reaches in this case almost point seven, seven, why for the cross correlation between the rain of stress and the turbulence and density is
09:43:32 fairly small, so definitely there's a critical role or the face dynamics in the drive of the rainbow stress in the turbulence saturated regime.
09:43:46 So, this is what we observe I come to the conclusion and discussion.
09:43:51 One issue which is raised here is whether this kind of observation in numerical simulation is accessible to experimental measurements.
09:44:00 This might be possible for edge plasma where lovely probes can measure the electric potential fluctuations, a different position, and then we can cross correlate and she part so I should I should stop, maybe.
09:44:30 some time for discussion. So, this is your last slide. Right. Exactly, exactly.
09:44:32 So the open issue which we we have right now or 334.
09:44:38 We want to weather, what we observe here for iron sky turbulence is robust and is reproducible for all the kind of tablets and whether it's multi scale or not and especially when taking into account electrons came turbulence.
09:45:09 Then, so far we have not explored this because we were lacking data, we will have not explored whether this phase instability was key in driving staircases, and possibly transport by years so this is where I'm going for sure.
09:45:12 And one open issue is also whether this this face which is absurd to to play a key role is transported.
09:45:22 And if so, what governs its transport.
09:45:27 Thank you for your attention.
09:45:29 All right, thank you very much Yannick interesting talk questions, ladies and gentlemen.
09:45:42 People are questioned out.
09:45:46 While we're waiting I'll ask a trivial question that you can elude Dave Well I'm ahead of David for now but but so let me finish.
09:45:59 Can you elaborate you mentioned, of course the goodie have given me which on the question of the phase curvature giving the stress in a homogeneous medium I'm glad you noticed that there was something we were amused by a you mentioned though an experiment
09:46:17 that purported to address that.
09:46:20 By shoe.
09:46:23 Sorry to say there are a lot of shoes in our business. Could you elaborate on that.
09:46:37 Yes, well, they observe some not. Who's they and on what device.
09:46:40 Oh, it's filling your device.
09:46:45 So not a token Mac and Jim Ian was involved, actually, in the, in the analysis of the data as well but I'm just curious what device and was it the linear device ID swept the linear device and Peking University.
09:47:02 I mean they're linear right in China, and I can't remember right now but I sure I can. Okay and Sunday night appreciate the reference that's interesting.
09:47:17 And the second point though is can you give us a quick synopsis of what they found.
09:47:24 I mean, of that. So, yeah. Well, the main claim that they do absurd regions where the reign of stress is driven by the curvature of the face and how you can deduce that
09:47:46 well.
09:47:51 Is it is it just because it's flat radio Lee in that region so there's nothing else possible perhaps on a no no I mean, part of the deal but they go quite deep into the analysis of the fluctuations and try to
09:48:08 to to evaluate the weight of the different terms in the reign of stress and end up with a conclusion that the leading one, the one governed by the curvature.
09:48:19 Alright well being, it will be nice to get the details if you could send me please. So, David your next.
09:48:30 So Danny Can you reduce model you are drawing an analogy between the interchange instability and convection is there.
09:48:41 Can I think of what what you've been talking about today in terms of phase dynamics can I think that in terms of a fluid problem, is there a nice simple model fluids problem I should be thinking of, just for, you know, because plasma physics is way too
09:48:55 difficult.
09:49:00 I I don't see why this kind of of physics, would be absent from neutral fluids actually.
09:49:12 So there's no reason why it should be absent, to my feeling you get the here the electric potential is choose to the stream function.
09:49:25 And so, considering the, the phase of the stream function in one
09:49:31 in one direction will be similar to the analysis that we performed here at St really a kind of another way of looking at the data.
09:49:44 And actually the the the simulation which I showed here, which is kind of pathological, I should say with respect to tokamak physics because the parameters are quite far from what we expect from artists they would correspond to an extremely small token
09:50:03 back in the jargon, this would be a very large truce tall.
09:50:07 But this simulation was published in 2003, and the authors did not look into the system in the way we looked at it.
09:50:19 Now, by splitting into magnitude and phase. So it's really a new way of looking at the data, which offers a way to to highlight the role of face right yeah yes I would say that it's, it should be should be observable.
09:50:40 If I may here, because in in years gone by in my wasted youth of doing resist of MHD with Ben careerists and so forth, David.
09:50:55 I mean one of the things we hit upon is, you know, in shall we say pure MHD systems where you have radio standing waves only you don't get any of these interesting phase dynamics and you don't get any Reynolds stress and way to put it is, there's no real
09:51:13 part of K radio right there's no radio propagation in the way if you must have a radio group velocity to see that effect.
09:51:23 And you know you course you quickly observe that any feeble Daya magnetic effect, or any, any weak electric field sheer effect will immediately give you the radio propagation and bring all of these things into the game.
09:51:40 So you might want to keep that kind of issue in my thing in mind right you need, you must have a you know a real part of at least for this type of model real part of K radio which you don't you don't get in certain very simple limits.
09:52:02 Okay.
09:52:05 All right.
09:52:07 low thar your next.
09:52:10 I am very nice talk, yani.
09:52:12 I had a question regarding the effect of the source term on Reynaud stress that you showed in one of the earlier slides. Are you available, any simulations in a drift regime rather than interchange that shows a similar effect so so that basically applies
09:52:28 applies to close to your clients or even maybe other cylindrical systems with quasi close to your clients,
09:52:37 you're referring to this.
09:52:40 Yes, the changing nature of the source, driving the.
09:53:01 Yeah. And then the next slide that showed the distress. The rain on stress, essentially taking off with the source, Yes or No.
09:52:59 No.
09:53:01 No. Well, I know that and you, you also know that when running for instance tablet simulations.
09:53:15 With bike imposing an absence of zero flows for certain time. And then once turbulence reaches saturated state then you switch on. There's also then they grow exponentially.
09:53:30 Right before reaching a saturated state whether this exponential growth is governed by the phase instability is to my mind, an open issue. I don't know what is clear, that's from the dissemination the GSR simulation so the really kinetic simulations.
09:53:55 The. So, when we run, normally the simulations, we allow those to develop from the very start of the simulations. And in this case, when they, they are excited, there's a whole foods they grow at the same time of turbulence of the turbulence growth.
09:54:12 They grow at the same time of turbulence of the turbulence group. And in this case, the main drive is governed by determinants intensity not the face shift.
09:54:21 Okay, so I think that there's really a difference here and this is kind of abuse, because
09:54:30 in this case the, the growth of the tablets.
09:54:35 Well obviously not obviously.
09:54:45 But this is what what we see. So, starting from a saturated regime of saturated turbulence, and then switching on the zone folks would be a nice experiment numerical experiment in more advanced simulations, such as JIRA for instance or some others to
09:54:58 check whether this kind of phase dynamics can also be at play.
09:55:06 Okay, thank you. Again, I can't resist the remark from me and I'm probably the oldest guy here so you know that, not for the curvature but that experiment was done by z Hamlin and harm and I you know in the late 90s, right.
09:55:26 When the whole business of modulation zonal flows was getting started right so what they what they did in GTC was let the thing go to a saturated state with the zonal flow turned off.
09:55:41 Then remove the the damping of the zonal flow and you see being the growth of the zonal flow and concomitantly a sharp decline in the amplitude of the drift waves.
09:55:55 So that, that type of switch on experiment for the flow is done.
09:55:59 Now it was it was not.
09:56:01 It was not done looking for this phase curvature effect, but something like that could be revisited. So, exactly i, this is what I had in mind actually but I did not mention the paper but that was this kind of numerical.
09:56:22 It has a long history already you know that's my. So nothing exotic so Jen's.
09:56:31 Yes, I just a comment and question I think that this experiment you talked about was from from Beijing University. If I put a put a reference on the, on the, what he called it the chat here.
09:56:52 And they did miss Lina machine is is,
09:56:57 as you also said Yannick.
09:57:00 Another nother point I think that this idea that the variation of the face is mattering for the Reno space is also used to, way back in in in in fluid dynamics in connection with Kevin and march type instabilities, because in Calvin Hill Mars type instability.
09:57:20 This GIFs Ramos race takes place so kind of negative viscosity that you can, you can formulate this negative viscosity, that drives the waves. And I think it was Raskin roofs many, many people around him, that that also.
09:57:42 I believe they also did some experiments, but they were not really able to make sure that in in in in detail but some simulations word on there.
09:57:51 So that's something that that has been around for a long time. it's also paid was, was the
09:58:02 genesis is even older than I am so good.
09:58:08 So.
09:58:11 All right. Any last question for Yannick.
09:58:14 I was worried that we'd fill up the time, and it's now 958.
09:58:24 So, all right so let's thank Yannick again.
09:58:30 For an nice talk, let me make a few comments, I mean there's I had hoped. Actually I had wanted to discuss a few things, but no time next week for this group, the transport barrier group.
09:58:49 It's a.
09:58:51 We have a interesting lineup the theme will be a little different, it's a little more focused.
09:58:59 And it's on the matter of symmetry breaking, and we'll have talks by Laura cope.
09:59:06 And by oz gherkin, and by Edgar will be back and different different aspects of symmetry breaking and if there's time I might give a one or two view graph on that related to turmoil rotation one of my favorites but that's that's optional I've talked a
09:59:26 lot. So thank you everybody.
09:59:29 I think it's 959 so we can call it quits.
09:59:35 And hope some of the subjects that were discussed today will be coming back tomorrow in Professor Hughes's group, particularly relations to, to the bomb for us to the well and Smith in young type of models and so forth but otherwise.
09:59:57 I guess we see you next week. So if not before so thank you everybody.