09:02:21 So let's, let's get going.
09:02:25 So as usual.
09:02:30 Slides Just give me a second.
09:02:41 Look at you know we have today's international language morning we had Russian now French and also.
09:02:53 All right. So good morning, good evening wherever everybody, wherever you are.
09:03:01 And this is the third edition of the transport barrier working group.
09:03:06 And the theme for today is symmetry breaking.
09:03:12 And, you know, we have our usual three plus 315 plus 15 talks.
09:03:21 And
09:03:21 then our speakers or Laura cope.
09:03:24 as gherkin and Edgar know block.
09:03:27 And then we'll have a, if there's any time left. In the past, the discussion of the has been quite lively and we haven't really had time for general discussion but in the remote possibility that there are, there is.
09:03:44 We have will have general discussion where we'll try to eat in foment instability.
09:03:52 With the help of GM and David, and a few thoughts for today, which you can just keep in mind.
09:04:03 What's the role of symmetry breaking in barrier formation dynamics.
09:04:09 A good one for us in the plasma or fusion side is the interaction of symmetry breaking with fluxes and stresses.
09:04:17 And last but not least, comparative role of external or geometrically induced symmetry breaking with dynamic goals or symmetry breaking and their interplay and plus all the previous questions.
09:04:34 So, we're talking about breaking statistical symmetries, we can if you like. Yes, but you. I mean, you were talking about breaking statistical cemeteries, or the inimical symmetries dynamical.
09:04:54 Uh huh. But if you want to broaden it to statistical this is fine. Okay.
09:04:54 And let us, I think, enough introduction.
09:04:59 And let's start with Laura cope who Laura I think has already been introduced. So needs no further introduction, so please go ahead and start sorry much pants.
09:05:13 I'm just going to start by saying that my internet connection seems to be a little bit unstable today so please let me know if I cut out at any point.
09:05:23 So I'm just going to try and share my screen.
09:05:36 There we go.
09:05:37 Okay, so can you see my, my screen. Yes. I started the correct view.
09:05:44 Yes, yes, yes. okay so. Hi. Thanks for the introduction. And today I've been talking about some work on my PhD at the University of Cambridge Roche with Peter Haynes, and in keeping with the theme for today's session.
09:06:00 The title of my talk is spontaneous so much of breaking in between plane turbulence. And, as we all know in this program. And this is a paradigm problem for the study of jet streams in planetary atmospheres and oceans, such as the jet streams in Jupiter's
09:06:18 atmosphere shown at the bottom of this slide.
09:06:23 And inevitably, there is a certain amount of overlap in this talk, my previous talk last month, but I have tried to incorporate a few additional details.
09:06:36 Okay, I guess, trying to slight part of my work. OK, so the problem that I'm studying, is that all two dimensional blue on aw periodic between, unquote, incorporating a stochastic balls, as an idealization of small scale turbulence.
09:06:57 So, the entire dynamics can be described by the, what is the equation given in the center of this slide, where we have a term due to the pizza effect.
09:07:09 We have a stochastic was linear friction, and then I'm using high viscosity for numerical stability.
09:07:18 And those three parameters of interest. So there's pizza, which is the background radium potential What is it, there's the energy injection rates, epsilon.
09:07:30 And there's the dumping rates, new.
09:07:33 And just a brief note about the stochastic forcing been using. It's approximately whites mommy's in time, and it's isotopic and homogenous and forcing a narrow amulets of wave vectors in folio space with a well defined, forcing wave number that corresponds
09:07:51 to small scale pulsing and relative to the size of the domain.
09:07:58 So this model, naturally gives rise to zonal jets, or donor flows, and it can be solved numerically. So to give you an example on this slide. I'm showing an example of a simulation emerge, a single jets spontaneously emerged.
09:08:15 On the left hand side you can see snapshots in time of the zone of velocity fields where it began a fluid flowing to the east and the west.
09:08:26 It will take the average of the plots on the left, then we generate the plots in the center, which showed the zone or beans or philosophy or the jet velocity profile.
09:08:37 And if we complete this profile as each instance in time, then we can plot these lucky time slots, Shane on the right hand side of the slide, which shows the spontaneous emergence of the jets in the system.
09:08:53 And I'm going to be showing a number of these latitudes time slots throughout this talk, and I'm going to emphasize that time will always be evolving along the horizontal axis.
09:09:07 So, using this system. I've run this model a broad range of practice, and the original goal was to try and evaluate the different types of behavior that exist.
09:09:21 So for example, we found many cases of randomly wandering behavior.
09:09:33 We also find examples of jets merging, and also new jets forming, or new creating. But previous that these has already observed these types of behavior.
09:09:39 As I mentioned last time. We also observe this new type of behavior. So we observe this spontaneous symmetry breaking, as in doing object migration behavior.
09:09:52 And in this particular example, you can see that a single Jetsons forms, and it's migrating either north or south, with a constant speed.
09:10:02 And it also has the capacity to spontaneously change its direction of migration.
09:10:08 So this is the first main result in the tour, which is that in the speed of the plane system, we observe this new type of behavior, I migration behavior.
09:10:21 It can be helpful to visualize the simulations I've been running in terms of a parameter space.
09:10:28 So on this slide, and along the horizontal axis. I plotted the vines wave number, which is a theoretical prediction of the number of jets in the domain.
09:10:45 And then along the vertical axis, I'm crossing the zone Australian parameter, which is a dimension this number, and can be considered to be a measure of the strength of the Jets, relative to the eddies.
09:10:54 Each of these circles corresponds to a single simulation.
09:10:57 And they've been color coded according to the number of objects that was set up in the domain.
09:11:05 Now, an actual question to ask is, which out of these simulations show migration behavior.
09:11:12 And it turns out to be these ones here.
09:11:16 So you can see that migration is happening to these sub intervals of the vines wave number scale.
09:11:22 So we can summarize migration behavior in terms of parameter space as follows.
09:11:30 For different lines fake numbers, we can observe, different numbers of jets migrating.
09:11:36 And as the same not should be parameter berries. Well, philologist a national fee, or stronger jets, they behave more deterministic Lee.
09:11:46 However, as the center should be decreases the Jets behave most accounts quickly, and they changed direction, more frequently.
09:11:57 Okay so, to further understand migration behavior. I've considered a set of reduced models using a technique known as the generalized cause it linear approximation, which I mentioned in my talk last time, and also speak to by escaping very nice introduction
09:12:13 to this. A couple of weeks ago.
09:12:16 So in terms of this particular system.
09:12:20 We need to consider all of the possible zonal scales in the system from the largest dental scale associated with the Jets. I even turned on me down to the smallest resolved zonal scales, all don't wait numbers.
09:12:37 And then we're going to separate these into two different sets, using a spectral fields them.
09:12:43 So the first set, which I'm going to call the moments. We're going to put the largest zonal scales or the smallest weight numbers.
09:12:52 And then the second set we're going to put the remaining zonal scales to do the smaller scales.
09:12:59 And then begin to define the separation wave number capital lambda. So, be the largest wave number contained in the low notes.
09:13:09 And then the trick is that we're going to allow fully nonlinear interactions between the late nights.
09:13:36 We're going to allow certain Lavinia interactions between the low and the high notes, and they collect certain interactions between the highlights, such that the original conservation properties of the system are retained.
09:13:35 So much I want to be speaking with you. We're going to decompose into the variables, and for example the screen function into the low and the high notes, which I'll debate using letters, l and teach.
09:13:49 And then, if we consider the original participatory equation with him in terms of linear and nonlinear operators. We can do by evolution equations for the low moods and the high notes.
09:14:03 And then the generalized closet in your approximation neglects these high, high non duality terms in the evolution equations for the high notes.
09:14:14 Now, this amount neglecting the same interaction terms from the original participatory equation. So we can do yb unique sense of what is the two equations that depend on the separation wave number, lambda which we can you Merkley integrate forwards in
09:14:33 time using the previously described procedure.
09:14:37 So to summarize the models. And first of all, introduce fully non linear model, where lambda was equal to the numerical resolution.
09:14:46 And this allows all possible nonlinear interactions between the zone or wave numbers.
09:14:54 As an intermediate such as models, when lambda is greater than or equal to one, we have these generalized causes linear models.
09:15:02 And then at the extreme end, but lambda is equal to zero, we have a closet in the system, which neglects all possible Eddie, Eddie interactions,
09:15:13 not a Christian, but we might want to ask is, how does migration behavior depend on the value of this separation wave number lambda.
09:15:23 So to answer this question, I posted a single set of parameters. And I've been a set of simulations, using five different values of London.
09:15:34 So on the bottom row in the fully nonlinear system. I've chosen these parameters, such a pair of jets on migrating.
09:15:44 And then considered for other values of London.
09:15:47 So on the top row. But, let's have a look at the the court system when lambda is equal to zero.
09:15:54 And in this particular case, we see that the migration behavior no longer exists.
09:16:00 However, for intermediate values of lambda. In the second, third and fourth rose.
09:16:06 And we see that the migration behavior does exist, and in fact the properties of the migration are also retains so for example the migration speed.
09:16:17 I mean fund these results are generic across a range of different parameters.
09:16:22 These results are generic across a range of different parameters. So, this is the second results in the talk, which is that migration behavior requires lambda to be greater than or equal to one. within this generalized cause in the framework.
09:16:37 And now the last week questions of interest in terms of migration. And the first is, why do the Jets only migrate one bandit is greater than or equal to one.
09:16:50 Secondly, can we predict the speed at which the Jets might rates.
09:16:55 And then finally digits also migrating more complex systems.
09:16:59 So focusing to begin with, on question one.
09:17:03 And I first of all needs to introduce as I did last time of the concept of zones, which are the generalization of philosophy ways to close with attitude and not share, he did not share why the standard theory of your philosophy race horses perturbations
09:17:21 of the system away from me background state, which may or may not depend on that achieved one.
09:17:42 if we substitute this decomposition into the equation of automation, and we linear minds, then we can divide the linear waste preservation equation given here. systems in which the mean background state does not depend on why, When there's no shear and
09:17:48 no jets, then this coefficient is a constant and this equation is linear in x and y. So the solutions are saying is boydell in x and y. So these are standard linear velocity rapes.
09:18:05 However, when this being bought brand state does depend on why. So when there is share, and our jets presence.
09:18:13 But this coefficient also depends on why.
09:18:16 And so, this time the solutions are still sinus problem x, but they're more generic in one. Okay, so be solutions, which we call zones. They project onto a range of latitude and await numbers.
09:18:33 And to show you an example of a say not. Here's a simulation where a single jet is present. And you can see that is coexisting with this way that has several ways number one.
09:18:45 Now, in the fully nonlinear system, we're allowing all possible nonlinear interactions.
09:18:52 And so energy can flow into all zonal wave numbers. And so, In this model, we do expect to see the formation of large scale so not in the generalized quality of models.
09:19:07 And we also allow energy to transfer between center wave numbers. And so we also expect to see the formation of saying on in this system.
09:19:18 However, in the system. We only allow energy to directly transfer back and forth between the jets and the zonal skills that are directly forced. And so we do not expect to see the formation of St John's in this particular system.
09:19:34 And this is a key points.
09:19:37 So the question is why do jets only migraines, when lambda is greater than or equal to one.
09:19:43 Also answer this question again to concert of three different examples. So the top world will consider two jets migrating off in the middle to jets migrating.
09:19:54 So, And then finally to non migrating jets.
09:19:58 And then in the central column here. I'm plotting snapshots in time of the zone. Last 30 fields where I've taken the square to me. I squashed it in zonal direction, and that I've been tested negative four times in order to visually enhance the physical
09:20:16 features.
09:20:17 And we see that when the Jets are migrating nor.
09:20:21 There will be several flow anomalies on the northern side of the Jets.
09:20:26 And when the Jets migrate south, we instead. See these anomalies, on the southern side of the Jets.
09:20:32 And then it has not migrates in case one of the Jets has these anomalies, both on the northern and southern side of it.
09:20:41 And these features can also be visualized in the porches to build.
09:20:45 So for example, when the jets and migrated north, we see being read positive for TriCity anomalies on the northern side of the Jets.
09:20:55 And, but the Jets migrate south, we instead see these blue negative anomalies on the southern side.
09:21:02 And then finally, this non migrating case shows both positive and negative Blue 40 City normal is on either side. One of the Jets.
09:21:13 But, in order to verify that these autism families are low waves, I zone ons rather than CO heaven policies need to consider the streamlines.
09:21:25 So, considering the example where a pair of jets are migrating North American side of the slide. I'm sharing snapshots once again of the zone of velocity and the participant fields in the box you can visualize the instantaneous streamlines.
09:21:45 However, it's useful to plot the streamlines in the frame of reference, moving with the Jets.
09:22:00 So in the fourth plot, I subtracted the face speed at the chance, multiplied by latitude why from the string function.
09:22:00 And you cannot serve at the streamline spawn not closed, indicating that these four TriCity anomalies, or wavelength features, rather than came here and policies.
09:22:11 I, they are sewn on.
09:22:17 But why do the chats migraines.
09:22:19 Well, to answer this question, we need to consider the evolution equation for the Jets, where we observed that they are maintained by the final stress loss.
09:22:30 So, I found an example of a simulation, when the first half a pair of jets and migrate to know.
09:22:37 And then the second topic spontaneously starts migrating.
09:22:41 So for each half of the simulation. I've competed time averages in frames of reference that move with the Jets, and the Reynolds stress was shown in pink and the jet musty profile.
09:22:55 We see that when the jets to migrate north, both the well stress balls and the Jets themselves are a symmetric about the axis of the Jets, the net force in the northern direction.
09:23:09 However, when the Jets stop migrating both the mental stress balls on the Jets are symmetric about the axis of the Jets, so there's no net force on the jets in this case.
09:23:22 So my question requires an asymmetric Eddie forcing and a corresponding asymmetric mean flow.
09:23:30 But so let's extend to these protocols directly caused by the zone on all the long way.
09:23:36 Well, to answer this question, we need to consider the contribution of the zones to the mental stress force.
09:23:44 And it's worth mentioning that in all of these simulations by brand, the dominant zones. Always have zonal wave number equal to one.
09:24:03 So we need to decompose the Eddie so the perturbations into the modes, corresponding to the zone obeyed number one minutes I either zone on, and the remaining external beta numbers.
09:24:07 And then, if we consider the evolution equation for the maintenance of projects. We can decompose the mental stress balls into the contributions from the St John's, and the remaining CERN or wave numbers.
09:24:22 And then, for an example of a simulation, where a pair of jets on migrating north.
09:24:37 Or we can decompose the full rental stress balls into the contributions from the Cylons, and the remaining zone or wait numbers, we see that the zones actually have a negligible contribution.
09:24:47 And in fact, the majority that the main part of the mental stress force is coming from the meaning Cerner wave number notes. So this is the third result in this talk, which is that the zones are playing organizational goal of the smaller scale Eddie's
09:25:03 in the migration mechanism.
09:25:08 Okay, so the second question is, can we predict the speed at which the Jets migrates.
09:25:15 And just to clarify the migration speed is defined as the change and latitude of the position of the Jets, divided by the time taken.
09:25:25 But we might take off the size of the migration speeds can be determined in terms of its external parameters in the system, where the function is to be determined.
09:25:37 So, using 130 different simulations. I've been rated the dependence of the migration speed on each of the external parameters. And it turns out, but there is an explicit relationship, given here, which, when we be a range shows that the migration speed
09:26:08 can be received as the product of the damping rates, New Times vine scale, which is an estimate of the spacing between the Jets. And we can verify this on the simulation data on the right hand side.
09:26:13 When plotting the migration speed, be against this relationship here.
09:26:19 Each of these circles corresponds to a single simulation color coded according to the number of jets migrating. And you can see that as a clear, then your relationship with a pre fractal That's an order unity.
09:26:32 So this is the fourth result in the top which is that the migration speed is given by an expert and expression, and it's proportional to the product of new times the vine scale.
09:26:45 And then finally the third question is digits. Also my grades, and more complex systems.
09:26:52 Well, the answer is yes they do. So for example, migration behaviors have been observed in general circulation models in both the pole bird, and the equator with directions.
09:27:05 And in terms of observations. Well, there have been poll words and tendencies observed in the Earth's atmosphere.
09:27:16 However, it is worth mentioning between each of these studies, but is a clear symmetric braking mechanism presence to to the spike or geometry. And this is absent in my pizza plane systems.
09:27:30 So this demonstrates that spherical geometry is not required for Migration behavior.
09:27:37 So just to conclude, and in this pizza plane system, we observe this new type of behavior. So we have to migrate.
09:27:49 Migration requires London to be greater than or equal to one. In the generalized closet in your system zones would use the plan, organizational goal, the migration speed of the Jets was found to be given as the product of the dumping rate times the weighing
09:28:08 wind scale.
09:28:09 And finally, using the system with the cheese that spoke will judge me or other similar to breaking mechanisms such as probably have a cemetery on not required, and for Migration tendencies.
09:28:23 I thank you for your attention, and I'll happily answer any questions that you may have.
09:28:36 Right, thank you very much, Laura very interesting talk questions ladies and gentleman, okay Adrian, go ahead.
09:28:41 That was great and clear Talk. Thank you. I have a question that I worried just comes from me not having worked with this stuff before.
09:28:50 So you separated. As to two questions about zone ons, I mean, my first question is, it seems like they must not be literally unstable, given that you said that you don't get them without generalized quasi linear so I guess my first question is, what's
09:29:06 their sort of frequency and growth rate like. and then the related question is that when you showed the rental stress stuff and you were separating out the contribution from m equals one versus contributions that other way of numbers.
09:29:19 It seemed like you were defining anything at m equals one to be a zone on contribution so is that sort of the definition of zona that it's anything m equals one, or to do the M equals one fluctuations include zone ons, and also other stuff.
09:29:36 OK, so the answer your first question was, two attempts to answer your first question. And let me just find
09:29:47 your progress slide.
09:30:03 So could you just clarify what you were asking the first time would you mind Adrian. Sure.
09:30:09 Arizonans unstable or are they, stable.
09:30:12 So, and they seem to be fairly stable and, and my numerical analysis shows that they seem to share the same zonal face speed as linear last few weeks.
09:30:27 Okay, so they seem to have a dominant zonal wave, and they seem to have a dominance longitudinal wave number. However, they also project onto a range of latitude, or wave numbers but there's always a dominance latitude and we'll wait number, which is
09:30:43 generally equal to the number of jets in the domain and the zone ons share the same zone or base speed. As you would expect from linear velocity wave theory, in the absence of jets.
09:30:58 Okay. And then the second question was about the composition wasn't it if I go back to this slide here. Yeah, so you're asking, why have I chosen, just don't always number one notes.
09:31:11 Yeah. That's because in all of the simulations, and it was very clear that there was just a dominant external wave number one made that characteristic with the Jets, and this was associated with these anomalous features that we observed this character
09:31:36 and feature this this wavelength.
09:31:27 And that I showed earlier on. So it just made sense to kind of try and figure out what was the contribution from these very large scale features and to the mental stressors that job and maintain the Jets.
09:31:43 I mean it's doable. You could also continue the analysis and consider others and awake numbers as well, and but my feeling is that the main segments you breaking mechanism is coming from these very large scale waves in the domain.
09:32:03 x. Okay, Brian, I think you're next.
09:32:08 What is the energy source for you say they look very much like Ross P waves, but they're not unstable rescue it so what is the energy source
09:32:20 for the Cylons, and so they arise as a result of nonlinear interactions in the system, which nonlinear interactions between the intermediate scale so the small scales.
09:32:38 That's difficult to say, I mean, I'm essentially I'm bossing small scales in the system. And then you're going to get a series of bonding interactions and exactly which ones are interacting with the small scales or quasi linear.
09:32:53 Are you saying that these things arise from the quasi linear interaction, or they arise from the interaction among the mentor intermediate scale.
09:33:03 presumably intermediate scales yeah so you do not get the zone on in the closet in your system for example.
09:33:11 So in the closet in your system, you can only have two Eddie's, for example of wave numbers and into minus m interacting together jets, with a weight number equals zero.
09:33:28 Do you have any reason for this velocity scale. I mean, you discovered it but does it come out of your theory of the intermediate intermediate scale interactions.
09:33:45 I mean, the dependence on the dumping rates, is something that I'm actually still quite puzzled about and if anybody has any ideas, I'd be really happy to listen and the dependence of the vine scale, in some sense, just make some physical sense because,
09:34:04 the migration speed is going to be proportional to the net force acting on the Jets. And so, water jets are going to have a larger net force acting on them, then the number of objects.
09:34:19 So, if the Jets are more closely. And if the Jets, have a much shorter distance between them against to have a much smaller flanks. And so you're going to have a smaller force.
09:34:34 And so, you would expect to corresponding with smaller migration speeds and but yeah the dependence on new, I really at the moment, don't have a feel for that unfortunately.
09:34:45 But, if anybody has any ideas, I'd be really happy to discuss that.
09:34:51 And lastly, the most problems have boundaries you know you could imagine the pole and the waiter about did these wandering things happen if you put boundaries.
09:35:02 So, I've looked at the case with boundaries. And, of course, the problem with boundaries is that they can only migrate so far they can't travel through the boundary.
09:35:15 and I believe so.
09:35:19 We did see an experimental talk, and by Daphne from must say a few weeks ago, and they were looking at experiments in Angeles, and I believe that they did see some sort of migration tendencies within that system and.
09:35:38 But of course, whether you can say that it's the same type of behavior as this periodic system is is very much to be determined.
09:35:47 I think, Okay, I think I'm next actually one of the one of the merits of the new system is it gives the chairman a break.
09:35:59 House, a few questions how sensitive, are the results to, to the cut off being around one are going higher your lambda.
09:36:12 So, and I found that we got provided but lambda is at least one, whether it's 123 10, whatever the results of a stain. Essentially, and is that because the zone ons are concentrated at one wavelength, do you think, is that the point, or one scale.
09:36:34 I think that because provided that lambda is at least one, you have this scattering mechanism between the same way.
09:36:42 So energy can transfer long linearly into any zonal scale, and the same as always seem to appear. The dominant ones, always seem to appear to be wave number one.
09:36:53 So energy can transfer into that wave number provided that bundle is at least equal to one. So as long as you get one in it doesn't really matter. It's okay.
09:37:04 I mean on the subject of proportionality to damping, have you swear things pop up in games I've played is, if you try to make reduced models, say, of drift waves and zonal flows, and that heads towards the infamous predator prey.
09:37:31 And here the game might be a reduced model just of the zone on and the jam the jet or something and I have a feeling that may explain that the that the dependence on, on the damping that you're getting.
09:37:46 It would be very thing, am I certainly not something I've actually, I've actually tried or attempted to do. And, but the idea of trying to come up with some sort of reduce system that retains only the essential ingredients as you described is something
09:37:59 that I have thought about.
09:38:09 So, it would be interesting at some points to maybe to see that and maybe to verify it. Yeah. and I mean the because it's.
09:38:12 That's usually how you know one thing controls the other and the other guy is controlled by the damping and so the damping sets the parameter for the whole story is kind of the pattern, I see.
09:38:26 And that so you might you might consider that it would be would be interesting to see sort of a more wacky question have you explored the possibility of any asymmetry or in homogeneity in the forcing.
09:38:44 And how does that change the story.
09:39:03 I have been so if you. I mean, you can imagine that the forcing could be quite crucial to the problem, in some sense, and. And of course if you if you do start to incorporate so much you're breaking within the forcing itself up, is it arising, you know,
09:39:06 of the dynamics of rising due to the unforeseen or what is it difficult to me, to know. And all I can say is that I have looked at a range of different forcing scales, ranging from the scale of the Jets themselves down to two very very small scales of
09:39:22 course and, and, and, provided that the forcing scale is smaller than that of the Jets. And I find this this migration behavior so there's so much breaking taking place.
09:39:35 And, but in terms of non isotopic or not so much in this distributions, and I couldn't say.
09:39:44 Thank you.
09:39:46 Next was David
09:39:50 low enough talk to you had where you had all your simulations and then you you grade a load of them out and you had kind of three batches with remind me where they the ones that were migrating still be the colored ones are the ones that are migrating,
09:40:07 yes.
09:40:10 We're not migrating Yeah, okay, I mean it's a funny stochastic thing, and I should actually point out that, and I can predict from the parameters, when the migration will happen.
09:40:22 But if I want us.
09:40:25 It may not necessarily happen. I may need to run an ensemble of simulations to see certain ones that show this behavior. So it's not the only stable attractor for particular set of parameters, which is why you don't see all of these three dots and sharing
09:40:40 migration tendencies within these batches.
09:40:45 I mean I guess that's what I was going to ask you you have a formula for the, for the speed of migration when they migrate when they're migrating but do you have, do you have a criterion for migration.
09:40:58 And I do have certain criteria. So for example, you seem to need people to chew on as I know.
09:41:09 I'm more worried and I'm more interested in the, in the grouping in case.
09:41:16 I'm afraid I don't know.
09:41:20 Right, okay.
09:41:22 Just to comment on your your result five. Were you surprised by a result five, I wasn't surprised by a result five.
09:41:49 the fact that you can see jets moving north or south isn't surprising. I think what is surprising is the consistency of it, but they seem to walk into these states, and they're fairly consistent in the States.
09:42:00 Okay, so in systems with spherical geometry, you see that they, the systems of migration exists. And, but I think the surprising other one as I said was just that you get this migration over such long periods of time, and that that was what was surprising
09:42:18 for me that there was nothing stopping it from happening. And also that you can get these like spontaneous changes of direction the current and, which was quite surprising.
09:42:35 Okay, thank you.
09:42:38 Okay, our next customer is Steve.
09:42:43 Okay. Thanks, Laura is a very nice to have somebody interested in your, your thing about things result for the zone ons play an organizational role.
09:42:52 So this is interesting. So, and it also speaks of Pat's, trying to get a minimal model.
09:42:57 So presumably lambda equals one.
09:43:00 So, it can organize the rental stresses by a low, high going to high interaction, which is there in the to the k equals one counts as low in this case, and can interact with a high to give you another high, and that can do a bit of organizing.
09:43:20 And also, you don't have the zone on there for the quasi linear so presumably there's a high high going to low to give you the zone on in the first place, so your minimal model could be to keep the mean, the cake was one, and I'm one hi mode, presumably
09:43:52 Yeah, it was. There's a, there's a small goal in the system which I didn't actually mention, but I'm actually putting an Oculus, so I'm actually forcing quite a bunch of wide ranging, and then all the numbers.
09:44:12 Okay, just put a pulsing in one hi wave number. That's true, that's true. Yeah, it's probably okay.
09:44:09 It's interesting.
09:44:10 Thank you. Thanks.
09:44:13 All right, Petros I believe is next.
09:44:19 Yes. Can you hear me.
09:44:22 Yes. Yeah, yeah, nice stopped. I just wanted to make two comments first comment to pat sort of question about the forcing it has been shown that the minute you change this, forcing and make it the initial topic within the framework of the simple QL.
09:44:42 Essentially, you can have migration. And essentially, that's exactly what you're showing, because what you're showing is essentially that the forcing the effective forcing of the mean of the, of the key equals zero, essentially has been has become a symmetric.
09:44:58 So essentially it is the effect that the change of the character of the effective forcing that essentially important you know sort of creates this type of migration.
09:45:09 My second point is that it has been shown essentially also to the, to the question of why the zones are essentially cables one, it has been shown through analysis of the statistical dynamics of the, of the cemetery breaking associated with the stability
09:45:25 of the statistical dynamics of this problem that we've done that, around 2013, that the flow essentially bifurcated and creates essentially a zonal flow, plus x equals one zone and in most cases.
09:45:45 So, most problems, this depends on the geometry, but in most cases you can show exactly that. You can make exactly the prediction for the, for the content essentially of the main flow plus on on type of component which essentially supported through the
09:46:03 nonlinear directions.
09:46:06 Very much essentially like in the closet linear type of fashion.
09:46:09 Yeah. Okay, thanks very much. I'm just a question there was there. No no I just made comments. I'll make a comment as well as do that so I did also look at the cumulative expansion framework that Steve Tobias, etc.
09:46:28 and use CT they call it, and I've looked at the system within that model. And I also see migration behavior within that system that is essentially acquisitiveness system, which is sort of somewhat puzzling and.
09:46:44 However, the migration tendencies tend to be different in characters what I'm observing in these feats of playing systems. And so for example the migration speed up the Jets doesn't seem to have any relationship with tool to what I'm observing in this
09:46:58 What I'm observing in this particular system, and also the spacing between the Jets, and also seems to be less regular, it's one of these important distinctions I'm not too sure at the moment.
09:47:09 But yet, something to investigate further in the future.
09:47:16 All right.
09:47:18 Well, I think we've had quite a lively discussion. So let's thank Laura again and we may come back to this if we have time at the end and we move to our second speaker, as they're vegan and Oscar has not been introduced.
09:47:36 So we'll do that now you can start sharing your screen in the meantime so as a girl began I guess his education in the Middle Eastern Technical University he did his PhD at UCSD, and then move to France I suppose a postdoc for a bit that ca and then eventually
09:48:00 became a member of CNRS at a co Polytechnique.
09:48:23 And when he's not doing that you'll find him quite interested in Turkish politics. So, after that, take it away Oscar.
09:48:28 You okay can you hear me. Yes.
09:48:30 Okay, great. Thanks for the introduction, Beth.
09:48:35 So, today I'll be talking about envelope motivations, and often nonlinear plasma waves. So since then,
09:48:47 we'll be talking about plasma waves let's recall the medium that we're interested in. So this is basically a medium made of charged particles moving around in under the influence of each other's electromagnetic fields, so they're constantly interacting
09:49:06 and somewhat surprisingly I mean field description right the electromagnetic fields are related to the, the particle distributions through Maxwell's equations work extremely well to describe this medium waves in society May June, occur as a result of
09:49:28 oscillations of these mean electromagnetic fields which are linked dispersion relation which linked the, the frequency and the vape number in this medium in the context of the tokamak were interested in drift waves which are basically the same as the
09:49:49 rose be waves, except the range of frame numbers are different and we usually use a convention, or x is why and why is x for the geophysical dynamics people.
09:50:04 Now waves in general can be described using index of reflection is an optics, or the plasma the electric version in the, in the case of the plasma problem.
09:50:16 And we know, in the case of optics. If the index of refraction is a function of the field intensity, we get nonlinear optics in in our problem, the fluctuations the waves can to generate through rainbow stresses large, large scale mean flows, which then
09:50:37 effect or shear apart these waves, in which make the problem intrinsically, the index of reflection of this problem intrinsically nonlinear through this connection through the main, main floor.
09:50:54 So this nonlinear index of refraction gives rise to amplitude equations that are well known in physics or mathematics, give rise to solid, solid long chains etc.
09:51:08 Since part of the, the idea here is to do self promotion, please see our almost completely irrelevant review to the subject that we have published in Journal of physics, A.
09:51:24 in 2015 with path, which some somewhat gives the, the basics of the ideas that are found here.
09:51:33 So the waves in plasmas linear.
09:51:37 Since I'm going to talk about envelope dynamics let's let's recall how the envelope works in a simple wave, a way within an envelope travels via face propagation which is the propagation of the, the face structure of the way, then the envelope moves with
09:51:56 a group propagation is subject to vape dispersion, which we can see here how a sort of dispersing and and and dispersing wave behaves.
09:52:09 We see the dispersal wave at the at the top which sort of gets dispersed because of the fact that the different free MMOs in such a rave travel at different speeds.
09:52:25 In, resulting in sort of defacing.
09:52:29 Now, we know from the basic Heisenberg uncertainty principle that a Gaussian localized wave pattern in real space, results in a Gaussian localized wave pattern and in case space.
09:52:44 So, this, this pattern actually does not change during dispersion what happens is that the phases of these different favorite vape number components in case space gets randomized, so as a result of this we get in spatial pattern we get this dispersion
09:53:05 but in case space we only have sort of phase randomization events this dispersion which causes the veil to spread around is balanced by non linearity, we get a particular kind of solid on which is envelope falling down.
09:53:25 And this. This happens because these different phases which are trying to travel at different speeds are clumped together by the non linearity.
09:53:38 And this is in some sense if you think about it this is similar to synchronization and if you haven't read the book by Strogatz, which is sort of like a best seller popular book on synchronization explains the phenomenal really clearly.
09:53:55 This happens through interactions with a mean quantity. So, each very phase in some sense, adjust itself is if it's faster than the bulk or slower than the bulk through its interaction with the main flow.
09:54:12 And it either slows down or speeds up depending on this phase which respect to the, to the main solid.
09:54:19 And this is the physics of this is similar to synchronization. How synchronization helps folded information of loss or loss or plus one system is also very interesting problem.
09:54:32 Because of its expanded degrees of freedom and face space.
09:54:38 So the idea here is, we can have these kinds of interesting phenomenon of envelope dynamics.
09:54:46 But before that, let's remember the importance of this for waves or balance.
09:54:56 Plasma turbulence is a peculiar problem, it's first of all it's internally driven which means there isn't there is almost always in instability which drives the system.
09:55:07 And it's in a coexistence state of weak and strong toward business usually it's either coexistence in different scales. So in one range of scares you have the turbulence in other have strong turbulence or you can have coexistence in different channels
09:55:29 ions you can have strong for electronic in a week things like this. Now, in this talk I'm not going to talk about instabilities or coexistence. If this sort of bothers you think of me as a mathematician, and that way you are not going to be pissed off
09:55:46 as to how much, how little physics I have in my thought that you'll be pleased that how accessible actually my talk is.
09:56:05 So coma growth turbulence in this picture just to position ourselves with respect to what is well known, is basically the setup where we inject energy at some scale this debate at another scale and usually between this source and the sink.
09:56:13 We have energy transfer and this energy transfer happens through non linearity.
09:56:19 And in this inertial range where there is neither production or dissipation usually we get the K to minus five perspective for 3d now via stops like toddlers.
09:56:32 In contrast left or balanced relies on resonant interactions. For example, if we have three wave resonances.
09:56:40 We have a triad Kpq interacting. We should have the resonant interactions between these frequencies evolved there's an issue about this good this guy systems may not permit exact resonances This is unlikely to be a real problem in practical systems apart
09:57:01 from large scales very actually the resonance far apart. so the frequencies, have to be very different.
09:57:10 And so the description of.
09:57:16 They've done business relies usually on fav action.
09:57:22 And for simpler, your fav system this is potential extra fee, there's a and very interaction evolves with a sort of complicated transfer integral which needs to be calculated based on.
09:57:37 If you have three way of interactions or four wave interactions, etc. This is not most of the work of calculating based on this.
09:57:47 So basically we are really relying on resonances so let's go back to how these resonant interactions work for China has given me my system.
09:57:56 This is basically we have a dispersion of the lesson which is a clean this portion.
09:58:16 Now, when we add some sheer into the system. One way to handle external sheer his views, sharing coordinates.
09:58:24 So when we look at this dispersion religion we would say or share it doesn't seem to affect this is personalization. But another way to look at it is to use these sharing coordinates, which basically transforms cakes and into a time dependent function,
09:58:39 related to share.
09:58:41 So when we substitute this kx into here into the sort of resonant definition of the resonance manifold, we can see that in time, when we apply some shear to the system there is no money for strings under the action of this year.
09:58:58 And when this happens, the system loses its ability to make these kinds of three way of interactions, and it has to resort to interactions through all of laws.
09:59:09 And it.
09:59:11 So this means a seed large scale love forces the system towards was linear like interactions, sort of symmetry breaking in terms of spatial heterogeneity being destroyed by the system by the formation of zone of flux.
09:59:29 Another way to look at this is going back to the vape Connecticut question so a homogeneous version of the wave Connecticut question is, as I wrote here.
09:59:40 Now, in homogeneous case can be described using any homogeneous version of the wave Connecticut question, which is written here. So nk is the day of action but in fact it's comes from the inner function this time.
09:59:55 And it has some in homogeneous terms related to the basically coming from the zone of flaws or external sheer flaws.
10:00:06 So what we can do is we can take a homogeneous solution and substitute into the inner homogeneous wave kinetic equation, which would of course, manage the in homogeneous terms, but for target with the seed modulation to see if it grows.
10:00:20 Actually, it grows because of the motivational instability of the system. This was done by Newell 2012.
10:00:29 And this can be seen also from the perspective of motivational instability as long as the system is motivational they are stable, because this, this is something that's expected.
10:00:42 But the question still remain how general is this, for example, is it possible that faith are the ones, always end up, forming your condensate effectively transforming itself to disparate scale in traction problem is I think is an interesting question.
10:00:57 So let's go to modulation instability.
10:01:02 One of the canonical examples of this is the nonlinear Schrodinger equation, we can define a complex amplitude for the, for the electrostatic potential for the system.
10:01:16 And, and then basically take a plane wave, expand. And that is assuming that we have, fellow city in this portion.
10:01:28 And basically force that this portion is to be balanced by the first one that appears in the expansion of the non linearity so this is normally a perturbation expansion, but I think nobody understands the adult with a perturbation expansion so I put it
10:01:43 as if I just don't expand anything but. So here, this is the first time that appears in the expansion of the nonlinear term and rich fiber here is the main floor basic z Crossfire grad fiber is the me mean flow.
10:02:01 And the main flow itself is done linked to the fluctuations through the first term and the expansion of the reign of stress so this is the first time in the expansion of the reign of stress and we can basically solve this and substituted into here to
10:02:16 obtain, assuming a purely normal perturbation both the envelope and mean, are only motivated in the direction we get the nonlinear Schrodinger equation know we can substitute the homogeneous solution of the non English reading a question which is a nonlinear
10:02:35 Now, we can substitute the homogeneous solution of the non English reading a question which is a nonlinear homogeneous solution so this can be substituted into the equation and satisfies it except report target with that upside.
10:02:44 And then basically write down a linear instability condition for the, for the evolution of doubt upside we obtain a dispersion elation for the zone of low frequency in some sense, and we get basically a growth rate if this right hand side is negative.
10:03:07 And from by sitting The, the gamma by DK x equals zero kx mean the zonal the number, we get the zonal wave number, normal flow vape number for the most most unstable mode for the system.
10:03:36 we can write the solar toner solid on Chain Solutions usually with some velocity, defined by the system. And we can introduce higher order corrections to the reign of stress which gives additional description as well as some damping proportional to the
10:03:43 fourth power of the, of the field.
10:03:48 So since this is size zero square we obtain upside zero for power as well.
10:03:55 So there is another interesting variant of the nonlinear short and Gary question which is the double derivative nonlinear shooting guard question.
10:04:06 If, for some reason, mean flow are also governed by the Chinese government question. Now in this case actually, the main flow, because it's the modified China has a give me a question.
10:04:18 The main flow was basically coming from some something like the older equation. But if we have for the main flow that China has I gave me my equation as well.
10:04:30 We get the double derivative nonlinear Schrodinger equation and we go through the same sort of development.
10:04:38 And this this thing also has interesting solutions, it says SMS, Tanya, etc. The whole
10:04:47 sort of framework applies to it as well.
10:04:50 It can be described as one interesting solution though for this case is that the solution can be described by the emergence of the elliptic integrity of the second kind, which is basically.
10:05:10 Remember the, the Jacoby is Snowden function is the inverse of the elliptic can trigger off the first game.
10:05:21 So this is the inverse of the electric integral of the second kind which is defined here, as some xi, xi is the moving coordinate basically with a velocity that is not actually arbitrary but I didn't write the velocity here so which is linked to its amplitude
10:05:34 and arrays, em here is defined in terms of the parameters of the model like this.
10:05:42 And what we can see here the solution at least the square of the solution is shown here in in orange.
10:05:52 And what it is is it's sort of like a site design so it'll wave, but it phase is speeds up here and slows down here so it's sort of forms these triangular structures.
10:06:07 And in this case the mean flow because of the structure of the equation mean flow is proportional to the second derivative of size square.
10:06:17 And the second derivative upside square in this case is these sort of structures that we find is the main flow.
10:06:26 And if we add on top of that, the to calculate potential vertices and decreasing density gradient, like this, which is chosen to make this exactly flat me obtain this kind of staircase structure for the potential to city.
10:06:47 And this is natural in some sense because of the contribution that comes from the flow to the background as.
10:06:58 So, this, the actual solution, which is traveling also in space has this form.
10:07:10 And notice that for this particular problem because the main flow is also like China has like our Nima potential vertices actually dominated by the density.
10:07:23 And the the paper I give here is really for streamers but the math I present here is for follow flows and in fact it's completely identical.
10:07:34 So, this is basically my talk, I, I want to point out that this this framework can be extended to gyro kinetics, which is sort of our choice of the most sophisticated model that we can use.
10:07:54 And we can get all kinds of interesting solutions, even staircases from the system.
10:07:59 And it has this aspect of symmetry breaking in the sense that the homogeneous it of the system is destroyed and we reform these jet black structures.
10:08:12 And it also forms finite time singularity which I didn't talk about it but basically you can cook some ish.
10:08:23 You can cook a framework where you start
10:08:30 with some arbitrary two dimensional motivations and see how the system will evolve in time.
10:08:47 This is described in this paper saw the saturation mechanism for this kind of basic amplitude modulation approach is sort of unclear, it can be higher order terms in rental stress versus mean flown on the narratives, for example,
10:09:10 rotation for example of the basics normal flow structure and things like that.
10:09:17 So the bottom line is that these amplitude equations provide the interesting excellence case for which, for, in which direction the system will evolve as the nonlinear does come into play so it's not really perfectly adapted to describe the whole evolution
10:09:35 of the system.
10:09:37 But it's very interesting to, to see, starting from the linear model latest state in which direction the system is going to go.
10:09:46 So this is basically my talk, I hope you find it somewhat relevant to the subject.
10:09:53 All right, thank you Oscar, now we have a double derivative, to go with double diffuse of convection now so do you have increased the DD count, some questions ladies and comments questions, comments and questions, ladies and gentlemen.
10:10:12 Hello. I bet. Yeah, go ahead.
10:10:16 Yeah.
10:10:19 Input then does the nonlinear plasma Vu has a big application in the foots article acceleration I think that you already seen.
10:10:34 But we are, we are not thinking about the modulation of nonlinear plasma. Exactly. Actually we will do the acceleration for child particle acceleration and the intensity populated we will very very high is a nonlinear plasma we celebrated it you.
10:10:49 Did you, did the some calculation, or about related to this that the modulation in nonlinear plasma way.
10:10:58 And before I think that is really good for the for the article actually is that I never think that the internet is so interesting.
10:11:10 I'm not sure if I understand the question but I think it's relevant for a nonlinear particle acceleration, and there are people that are working on this may be the first examples I've seen on this person similar problems actually.
10:11:28 Yeah. My question is related to how because not only in the plasma wave is good for the particle acceleration for after the gv and TV TV world and we'll see.
10:11:43 Yeah, it's, I agree.
10:11:48 All right.
10:11:49 Okay, let's Jen's.
10:11:54 Next.
10:12:00 Yes, thank you very much so school for interesting talk enjoyed very much is bringing the Iranian excluding the equation into play. That's one of my favorite equations.
10:12:13 But what what I wanted to ask you about is the during the last part you talked about the double derivative non transcoding equation information of singularity and find out of time, but how to what what is the sort of the physics quantity that is blowing
10:12:42 in debt and then what will that how that manifests itself in Indian physical system.
10:12:40 Now, it's an interesting question actually because this, this equation you can write the Hamiltonian for the system, which is non trivial, for example, derivative nonlinear shredding every question is not like that.
10:13:17 This is when it's double derivative, you can easily write the Hamiltonian for that system, and then show how the using the Hamiltonian, depending on the initial conditions you can show that the system is going to going towards some
10:13:16 singularity solution. And this is from this picture, it's, I guess. One way to see it is to calculate the variants of the sort of the expected expectation value of this side.
10:13:33 When we take site is there is sort of a function and calculate x, x squared times square and leverage it to see the variance and look at the evolution of the variance, we observed that the evolution of the variants, if they have initial Hamiltonian is
10:13:53 negative.
10:13:54 is always.
10:14:12 It gets always reduced yes yes so that immediately implies a finite time singularity solution.
10:14:08 Yeah, that that that I, I see but I was thinking of what what is what's kind of physics to relate to that final singularity what what is that the velocity that goes through, I guess.
10:14:20 I mean, we can see the structure in some sense here. So this system tends to generate these curvature effects here. So, the curvature at the top of the, the beak becomes rather sharp, which generates extremely sharp jets and these jets than sort of feedback
10:14:46 on on that structure because the second derivative is proportional to the, to the jet and the jet accelerates and then this feedback and there's this feedback mechanism, which happens, giving rise to the, to the final times.
10:15:05 But this is actually about this more interesting I thing is when we start from 2d modulation, which I didn't really talk about here, we see that this.
10:15:16 This effect is and I saw trumping so the system tends to go. For example, the initial modulation that we impose on the system which is isotopic modulation, it breaks down, preferring one direction.
10:15:34 So then we go back to this kind of scaling which is all of low scaling and then we can continue with the finite time singularity. So in other words, even from isotopic to the modulation, the system tends to generate these jets and make them collapse in
10:15:49 some sense, of course, in reality the collapse the system cannot describe.
10:15:54 I mean, at some point it's very breaking and you need to talk about what happens there.
10:16:02 Thank you. Yeah, thanks.
10:16:05 I think we better move along smartly here. Any further questions for Oscar.
10:16:13 Can I ask a good question.
10:16:15 And go ahead get here.
10:16:17 Yeah, so if you allow for transfers modulation.
10:16:22 In this plasma problems.
10:16:25 Do you get the Davis Johnson system, or you get something else.
10:16:33 I'm not sure, not familiar with the system
10:16:36 are talking about so I can't really say anything.
10:16:41 We can discuss this. Yeah, might be interesting to discuss anything else.
10:16:52 Alright, I think we're we're sort of in. In, where we want to be in time so let's thank our again.
10:17:00 And our last speaker for the scheduled speaker for the morning is Edgar Knobloch who I think everybody knows it's already been introduced and he didn't need to be introduced anyway so go ahead.
10:17:17 Okay.
10:17:51 Can you see my screen.
10:17:54 Well, it looks like it's covered by something we see a black screen.
10:18:00 I don't know why.
10:18:02 I see it. My.
10:18:07 We saw it before.
10:18:09 Not sure what what changed but earlier we saw it, I tried to put it on.
10:18:16 Now we see it.
10:18:17 Okay, maybe I'll just do it.
10:18:21 Okay.
10:18:22 So, thank you for the invitation to give a token the session.
10:18:28 So this is a session on on symmetry breaking so I will say some things about both spontaneous and for similar to breaking.
10:18:38 And I don't know what the relation of this talk is two staircases, but I think it will relate to some extent to what we've heard, both from Laura, and also in the previous talks in the session.
10:18:53 So I'm going to be talking about binary fluid, convection.
10:19:36 Sorry, I don't know how to
10:19:44 KVYY this is not working.
10:20:06 Was it displaying additional pages earlier when you opened it.
10:20:12 Well it's, it has no, it has all of my pages in on my screen but I can't advance that
10:20:28 I'm sorry guys, um, I don't know what it's going on here.
10:20:45 Can anyone see the screen here. Well it's black still black. Yeah.
10:20:53 Sharing the screen of the window and go, Oh, that's, that's me that's the issue.
10:21:09 I don't remember we can see something.
10:21:12 Well, that's the first page but I can't advance it. That's the problem.
10:21:19 I think at least disconnecting and reconnecting might Yeah.
10:21:29 We have time Why don't you try that. Sorry.
10:21:32 I'm just gonna give you all this and I'll start again.
10:22:17 Sorry, hang on just one second, there's something that's happened, I don't know what.
10:22:36 My goodness.
10:23:34 I'm very sorry guys. Something might talk has disappeared for some reason that was the problem why couldn't.
10:23:49 Was it on a network connection or. No, it's nothing to do with the network.
10:23:59 It's just not.
10:24:02 It's not later. I don't understand some problem with.
10:24:14 Well, I mean, do you have an iPad handy you want to fry an iPad lunch type of talk, you know, and I'm
10:24:32 or put it up, you know, but if all else fails, I mean, just explaining it I think it's got a little pictures so, yeah, OK, so I'm going to read something strange has happened because it was working yesterday when I'm prepared it so.
10:25:01 if everything else deals that you're running Windows you always have the full reboot podcast, but I'm not doing windows, I'm sorry.
10:25:35 but maybe we can make or break for five minutes and the editor can find what the problem is there's no pressure.
10:25:47 Certainly anyone who wants a break. Take a break now. Okay.
10:25:54 I'm really sorry about this.
10:26:05 I'm just, it's understandable these things happen. Yeah, don't worry about it. It was working, yes.
10:26:07 But let me just play with this for a second. Okay. Yeah. All right. Okay, add your is going to play if you want to break, break.
10:26:15 If not you can sit here.
10:27:53 Okay, so let's, I think it's going to work now. So let me try again. Okay.
10:27:59 All right, everybody.
10:28:04 So share screen.
10:28:55 Sorry for a moment we could until you went this weekend. This weekend, this we cannot.
10:29:02 You can see it now. Right. Correct. In see it. Yes.
10:29:07 Okay, so let me now we ah.
10:29:10 Excellent.
10:29:11 Alright guys, so I'm going to say a few words about binary fluid convection so this is a system that's very closely related to W diffuse if convection.
10:29:21 I'm going to be talking about it in the connection.
10:29:26 In the context of what's often called military wt physical interaction.
10:29:31 So for those of you who are oceanographers. That means that I'm going to be talking about cold fresh fluid over warm salty fluid.
10:29:43 In my case, I'm not going to be introducing these concentration differences through boundary conditions, but they will be produced spontaneously. In response to an applied temperature gradient or temperature difference across the system through a cross
10:30:02 diffusion effect called the sorry effect.
10:30:06 So what happens when you take a mixture like this and you put it in a temperature gradient.
10:30:13 You can either have the heavier components, which would be sold in this case, migrating towards the cold boundary that's called the positive sorry effect.
10:30:22 Or it can migrate towards the warm boundary and that would be the anomalous or negative sorry effect.
10:30:30 So this cross diffusion is a general phenomenon in in all of the systems, and it occurs, as I say in salt water for example.
10:30:52 It's never considered under oceanographic conditions. But it does exist in the Barclay setting, and it was first observed I think by cold world back in the early 1970s and experiments on thermal Highline convection in closed containers.
10:31:06 So the basic equations here are the ones at the top. They've been non dimensional lies relative to the thermal timescale. The important thing here is that the buoyancy force which is the second term here.
10:31:21 Of course it depends on the temperature fluctuation T, and on the concentration, see through this cross diffusion effect s, which is called the separation ratio, and I'll explain what that is in just a moment.
10:31:36 So both temperature and concentration of course contribute to the point so forth.
10:31:41 Of course, the temperature is infected and defuses and the concentration here. See, is also affected, and it defuses but here's the cross diffusion of fact.
10:31:54 This is the effect of the temperature generating concentration changes.
10:32:01 And so these equations have some directionless parameters, is the usual pronto number.
10:32:08 There is the inverse Louis number the ratio of the thermal diffuser with it to the excuse me to the scientific method of thermal defense everything, small number, there is the number that drives the whole system that's the temperature difference across
10:32:27 imposed across the system. And finally there is the key parameter.
10:32:30 The separation ratio here, which is proportional to the sorry coefficient and that's the cross diffusion coefficient that I mentioned, and that is going to be a negative connotation here.
10:32:41 Remember negative sorry coefficient means negative separation ratio and that in turn means that heavier component migrate towards the warm boundary. The warm boundary is the low boundary because I'm heating from below.
10:32:58 Now the main difference between this system and W diffuse if convection is through the boundary conditions because here.
10:33:05 I'm assuming Amina close bonds between two impenetrable boundaries that top and bottom.
10:33:11 And so the boundary condition is on the mass flux and the masculine, is a contribution from both the concentration.
10:33:19 And from the temperature through the cross diffusion effect so the boundary conditions are imposed on this quantity eater, rather than on see.
10:33:28 I'm going to assume I have no sleep boundary conditions and fixed temperature boundary conditions that have been born.
10:33:34 So that's the setup.
10:33:37 So I'm, as I said, this is the military case.
10:33:41 It's the fresh cold liquid above warm.
10:33:48 Salty liquid generated via this cross diffusion effect.
10:33:53 And the first instability is a military instability.
10:34:00 Yeah.
10:34:01 First instability that you see is a complicated spatial temporal state called expressive chaos.
10:34:10 That is described by amplitude equations obtained a long time ago by Brotherton or Spiegel, and the similar equations applying to present system. What you see here is some initial disturbance evolves.
10:34:24 Through this kind of focusing instability and then he breaks up into this complex spatial temporal state.
10:34:30 And this occurs, you know. Here are two examples that different parameter values of what the spatial temporal state can look like.
10:34:39 So in our case, this is what we find. So this is a plot. On the left of the temperature at mid level through in my layer as a function of time increasing upwards.
10:34:51 And you see this complex spatial temporal state that occurs. Once you pass the onset of oscillations. But then the whole state undergoes that kind of focusing instability, much as what occurs in the focusing MLS that we heard about in.
10:35:10 In the previous talk, and the whole thing concentrates into a kind of localized structure that sits in one part of the domain, this is with purely boundary conditions.
10:35:19 But in this case this localized structure.
10:35:23 You know it's not a stable state of the system, and over time with the grades, and he collapses back into the state spatial temporal chaos.
10:35:33 And of course that's unstable to this focusing instability so the process repeats. With this kind of structure occurring perhaps in a different part of the domain.
10:35:42 So, the net outcome of this is that you get relaxation oscillations.
10:35:49 Close the onset of this instability instability.
10:35:54 You get this gradual erosion of the structure that I mentioned, then it collapses into the chaotic state, here is the focusing instability. Then it degrades again over time lapses back into the Celtic regime, and so on so forth.
10:36:09 This is not a periodic signal, because of course one of the states that's involved in this is a chaotic state, but you get this kind of chaotic relaxation selection.
10:36:19 So this is what occurs at a particular value of devalue number 1774. If I increase the parameter value just a tiny bit 1775. Now this localized structure is stable numerically for all time forms via this focusing instability of the spatial temporal chaos.
10:36:41 And then it just sits there for all time and notice that the motivations in the background I've been kind of sucked into the localized structure.
10:36:52 And so the background state is actually.
10:36:55 It looks stable right.
10:36:57 Even though it is passed the onset of the instability.
10:37:02 So it should be unstable we'll talk about that in a second.
10:37:06 So here is the bifurcation diagram.
10:37:13 I'm showing the way number on the horizontal axis, on the vertical axis is a measure of the amplitude of emotion through the quantity called the Nestle number that's the dimension less measure of the of the of the amplitude of the motion here is that
10:37:30 initial or filter instability, that's a half bifurcation. And here in these daughters the spatial temporal state this disperse if chaos state that I mentioned.
10:37:40 And out of that state through these relaxation oscillation emerge. These stable localized solutions.
10:37:48 There are two branches of them I'll show you that in a second.
10:37:53 I'll do that right now, what these look like.
10:37:58 So here I've split up that picture into the two branches that intertwine.
10:38:05 So there's going to be a branch of so called odd states, and a branch of so called even states, and I'm going to show you what the state's look like at these three successive points along these branches, A, B and C.
10:38:20 So this is what the old states look like. So an old state has this role.
10:38:28 You know that in this case or counterclockwise on the right. And it's also counterclockwise on the left. So the one on the right, and trains heavier material from below.
10:38:38 And the one on the left and trains lighter material from above. And so you get a kind of concentration gradient across this mixed region, which results in transport of the heavier material across the structure so this is a kind of little pump that transforms
10:38:53 stuff from over here, and puts it over here.
10:38:57 And when I go to point B.
10:39:01 The upcoming bit unusual situation and then when I go to see.
10:39:05 I've added a roll on each side of the structure. And so now on the right arm and training from above on the left arm and training from below. And so now operates in the opposite direction.
10:39:18 So these are little stationary pumps, and they are stationary by the way because this state has a symmetry and that's the connection to symmetries it's symmetric under left right reflection, followed by up down reflection reflection in the midpoint, and
10:39:35 In addition, we have even states that were the roles on the two sides, rotate in opposite directions. So in this case, they can train material from below.
10:39:41 that forces the structure to be stationary.
10:39:49 And so in the mixed region you have an above average density. If you go from A to C, you've added roles, one on each side.
10:39:58 And now you're in training from above and so the
10:40:03 concentration inside this mixed region is less.
10:40:07 Now these states are also stationary but they're stationary for different reason they're stationary, because they are symmetric. On the left, right, reflections reflections that I'm calling our one.
10:40:22 I just want to say one word about why these structures could be stable even though they connect to an unstable background.
10:40:30 And the reason is that they are present in a region below so called absolute instability.
10:40:38 And those of you who are plasma physics is know about how to compute the onset of the transition to absolute instability.
10:40:46 You have to look at a double root of the disclosure relation in the complex k plane.
10:40:54 And we have done that here it's a little more complicated because for each combination of the real part. kr of the wave number and the imaginary part of the wave number.
10:41:08 You actually have to solve the boundary value problem was the boundary condition top and bottom. But nonetheless, this shows the dispersion of relation in the complex plane at this value 76.3, and a little bit later at 17 86.5 when you see that in between
10:41:23 there was a double root of the dispersion election.
10:41:30 And that demonstrates the presence of the transition to absolute instability in the system.
10:41:37 And if I just go back to my bifurcation diagram here.
10:41:47 You see that these localized structures are indeed in the region below the absolute instability threshold. If you, if they were above the absolute stability threshold that region, the question region between these localized structures would fail with
10:42:05 waves, and the picture would look very different.
10:42:08 So that's our understanding of why the states are actually stable, even though they connect to an unstable background.
10:42:16 And if you go a little bit, you know, above that region where these localized structures are present, then you have a dynamical invasion of the structure, and the structure grows by dynamically you creating new roles as a function of time, and you get
10:42:33 a pair of friends that invade the lemon region.
10:42:37 This is how the system from
10:42:43 transforms into the fully convicting state through the propagation of these from.
10:42:50 Sorry.
10:42:54 So now I want to say a few words about a model that captures all of this complexity. The model is extremely simple.
10:43:01 It's the swift homework equation so called. It's an equation that is first order in time fourth order in space, and has it by stable nonlinear narrative is function f of view is going to be a positive number be three times up, so that's a destabilizing
10:43:18 term, the negative be five.
10:43:22 So, this term is squinted term is a stabilizing term, this equations has the same symmetries as my binary Floyd connection problem, it has that symmetry r1 which is the left right symmetry.
10:43:37 And then it has the symmetry are two, which is a cemetery which is a reflection in the mid playing of the layer.
10:43:44 And we've already seen those cemeteries have that important role that they prevent motion of the structures that we saw previously. In addition to that, our equation has gradient structure with a free energy given by this expression, and the stationary
10:44:01 states that we saw, and then we'll show in a moment, correspond to critical points of this free energy so we're really looking at the behavior of the free energy in this kind of free energy landscape to identify the different stationary states of the
10:44:16 system.
10:44:18 So this is what the bifurcation diagram looks like for that equation, you have here a branch of even solutions, like, like the ones that are showed you in the collective problem.
10:44:32 Here you have a branch of odd solutions.
10:44:34 And they are intertwined in this fashion that you can see just like in the previous fluid dynamic one example. In addition there are these cross links that correspond to branches of asymmetric states.
10:44:50 These states are always unstable. And if I break the gradient structure of the problem these asymmetric states will drift for the same reason that Laura described in her talk earlier.
10:45:06 And if I go outside of the region where these local states exist and they undergo this kind of
10:45:16 stick slip motion of these phones, describing the success of new creation of additional sales in one can predict the speed of these fronts from theory which I do not have time to describe here.
10:45:30 So everything works very nicely in the equation. Similar things can be done for the fluid problem.
10:45:39 Now, I'm going to introduced for similar to break into the problem so the structures we've talked about so far formed by spontaneous and winter break.
10:45:48 But now what I'm going to do is I'm going to break the symmetry art to, that's the Uptown symmetry reflection in the middle of the system.
10:45:57 And I do it through adding this little term to my equation.
10:46:02 So, this term is to affect it breaks the you goes to minus you symmetry.
10:46:08 It breaks the you goes to minus you symmetry. And so that means that the on states can no longer be invariant. They're no longer exactly odd because you don't have the art to symmetry, and therefore the old states will have to move.
10:46:22 And they can move because this term also breaks the gradient structure of the equation.
10:46:29 And where are these states that move well they live on these branches the CZ shape branches and these three branches that form fire. The destruction of the previous picture due to the presence of nonzero epsilon here.
10:46:45 So I have a lot of different stable solutions, which will drift.
10:46:53 And so I can take those solutions and I can call on them, no head on collision like this. And so I here I've taken two identical solutions traveling opposite directions, they collide, they stick together.
10:47:05 This is not an integral system so it's not like a TV or MLS, and they form a localized structure of generated by this highly inelastic collision.
10:47:20 I can take two different solutions again colliding head on.
10:47:26 And, and they formed and localize solution by sticking together in this way.
10:47:33 I can take a more rapidly a skinny solution that travels more rapidly.
10:47:39 The catches up to a more slowly traveling solution.
10:47:43 Eventually they catch up and they form a localized structure that drifts slowly in time time increases upwards here.
10:47:54 I can also collide, moving structures of this time with stationary structures that said the symmetric structures.
10:48:01 And here you can see a collision between one of these moving structures, and one of the stationary structures again, deeply, highly inelastic collision.
10:48:13 I should mention that in this situation where the front.
10:48:17 On the opposite science or opposite.
10:48:20 That leads to attraction.
10:48:22 If I have an interaction between a front that looks like this and a similar front for the stationary solution. So this is a different station or solution that this leads to repulsion and so this structure actually pushes on this and the whole structure
10:48:36 after the amalgamation drift slowly.
10:48:42 Because of that route positive effect of the incoming way.
10:48:46 So this is a game that you can play by introducing this parameter epsilon to get these localized structures to actually move.
10:48:55 And we can play the same game in the binary fluid problem in the fluid problem, right.
10:49:00 And so here it is, how am I going to do that. Well, one way I can break the middle and reflection symmetry is to allow heat to leak out of the system through the boundary.
10:49:11 And I'm going to measure that effect by this parameter beater.
10:49:16 So when beta is equal to one, the temperature at the top is zero, the temperature fluctuation is zero, just like at the bottom. I have mid plan reflection symmetry.
10:49:27 So I have the stationary, or then even states that we've talked about.
10:49:32 And here is that bifurcation diagram, very similar to what you saw in this new convert equation, including these cross links and examples of these asymmetric states are shown here on the right.
10:49:43 I mean this case, you know this is not a gradient system this is the full fluid mechanics. And so these asymmetric states will actually drift horizontal for the reasons for that Laura talked about in her talk.
10:49:59 So here is an example of such a drifting solution that I've made drift by allowing beta to be different from one, allowing it to leak out of the system.
10:50:12 So this is the solution that would have been strictly odd or symmetric with respect to our one or two in the beta was one case but now it is not symmetric under that cemetery, and so we drift.
10:50:29 And here is an example in this based on plot what that drifting state looks like.
10:50:34 Was the stationary even solutions don't care about change the upper boundary so they remain remain stationary.
10:50:44 So I can play the same game with these, so I can collide, two identical solutions of this type, head on. So this is now never stops. Okay, never stops.
10:50:56 And, you know, they collide, and they stick together in the same way that you saw in the homework equation.
10:51:03 So little more complicated, you know, this is the system supports wave so they some radiation that comes from the collision but the outcome is essentially the same.
10:51:37 If I, if I collide in moving structure with a symmetric stationary solution with more complicated because it's an asymmetric solution there are more waves generated but ultimately, you get a localized solution of the time that you saw in this way from
10:51:36 Berkeley. I can so that last case was the attractive interaction here is one of these repulsive interactions with the front over here is the same as the front over here.
10:51:47 So this solution pushes on this solution that would have been stationary.
10:51:57 But nonetheless, this you know catches up, and then combines again. This highly invested collision just like we saw in this rubric.
10:52:09 So this is all very interesting to me because of the following and this is my last slide.
10:52:19 You know what's remarkable is that this sweet homework model
10:52:25 is really a very good model for a system that is much more complicated.
10:52:30 And you might ask you know how is this possible, because you know this sort of overarching question is very simple.
10:52:37 It even has variation of structure, which was the heaviest Stokes problem does not have.
10:52:45 I have no w diffuse effects included in this model right, there's no physics of the debate diffuser system in that equation, and yet the binary fluid system behaves almost exactly the way that I would predict by applying by studying this number two question.
10:53:04 And moreover the question can even be derived by some kind of systematic Asim dotted procedure.
10:53:14 You know, that you could apply to the Navy stokes equation.
10:53:18 And yet, it does so well in in predicting what happens.
10:53:24 So I think of this equation, not as a physically motivated model but it's a mathematically modified model. And it includes the to essentially regions that I think are relevant in this kind of situation.
10:53:37 And one is that there is a final wave number of
10:53:42 of the state of the, you know, that's generated in the system. And that's wavelength of the roles that you see.
10:53:52 And the fact that it is a bi stable system.
10:53:56 Now the BI stability here is not so important because you get kicked into this final and produced aid via this focusing instability
10:54:05 in the binary fluid system.
10:54:09 So the question really that's raised by this I think is in the words of pat diamond the other day is, you know, which case are we in is, you know, all models are wrong but some models are useful is this word Thornburgh model a useful model for understanding
10:54:25 these binary complex complicated binary fluid mechanical problem, or is this the situation that you know some models are too good to be true.
10:54:38 I've tried to show you that.
10:54:42 By incorporating certain mathematical features into a model.
10:54:45 You can do actually pretty well, even though the basic diffuse of physics is not present in the model.
10:54:52 And so I think this is a topic for discussion.
10:54:55 So thank you.
10:54:56 All right, thank you very much at your, it was well worth waiting to get unraveled they're very interesting, number, number of very general issues relevant to.
10:55:11 So questions.
10:55:18 Well, they're slow I'll take, we can run over a little bit, I mean it's not going to be the end of the world. This was extremely interesting.
10:55:26 I mean, The last point let's start where we are now.
10:55:31 I mean, in the thing that's striking. I mean, while you were speaking I'm course of running a quick comparison of the swift Hillenburg, and the calm Hillier riches in some sense that the rat.
10:55:45 The, the route almost the the poster child of this workshop or has been it's a program at some point and the thing that's striking here is that there is no negative diffusion in the swift Totenberg no explicit negative diffusion.
10:56:04 And the question is it, it raises a question I've wondered about what what is really fundamental for layering. Is it the BI stability or the anti diffusion.
10:56:20 I think, you know, a few weeks ago there was people were saying well what you need for for layering is anti diffusion with some regularization.
10:56:30 And I was wondering, it seems to me I could get layering with by stability.
10:56:35 Right and the layers would be the two different states. Do you have a thought on that. Well, yeah.
10:56:45 So in the circle Murphy question, or some, you know, energy is put in through that parameter are.
10:56:55 It's not a diffuse if term it's just, you know, it's, it's like, you know, negative friction if you like.
10:57:03 So it's put in a growth growth rate often often instability. Now you can modify this equation and study so called conserved swim convert equation, which is like the Hillier equation but it has an internal an intrinsic landscape within it on here yet equation,
10:57:27 I'm here yet equation, of course, you know forums, you know the reason you get this coarsening to larger and larger scales is fundamentally because
10:57:38 you know there's no intrinsic link scale that would arrest that course. Okay.
10:58:00 And so you can do similar things to what I've talked about for the conservative homework equation which would be some kind of interpolation, if you like, between what you've described.
10:57:58 You know, in terms of the connected and I mix and what I was talking about here.
10:58:03 You know where I was not conserving anything.
10:58:06 Right.
10:58:09 But I mean, yeah, I mean it's done actually. Haha, that's interesting.
10:58:16 Well of course in the Bly model you have you have the two scales, but in a sense the second scale in the emergent scale adjusts Of course during the during the problem as the gradient during the evolution as the gradients change as the profiles change
10:58:33 would be a comment so can you comment on the front propagation you mentioned just qualitatively you mentioned it to the stick slip, I was curious about that.
10:58:46 Yeah, so, so the way that you know you localize structure. Once you remove the parameters outside the, the range of existence of the structures.
10:58:59 The way that the system responds is that it new creates new structures in time.
10:59:09 It's kind of discreet sequence and there's a particular time interval.
10:59:14 Know between each new creation event.
10:59:19 And so that time interval is what determines the speed of the invasion of the laminate state, the background state, and creation Tom can be computed.
10:59:32 And it depends, you know, on how far your parameter value is from the folds in.
10:59:46 You know in this diagram. So if I go over here to depend on the distance of your parameter value say here.
10:59:54 And the fold, and new creation time goes like one over the square root of that distance. So it gets longer and longer The closer you are to the phone.
11:00:07 And it's possible to compute all the pre factors and so on and, you know, very good agreement between simulations and predictions from this kind of deep in it's really a dependent problem.
11:00:20 yeah depending on.
11:00:24 It's kind of dependent theory.
11:00:25 Yeah would remark, depending and is of great interest to us in plasma physics where things are pins to geometry rational surfaces and all that. So, all right we better.
11:00:40 Other people are in line here so I guess Brian is next and then David was my observation. So go ahead, Brian.
11:00:48 Yes, you. You started out you ended by disavowing a physical interpretation. But you began by speaking of an experiment in which this equation had been inspired.
11:01:06 Can you can you can you go back and say that well what's actually happening in the, in the physical case that led to this.
11:01:17 Well, okay, yeah.
11:01:29 Well, let me just, just go back and look at these, these equations.
11:01:35 So the, you know, the usual WTF is a system that people started in oceanography for example or an astrophysics would be the, this set of equations so you know equation for the velocity field never stops equation for temperature and equation for the concentration.
11:01:53 And I'm using see here for the salinity, for example, because I want to keep the symbol s for the separation ratio, that's all.
11:02:02 So the buoyancy that drives everything ultimately know depends on both the temperature and the concentration.
11:02:08 The question is the concentration imposed externally through boundary condition, or does it, you know, develop spontaneously, in response to the apply temperature gradient.
11:02:19 Yes How can that happen what is the physical mechanism, give an example. Well, so this is what described by the story coefficient is cross diffusion coefficient, and it would be, you know, that's really what's after non dimensional non dimensional non
11:02:35 non dimensional ization is responsible for this term here.
11:02:40 Now physically how that happens is that, you know, if you have, Let's say you have a cold boundary, then the molecules are not moving.
11:02:52 You know, so rapidly as they are near say a warm boundary, right.
11:02:57 So that means you will, you know so that temperature gradient will then create,
11:03:04 you know, a gradient in the distribution of the heavier molecules, because the, the kinetic energy, assuming your network partition depends on the mass of the of the particles right so the.
11:03:18 So how, you know, what the concentration is going to be is going to depend on the local temperature
11:03:28 right i mean that's the basic principle that makes the salt to diffuse differently than the than the heat but in there, it's molecules that are doing it or if you're saying that there's another mechanism that wouldn't be the difference in the
11:03:46 diffuse of concept for the salt versus the, well I'm not talking about the difficulty of salt, I'm talking about the fact that there is a current have sold.
11:04:05 That is set up in response to an applied temperature.
11:04:05 In response to right so I'm not forcing it via us no boundary conditions on the salt concentration.
11:04:14 But it is a response of the system to putting it into a temperature a
11:04:33 Okay, that's that's the effect.
11:04:36 And it is, you know, it is present in, in salt water. I mean, but for some reason, it is not included in any oceanographic study and I'm just kind of curious, in general, why people do not include this term because you can measure this coefficient.
11:04:56 I mean this separation ratio can be measured.
11:05:04 You know in in salt water.
11:05:09 All right, David.
11:05:13 Yeah, I was just wondering about the differences. I mean, kind of what you've been saying but the differences between standard double diffuse of connection, I mean if you.
11:05:23 I mean, presumably, you can message these equations into something looking quite double diffuse it if you kind of add, Add the TMC together appropriately.
11:05:33 Is that right yeah yes you can. But, but then you know the issue is, is how you're going to deal with the boundary condition, yeah okay okay that was my question so so the actual, the actual equations we couldn't make double diffuse it if you like.
11:06:02 can dive in analyze this problem, I mean this is, like, you know, you have a diffusion matrix and you can die. Exactly. Absolutely. You can do that but then you're going to have, you know, then you have to decide what are the, the right boundary conditions
11:06:06 on the mixed mixtures of DNC right.
11:06:07 And that's, that's fine if the boundary conditions on TMC identical, you know, top and bottom right, that they will continue to hold for these mixed variables.
11:06:17 But if you have one set of boundary conditions for tea and a different set of boundary conditions will see, then you know you can do it.
11:06:24 Yeah, yeah. So, so going or the. So going back to your to your last point, you know why swift Holmberg does well.
11:06:35 So, so it does well as a model of binary connection but it doesn't do well as a model of the Milan connection is that is that right. No, I would, I would, I would argue that, not because I haven't done that so I want to preface.
11:06:52 What I'm saying with that I have not done the corresponding calculation for standard w dishes if wer metric but.
11:07:00 However, you know, the physics of this is really very similar to why because you have no fresh, you know, cold, liquid above you know warm liquid. Here, the only difference is that it's generated via this cross diffusion of fact, rather than through imposing
11:07:24 Here, the only difference is that it's generated via this cross diffusion effect, rather than through imposing boundary conditions that open bar.
11:07:28 Right, so I would expect that essentially everything that I've described here for this setup would also be true for standard definition is the connection where you impose a concentration difference across the system, kind of external.
11:07:49 Okay, thank you.
11:07:52 Hey, one last short question we're over time now by a bit dm okay it was actually clarification with what you said to to Brian, because the way I understand how you described the how the temperature plays a role is would be similar to when since I acquire
11:08:15 Jenny comp or whatever if you put a cold spot.
11:08:18 In this pod system, it will attract particles, so you will generate a current just having that cold spot.
11:08:26 And you will generate blocks and maybe an instability. So, again, this is what you're saying and this I understand, physically, that I had the impression that you were also saying that you could have an inverse instability, with this so a thermometer.
11:08:48 Basically saying that a cold spot.
11:08:56 Would retail. Yeah, that's right. You could have it both ways.
11:09:06 Could you could you explain the the does it give us right yeah so interpretation to that. Yeah, yes yeah so that's that's a very good point. So, so, in gas dynamics.
11:09:14 You can actually you know, compute, you know expressions, you know, from kinetic theory for these coefficients discourse diffusion coefficient, and in guess dynamics, you will find that the sorry coefficient is positive.
11:09:30 And so the concentration will be higher near the cold spot as you say.
11:09:36 Now in these strongly coupled systems like liquids, those calculations don't apply.
11:09:43 And you typically have to do an experiment to measure this Oracle efficient.
11:09:49 And these, the value of the circle version will actually depend on the mean properties of the system the main temperature the main concentration and things of that type, and around, you know, many systems, and salt water is one of them, but also the people
11:10:08 have done experiments on on alcohol water mixtures and things of that type helium three helium for mixtures.
11:10:15 And in many of these systems you can find conditions under which the story coefficient is anomalous or negative.
11:10:22 And so the heavier component actually migrate to the warm spot.
11:10:31 Okay, so the interpretation is that it's the internal interactions between particles that would lead to to that normal anomaly or is it a property of the interactions between the particles.
11:10:48 It's absolutely a proper interaction, not a consequence of the interaction among the particles right in this apply temperature field.
11:10:56 But it's, you know, it's, it gets very complicated when you have, you know, a dense liquid.
11:11:04 You can, you know, do you know some kind of, you know,
11:11:11 capital gains Kappelman score calculation, right, to, to, to actually predict the values of these coefficients reliably.
11:11:18 So you have to measure it, right.
11:11:21 It's just typically what people do, they just measure it.
11:11:27 Okay.