09:03:11 Okay. And so, I was asked by David to give a little talk on layering mechanisms is focusing specially on the gamma instability.
09:03:23 And so this is a very, very much of a whirlwind tour of some of the things we did with a lot of colleagues and students over the year for the gamma instability in double digits of systems.
09:03:36 So before I start, I just wanted to give a really quick recap on what the gamma instability is and you heard this from Timor Rocco's presentation, but the emphasis I want to give here is that it's really a very generic phenomenon that has actually nothing
09:03:53 to do with double diffuse convection at heart, it's something that's very generic. So if you consider two fields.
09:04:01 And I'm going to call them TNS here just for, so that we recognize them but they can be any to scalar fields. So, TNS our functions of time, and a spatial coordinate and the evolve, according to conservation laws, so one conservation law for temperature.
09:04:21 Can you see my cursor.
09:04:23 Yes, yeah okay one conservation law for temperature. One conservation or for salinity, or for s. in. In each case we have this flux here ft, and Fs.
09:04:36 And the key assumption here is that the flux is are related to the respective temperature gradient and salinity gradient via a function here. Now in fluid dynamics we often call these numbers so let's call this initial number.
09:04:53 So, and the key is that these functions are functions of the ratio of the gradients only. So that's an assumption. So, if this is the case, if these flexes or functions of the ratio of the gradients only, then the system is layering and stable provided
09:05:13 the ratio of the flexes ft over Fs is a decreasing function of the ratio of the gradients DTDZ over DSDZ.
09:05:25 And as you can see here there's no mention of double defensive convection here, you know, it's completely generic and it happens. So whenever you have to flux laws that have this property here, where the new sub number is only a function of the gradient
09:05:41 ratio, and the flex ratio is decreasing function of that gradient ratio. So in Wi Fi connection that gradient ratio is called the diff density ratio.
09:05:53 But just bear in mind that this is what it is when I talk about it.
09:05:58 Okay, so in Timor's work in 2003 specifically.
09:06:04 He defined two functions so one is the standard we define New Silk number so relating the temperature flux to the temperature gradient.
09:06:11 And then another function he defines is gamma, which is the ratio of the structures so the gamma is the ratio of the temperature flux to the salt flats and so you can write Fs as gamma minus one f t.
09:06:25 So with that definition, you can compute the growth rate of the gamma instability by solving this quadratic function to more show Diana's talk, I believe.
09:06:35 And so lambda is the growth rate of the instability k is the spatial wave number of the instability. And so you can see it's a simple quadratic that you can solve analytically, and the coefficients a and b are simple functions of the new sort number function
09:06:51 and the gamma function, as shown here. So, a is the gradient, or the d by dr of our new and it's already by Dr. New gamma NB is minus RNA squared times d by dr gamma minus one.
09:07:08 And so these assuming you know these function new of our, and gamma var you can compute a and b.
09:07:27 Now the key here is that if you have a quadratic so this is a lambda squared plus a lambda k squared plus BK for that quadratic, it's a parabola that opens up. And so you can see that everything that whether that has solution or not it's going to depend on the
09:07:33 intercepts BK for.
09:07:35 And so, if that intercepts be, which is the sign of the intercept is controlled by be if that is negative, you're in this case and in that case you always have a positive growth rate lambda.
09:07:46 On the other hand if b is positive, then you can't have any.
09:08:01 I'm blanking out because what I said is wrong.
09:08:05 I think this drawing is wrong because A is positive is positive as well. So actually it should be looking like this, sorry. That's an extra an extra complication I forgot to mention, so this drawing is a little bit incorrect.
09:08:19 So if he is greater than zero, there's no unstable mode.
09:08:24 Anyway, so this be smaller than zero is equivalent to seeing that gamma has to be a decreasing function of our and that's the standard government's ability statement.
09:08:34 So, so what do we do with this in practice well, it to apply the theory, we need to be able to figure out what these coefficients a and b, which are the coefficients of that quadratic.
09:08:46 We need to find out what they are. And so in practice what we do is we run DNS of in this particular context fingering convection or military double defensive convection at different gradient ratios density ratios, so you can run a little box of WC convection
09:09:05 at density ratio are another little box that density ratio are plus Dr.
09:09:11 And then, in each case you measure the temperature flux on the salt flux. So you do our at our you do our plus Dr. From that you extract the initial number, and you extract and gamma, again, R and R plus Dr.
09:09:25 And then you just use finite difference in to compute these derivatives here and here and here to compute a and b, and from a and b, you can then compute the lambda of case so the solution of this quadratic here.
09:09:40 And that you can do it on it.
09:09:44 So, so this is basically what we've done for many years and I'll show you some results right so we did this the first time we did this and this is basically how I met Timor working on this paper with defined stomach.
09:09:59 And the promo and my student Adrian Traxler.
09:10:02 So, these are simulations I believe you saw so this is a large box fingering convection at Princeton number seven. DPCVT ratio one third, and density ratio 1.1.
09:10:15 Initially this boxes are just fingering and stable and eventually you end up with layers. And there's some intermediate step here which I'm going to gloss over.
09:10:24 And so what you can do is you can measure the flexes in this box at density ratio 1.11 you can measure some density ratio that are nearby, you can compute this new sort number this gamma you can compute the derivative of nice number and go on with respect
09:10:40 respect to our, and then you can predict the growth rate of the layering modes that come into the system.
09:10:47 And this is the result.
09:10:50 And it has a lot of information so this is time on the x axis and energy on the y axis, and for layering mode you typically measure the potential energy in the layering mode.
09:11:03 And that in the simulation for the mode that ends up emerging which has two layers as you can see here for that particular mode which is this blue line this is the energy in the mode that dark blue line.
09:11:15 And the prediction from the gamma instability theory is the pale blue light, and you can see the growth rate predicted as a little bit higher than the actual growth rate but it's very close.
09:11:27 And so one of the things that's going to be a theme in what I'm about to say is the gamma and stability theory is not just there to tell you whether layers form or not it's there to tell you how fast they're supposed to form.
09:11:40 And one of the beauty and really where I really believe it works is because the to fit extremely well.
09:11:46 So that was in this fingering convection at water type parameters, much more recently, Raphael, who you own an accurate my book contacted me to try and understand some of their simulations at very high Prancer numbers now this is Principle number 200.
09:12:05 And the goal there was to try and model. Salt.
09:12:19 So these are simulations are printed number 200, the facility ratio point one and density ratio 1.5. And again you can see that you start with fingers, they become nonlinear, and eventually earlier forms.
09:12:33 So you can play the same exercise.
09:12:36 computer flexes in the small box simulations once the fingers have become a linear computer coefficients of the quadratic computer growth rate of the mode, and this is again the energy in the layering mode, that, and the blue one is the one we observe
09:12:54 eventually forms in this blue dashed line is it's predicted growth rate based on the gamma instability and again the predicted growth rate is a little bit higher, but not much I think they're off by about a factor of 1.5 1.4 1.5 so it's definitely within
09:13:13 the right order magnitude.
09:13:18 Now we're getting to the more interesting thing which where the theory starts having predictive power. So, this is some work that we did, shortly after the work with Timor back in 2011, this is with my graduate student Adrian Traxler.
09:13:34 So she ran a large number of simulations of fingering convection at very low Principle number, so that more appropriate for stellar interiors.
09:13:43 And these are the Gamma's so again the ratio of the temperature to two. In this case, some kind of,
09:13:52 you know, chemical chemical elements. So temperature, the ratio of that flux of temperature to the ratio of that flux of chemical elements.
09:14:00 And that's as a function of a reduced density ratio, but you can see it's pretty much proportional to the gradient ratio.
09:14:07 And so, gamma you can see in this case is always an increasing function of the density ratio.
09:14:14 And the prediction is that no layer should form in that system.
09:14:19 And indeed We never saw any Laver's form in that system.
09:14:23 And so one of the predictions of that model is that in stars layers do not form by the Garmin stability and fingering regions and fingering regions in Star remain in the state of small scale fingering convection.
09:14:39 Now moving on, you know from the talks we saw a few weeks ago that there several flavors of a Wi Fi connection, one of which being the military double defensive instability.
09:14:50 And so roughly around the same time a little bit later.
09:14:53 We worked with Erica Rosenblum and Giovanni Meru who were two students.
09:15:00 And we did numerical simulation of a military double defensive convection low percentage number. And we measure the inverse gamma is a function of the inverse density ratio.
09:15:10 And if you think about it, gamma, gamma must be a decreasing function of our translated in inverse gamma spinning a decreasing function of inverse density ratio.
09:15:20 And these are all of these curves that we got from the small box simulations for lots of different values of the parental number and the facility ratio.
09:15:29 And you can see that the gamma curve is non monotonic it has a decreasing part and then it has an increasing part, and all of the large symbols are simulations that ended up giving layers.
09:15:42 And there's a very strong correspondence between simulations where we ended up seeing layers and simulations where gamma is a decreasing function of our, and in here, you know we didn't complete that curve, but here it's actually going back up now we
09:15:58 have some later simulations. So here, indeed, there's no layers, even if even though it looks like it might be decreasing here.
09:16:06 Anyway, the point is there. Again, the government's ability works in being able to distinguish when layers are supposed to form and when they're not. And this is an example of one of these simulations.
09:16:16 I think for density ratio 1.2 parental number and how point one.
09:16:21 You can see the initial wave like instability, then that develops into layers and so you see here, we have three layers.
09:16:31 Eventually, these merge and we saw that TMR has some merger theorems for layering instabilities after the layers have formed.
09:16:42 I'll just let this play.
09:16:45 And I think David talked about these layers and then the merge mergers as well and blueprints and number, solitary double deficit instability. There we go.
09:16:56 Military double deficit instability. There we go. And this is again.
09:16:59 Now the this is the spectral power and density or in other words potential energy of the layering modes.
09:17:05 And the one we initially saw had three layers. It's the one that's the blue one, but actually there was a four layer one that was growing at roughly the same rates that's the key for, for each of the modes that we see I plotted in in dotted line the actual
09:17:22 predicted growth rates, and you can see that they grow pretty much at either the predicted rate or slightly slower, which is again very common.
09:17:32 And the modes are too large, they will grow too slowly to be seen, so we don't see them grow, which is here.
09:17:39 So again, the comments similar to the gamma instability model here works extremely well.
09:17:48 This is the same thing as this, but just with the magnetic field now in that is threading the box in the direction of gravity so a vertical magnetic field.
09:17:58 One of the interesting things is you can see how the presence of the magnetic field modifies this gamma curve.
09:18:04 And what we see it makes it steeper at low density ratio and then the minimum is earlier on.
09:18:11 So effectively add these points. The curve is flatter and that corresponds to the colors here corresponds to the colors here.
09:18:20 So, the ones without magnetic field ended up forming layers much faster and when you add the magnetic fields the layers form slower and slower but still eventually form.
09:18:30 And this is just increasing that the magnetic field from zero to larger and larger about us. And this is work with an undergraduate student of mine I mean she signed me which are currently completing.
09:18:43 And finally, I already talked about this in my presentation, a month ago, you have again the same gamma style instability now in sedimentary fingering convection.
09:18:55 So, this time there's a correction due to the sedimentation. But again, you predict when layers form or when they don't form.
09:19:05 This yellow curve here is a fingering instability, with a fairly rapidly descending sediment. So again you can see gamma here is a decreasing function, the density ratio.
09:19:18 And this is a plot showing again the power in the density or the square root of the potential energy in this case is a function of time.
09:19:27 The yellow curve is the layer that ended up forming in the simulation, the yellow line is a fit to that growth rate.
09:19:35 The blue line is the predicted growth rate from the Garmin stability theory so in this case it's one of the rare cases where the predicted growth rate is lower than the actual growth rate.
09:19:46 And we attribute this to the fact that in this particular simulation the fingers extremely long.
09:19:52 So the assumption that the model is local in space probably no longer works very well. And as you heard two more described in this lecture, there might be some corrections to the GM instability theory that needs to be done.
09:20:05 When the fingers are extremely long.
09:20:08 And so that might actually end up helping explain that discrepancy so this is something we should look at in the future.
09:20:14 Anyway, so that that's pretty much all I wanted to say so, the government stability is nice because it doesn't just predict one layer should form but also our rates, they should form, and that's quite helpful when trying to compare it with experiments
09:20:26 and I think any kind of layering model should have that capability and predicting how fast Leaders Forum and not just if.
09:20:33 So that's it.
09:20:37 Thank you Pascal was nice.
09:20:40 Plenty of time for questions.
09:20:43 Please just
09:20:47 raise your hand.
09:20:49 Okay, Adrian.
09:20:54 So a lot of cases, people have three gradients, that they care about you're not just temperature and celebrity but there might be a third quantity.
09:21:04 This happens in like I've seen some genetic work where people care about you know density gradient temperature. Temperature versions.
09:21:11 Have you ever thought about, you know, you're presenting the game and stability theory is really general right whenever you have to fluxes that depend on distance ratio.
09:21:21 Have you seen any work looking at extending it to three fields know that's a really good idea you should do it.
09:21:37 Right, Pat.
09:21:39 So, a couple of questions.
09:21:44 The most obvious is I mean the gamma way of doing things approach strikes me is working with transport equations right you're starting from equations for fluxes and all.
09:21:59 What is in some sense known or what is the thinking on how you get from the basic primitive equations to transport equations of the right structure to get the gamut and stability.
09:22:17 Well, I mean, in Timor's original work.
09:22:21 The proposal was just to take a horizontal average, because for WC convection.
09:22:28 You can just horizontally average your primitive equations on getting these flux equations right now, what we have done to extend that work and it's not.
09:22:37 I didn't discuss it here but you can also work it with a local average. When you imagine you consider a patch of fingers that is now has a spatial structure, and you average over that local patch, or you can consider a narcotic average I mean anything
09:22:52 that kind of gives you rentals averaging any of these kind of things work actually an easy to do it you get to now equations that don't just depend on one coordinate z but they can depend on other coordinates the and you can extend all of these mean field
09:23:08 theories and get some other interesting results in particular regarding the formation of internal gravity waves from fingering convection.
09:23:17 But what physics in the basic equations is bet is in some sense, underpins the flux ratio being a decreasing function of our, and that's that's what I'm trying to ask.
09:23:31 All right. Sorry. Okay, so for fingering connection is pretty easy it's just because the, the salt being defusing the last few sips and the temperature.
09:23:42 Once you start transporting it you transport so more efficiently than you transport temperature.
09:23:47 And so the flex of salt is always
09:23:53 hot effects of salt is higher than the flex the temperature so that tells you that gamma is smaller than one, and that effects.
09:24:00 So ft over Fs initially becomes more important so I'm not explaining this way.
09:24:08 What if you if you were to be fully convective.
09:24:13 Right.
09:24:13 Then, temperature and salt would be transporting equally efficiently.
09:24:17 Then as you become more salt. More fingering unstable, that in balance between the salt and the temperature become more. And so, relative to temperatures salt becomes more transport and more efficiently so that's why government decreases.
09:24:34 And then, the reason gamma begins to increase again is that now the diffuser flexes of temperature and so start kicking in and you have that increase again.
09:24:44 Sorry I didn't explain that very well, but what, that's, to me that's something I want to understand better the other question is what does gamma predict about the scale of the Larry.
09:25:00 Well, yeah and then I have another go ahead. Sorry. It doesn't because as you know and I think the more discussed that the gamma instability has an ultraviolet catastrophe.
09:25:11 Right, right.
09:25:14 The government instability on its own, the way it was originally proposed has this ultraviolet catastrophe problem. Nine practice.
09:25:22 What happens is that the main field assumption is violated when you go to scales are too small because you're still supposed to average over a few fingers.
09:25:31 And so you need to take that into account and so from a physical perspective, it makes sense that at some point yes instability breaks down, and then P more recently developed a theory that takes that into account and then you you you resolve the ultraviolet
09:25:46 catastrophe using his new model.
09:25:49 Right, I'm just curious what you think. I mean, in these other problems usually what happens at that point in the story is, there is an identification of an emergent scale.
09:26:04 Right, such as in in Bly the, namely the asthma. Right.
09:26:10 And what what is the lurking the lurking scale, in some sense, the whatever team or is doing which I don't understand must boil down to identifying a another scale and the problem.
09:26:26 What is it.
09:26:28 So there's two possibilities.
09:26:31 So, in the original work that we did with tomorrow, which had to do with oceanic fingering convection. Right.
09:26:41 And let me share my screen again.
09:26:45 They were two things that happened and I glossed over the first but it turns out to be really important for determining the emergent layer scale.
09:26:53 There's this intermediate step here in which the system forms internal gravity waves. Okay.
09:27:00 And, and I can get into why they form, but basically these internal gravity waves.
09:27:07 By contrast with the gamma instability these have a well defined scale.
09:27:12 And they end up at least that's how I interpreted they end up filtering any layers that would normally grow on scale smaller than these waves. And so what we see our layers that emerged that our own skills that are commensurate with these emergent internal
09:27:28 gravity waves. So that's one mechanism that might set a layer scale. And that would be intrinsically related to whatever instability happens before the layers form.
09:27:39 Now, quite remarkably in these simulation by referral.
09:27:43 The collective instability that internal gravity wave instability does not occur, and it's one of the only known systems where the layering instability takes place without a prior internal graduate instability.
09:27:57 And so, and in this case you can see that there is still an emergent scale for the layers. And now this one, I believe, is due to this effect that the fingers have, you can't average over too short to smaller number of fingers otherwise it doesn't work,
09:28:14 And then empirically, I think there's evidence good evidence that you just need something like five finger lengths to be in a regime where you mean field theory makes sense.
09:28:28 And that's consistent with what we see here we have about five finger length in one layer.
09:28:34 But there's no good.
09:28:37 So, the point I'm trying to say is this is not a new scale that emerges is just five times whatever the finger length is and so order of the number of fingers that you need.
09:28:56 Thanks David.
09:28:59 I'm
09:29:01 everything you just said, kind of put me off my mark so I as great as the theory is one problem I suppose is that it's still semi empirical you need to actually be measuring these boxes at the start.
09:29:17 And that's just getting me thinking about how does the errors associated with that measurement, it sounds like you need to be taking the measurement, at a certain time in the simulation, you can do it right away as the fingers are forming.
09:29:32 but when they're better developed.
09:29:34 And you also need to be considering averaging over enough fingers. Can you just give me a nice sense of how sensitive your results depend on how you actually measure those boxes.
09:29:49 I mean there's several levels to that question. So one of the, one of the things that we can certainly say is if you, if you run these empirical determination of the flexes in small enough boxes, the layers never formed, and so you're always in a system
09:30:06 that has constant gradients and so the problem is well pose you wait until you're in a statistically stationary state you measure your fluxes and then you've got a good estimate for that right, and a harder question is the sensitivity, so it flexes are
09:30:23 intrinsically variable so you have your measurement of success with some kind of error bar because the flex is a variable.
09:30:29 And then you plug that into the A's and B's you calculate your growth rate. As it turns out, and just, and the, the computation of the growth rate can be quite sensitive to these A's and B's it's not quite a Neil post problem but it's very sensitive problem.
09:30:48 So you can have pretty large deviation in the predicted growth rate depending depending on tiny errors in these measured flexes. So that's why I'm not necessarily worried about being off by a factor of, you know, 50% in the predicted layer of growth rate.
09:31:06 But I'm not sure I understand what you mean by, we have to wait until a certain time in the simulation. Typically if you have small enough boxes you can actually measure flexes in a meaningful way.
09:31:18 Yeah, that's why I think what you said there that helps answer your question so you're not actually taking the full simulation calculator blood system that you do these little toy tiny box simulations first get the fluxes make the prediction but then
09:31:31 you run the big simulation, usually. Usually, I mean, sometimes, you end up with layers even in small boxes.
09:31:39 Occasionally, you get that. And that's a bit annoying because then you're indeed in the situation you described where, you know, you basically have to move your fluxes in between the time or the instability saturates and then the layers form and sometimes
09:31:52 that time is not very long. So occasionally that's a problem but in most cases it's not.
09:31:58 Okay, thanks a lot faster.
09:32:05 On the issue of an emergent scale.
09:32:09 It seems as simple physical reasoning that the layer thicknesses would just keep growing indefinitely until boundary conditions interfere and also wondering if empirically This is seen at least in some instances, if someone like the notion of bubble growth
09:32:24 in a, in a farm where the big bubbles keep gobbling up the smaller bubbles so imagine you have two layers of different railing number, just sharing a common interface.
09:32:35 Well, the one, the larger layer has a higher railing number it has a higher intensity of turbulence that's eroding at the interface. So, it would tend to grow in the same sense gobble up the smaller interface so that would just be sort of a physics argument
09:32:55 for it to grow indefinitely and since things scale is power laws like the muscle memory scales power rail a number that would even suggest that the growth in time would be some power law and time.
09:33:06 So I'm wondering. Has this been considered and what's known about this, or observed in the future.
09:33:15 Well, so in terms of has this been considered there's a really nice paper again by two more. He should be here.
09:33:21 So I'm just talking about his work is a really lovely paper by Kimora looking at different merger mechanisms in layers in in start cases that describe when the staircase is stable versus when the staircase is unstable, and indeed in some cases it's about
09:33:37 the flux was on the cross an interface and in some cases about the flex was within the layers.
09:33:43 And I don't want to get into the details but basically that that's a very nice way of rationalizing and trying to understand when layers coursing, and when they don't from an empirical perspective from our simulations.
09:34:00 What we have seen is that
09:34:05 in layers associated with the military double defensive convection.
09:34:10 Then the murder rates are typically very fast and procedures proceed presumably in the manner, that's very similar to what you described the fingering staircases on the other hand, have a tendency to just stay the way they are.
09:34:23 They don't seem to merge on any scale that's on any rapid timescale, so if they emerge, it's on a much longer time scale associated with interstitial flexes, which are
09:34:38 what relatively small compared to what you would say, if you consider a layer based flexes.
09:34:45 That seems actually logical physically because the fingers have their own structure, such that the transport of the layer may not be governed by the overall bulk property that scales with layer thickness so it opens to the picture.
09:35:00 Yeah.
09:35:09 Good.
09:35:11 Yeah Bosco, and thank you for a very nice overview.
09:35:15 There's one thing that I don't completely understand it was my impression that you need to do 3d simulations to get layer formation. In other words, that there's do not form if you do it to the computation.
09:35:32 And yet, in the gamma mechanism that you presented. No, there is no distinction between 2d and 3d, can you comment on that. I think your first statement is incorrect.
09:35:43 So for example, the, the original paper my Ratko in 2003 was 2d.
09:35:50 And in fact that's that's how I ended up getting with two more saying are you sure this is going to happen in 3d then Stephen came into the picture developed this code and then the rest is history sort of thing, but the the original paper by tomorrow
09:36:02 is Tuesday, and he found layer formation in 2d.
09:36:06 and he found layer formation in 2d. The paper by rough out we own that I discussed that high Principle number that one is 2d, a lot of David simulations and believe are 2d, and they will show layer formation so that's not that's not an issue.
09:36:18 And so it's consistent right and one of the cute things is that the the flux is in 2d, and the corresponding flux ratio in 2d are different than the flux is in 3d.
09:36:30 But if you compute the gamma growth rate in 2d with the 2d flexes you get the correct player formation rates in 2d and then you do the same thing in 3d you can correctly information right in 3d, all works.
09:36:45 It all works. Right. Can I ask something Pascal when you first introduce gamma, you, you have countless dT by D.
09:36:57 You had a dT by these no but no sorry that's all. When you first introduce our you had deep it said over the is by design, which is a function of said, I guess.
09:37:05 But when you, when you present things, so that then becomes a sort of Zed dependent criterion doesn't it. I mean, if there's always a function of said why might you not have it satisfied at certain places and not satisfied, others.
09:37:26 Absolutely. So from a very large scale perspective you think of a notion right and you have different density ratio a different steps in the ocean, you can absolutely get your form in one place and not in the other.
09:37:37 Now, the gamma instability theory itself is a linear midfield instability right so you consider.
09:37:45 You start by considering constant gradients of temperature and salinity so your initial density ratio is constant.
09:37:52 And then you look at perturbations away from that constant density ratio and then you do the linear theory and then you end up with the prediction for the gamma and stability, but in a real system where you have might have very large scale gradients of
09:38:05 temperature and salinity absolutely you can have layers form in one place and not in another.
09:38:14 did not try answered your question, I mean it would be really nice if we had enough computer power to get really great thing to test right you have a very large scale modulation of DTDZ and you should be able to see that layers form in someplace and others.
09:38:29 others. I guess my question might be wrong but I'm what I'm thinking. My question is that you haven't are not. And that's, that's are not that is based on what you said at the beginning, right, the.
09:38:40 And then as the as the simulation involves are defined as dT by design over the spine is it becomes a function of Zen.
09:38:55 And, but then you have. So in, in places are might be this and in places it might be that, but then you've got your small bumps calculations which which say that there are equals, whatever.
09:39:07 But that whatever might change in in depth in a real simulation.
09:39:11 Yeah.
09:39:13 But that's kind of exactly why the gaming similarities there in the first place right it's this positive feedback between.
09:39:22 So So you have your initial arm it's constant and then in some regions, it becomes a little larger in some regions becomes a little bit smaller so your your your flexes change in the different regions and your flux ratio is going to change in different
09:39:36 regions. And that's going to either initiative is going to make your initial perturbation grow or decay depending on that. And so,
09:39:47 I mean, I don't know if this is helpful but
09:39:55 the government's ability in some loose sense can be used as an anti diffusion process right. So, and it can be seen as a safety compute the evolution equation for our so now you consider our as a function of T amp z right and you actually combine the
09:40:13 T equation and the equation and you create an equation for our.
09:40:17 So that's the diffusion operator. And the question, the coefficient in France has a de gamma Dr.
09:40:34 And so do my dr is negative, you have an anti diffusion, and that's this positive feedback loop that creates an anti diffuse behavior for the density ratio, but this precisely because the density ratio varies with z that you get the instability in the
09:40:51 first place.
09:40:51 OK, so again I'm not sure I answered your eyes okay and I just need to think for the
09:41:01 echo if you got another question Are you still.
09:41:04 Is that a remnant and actually yeah let me just ask quickly Pascal so is it easier to get layering in 3d than in 2d because of the way that these flux is, you know, depend on dimension.
09:41:31 Uh, I don't I'm not sure there's a general rule.
09:41:38 My experience is that ballpark. Right.
09:41:42 Whenever gamma is a decreasing function of our entity it's also a decreasing function of our in 3d, I mean, within some slight changes in the, in the,
09:41:54 and then the typical ratio, even though the flexes can be say, typically in 2d the flexes are a little bit larger.
09:42:01 So even though the flexors are individually larger their ratio is roughly the same or not exactly the same. So I don't think it's theirs, it goes one way or the other.
09:42:15 But I don't, I haven't thought about this very deeply.
09:42:18 Okay, Thank you.
09:42:21 Small comment on that which is that in the to the ones we've added some to the ones with small segment or the same as one of your 3d ones Pascal and we did find layers into the way you didn't in 3d now whether it's because we could, you know, carry on
09:42:39 for much longer that kind of thing that might help Yeah, definitely.
09:42:46 That's one data point to answer his question.
09:42:55 Okay. Well that was a excellent discussion. Thank you Pascal.
09:42:58 Okay, we should move on.
09:43:04 Right. So our next speaker is huge and ma so Eugene has been participating he hasn't actually given a talk as yet so I should just say that he's, He's a graduate student at the moment in physics department in Toronto, working with the death penalty, so
09:43:20 pleased to have him give a talk today so Eugene whenever you're ready.
09:43:26 Thank you. Thank you for your introduction. And so I guess I need to share my screen now.
09:43:35 Yeah.
09:43:46 So, I'm
09:43:46 okay with you.
09:43:47 Yep. looks good. Yeah. So I thank you for introducing me and I'm very glad that Pascal has give really good, some summary on the Garmin stability and we have really a lot of discussion just before and I guess the talk that I'm giving now will be relevant
09:44:09 with at least some of the problems. For example, the ones that David just asked, which is related with a variation of our role, because now this work is related with a higher with a high vertical scale so it's might help to answer those questions.
09:44:29 But anyway, yeah,
09:44:32 I want to start my talk here. So I'm the as David content I'm a graduate students and University of Toronto. And this is a work with me and my supervisor, take out here, which has recently been accepted in the Physical Review fluids concerning the gamma
09:44:52 instability in the homogeneous environments and salting grains Derek is traveling.
09:45:01 And so, I want to firstly bring bring your attention to the thermal, a nicer cases in tropical ocean, which you can see two examples here.
09:45:10 Because what's happening in the topic ocean or the meat, like to the ocean as you guys the strong solar radiation at the surface so you got the warm salty water line above the code fresh water.
09:45:25 So this is the condition which favors the salt fingering stability. And we know that it has the potential to drive the system into the layering stage experimentally we find that this really depends on a very critical items number which is the ratio.
09:45:46 And these layering structures can be fine, as long as not as long as but only in the places where, where our role is smaller than 1.8.
09:45:57 So, as described by Pascal just before the cell fingering thermal he lives here cases can be very well captured in the gamma instability theory. And here I will just maybe repeated briefly here, where for the coming stability team more assumed uniform
09:46:27 gradients, and floods greed and loss where the flux is are only determined by this arrow, and he made the denier stability analysis, and the conclusion is the linear gradient profile will be unstable.
09:46:34 When the gamma is a decreasing function of our role as well as just content before. What happens in the pit can be showing this picture where the perturbation initial perturbation of the gamma Mo's will grow in a strong and to its reaches a certain magnitudes.
09:46:53 And then the density ratio will be gravitationally unstable at a certain region. So, this is how the.
09:47:02 Their forums and Pascal has showed a lot of DNS simulations which are, which shows a very great consistency was this theoretical model, which shows that this is really a powerful theory and further on, are the common dad's this common stability as, give
09:47:22 a very well explanation of why we observed the Syrah cases in only in the region where our role is smaller than 1.8. And you can probably see chemo radicals flavors talk to check on this specific mechanism.
09:47:39 But here, I want to talk about something beyond comment instability, because even though this is a very powerful theory there's something which cannot explain.
09:47:53 And I will talk about one other thing here, which is again if you check these data, you will find one trend, which is, if you, if you really check the step sizes and different vertical regions, you'll probably find that the step sizes are smaller in the
09:48:19 low grade and in high grade and region, compared with a low grade and region. Right, it can be just tighten this finger as well as in the data from Tennessee, in a hybrid and region the staff sizes are small, in the local region the stats, step, step
09:48:31 sizes are really big.
09:48:33 This question. Firstly clearly this cannot be answered by gamma instability. And this is because as we introduced the team or has made an assumption of the uniform gradient in the original.
09:48:48 The theoretical framework of common stability, so it cannot be used to answer this phenomena which is related with an in homogeneity of background gradients.
09:48:58 So we definitely need to investigate investigate more on this and do as far as we know this is actually a very ubiquitous trend that that can be observed in nearly every thermal realized.
09:49:15 Observe the notion, but there are currently no explanations on this.
09:49:21 So firstly,
09:49:25 you're there any question.
09:49:29 Okay, so I will continue. Firstly, I want to like, comment, like, it is a trivial question. This might be regarded as a trivial question if you look at the definition of the background gradient, which is, which is the ratio between the between the delta
09:49:48 theta which is the temperature differences of crossing their face and the HL wishes the layering that if you just look at this formula you make an impression that if you have the low gradients, low background gradient is naturally leads to a higher step
09:50:03 sizes that right. But however this this explanation this simple explanation is not good enough, because if you check this data, data, you will find that data.
09:50:22 Data is actually pretty large in the low in hybrid and region, and pretty slow in the logarithm region, which makes this simple, simple, simple explanation doesn't work because both numerical and denominator are small, in the local region in low rating
09:50:33 in low rating region compared with hybrid religion so you cannot get a definite answer for each out. That's why we need more complex theories to describe this trend.
09:50:45 And in this work, we use two methods. The first one, we try to extend the origin of garments, the ability from the system, which has only the homogeneous background gradients, to the system with homogeneous gradients.
09:51:00 And the other method we use is that we make the parameters the modus in relation to answer the dynamics of autism in a nonlinear region.
09:51:10 Okay, so in the MENA region, as I have commented we try to extend the gamma instability into the nonlinear region. So I will directly jump to the conclusion here by showing you, firstly, this is the original stability equation in the work of radical in
09:51:28 2003, and this is the stability equation which is derived by considering the variations of background gradients. These terms are just the new terms that is related with the operations of the background gradients.
09:51:45 We showed that we will, I will just try to be simplified here that these restrooms are actually not important, in the, in the landscape that we're hearing about and they and they are are very, some of the terms are just vanishes in in those equations.
09:52:04 So the conclusion is the modes in the garments stability remains the same after we introduced in homogeneity so that this means that the original Dharma is stability theory can be kept.
09:52:18 Even if we're introducing homogeneity and this will just keep this conclusion in mind and now I'm going to talk about simulation.
09:52:29 So, because gamma instability is do a linear instability so we will not be able to understand what happens after the system goes into the nonlinear region.
09:52:40 That's why we need simulations. And this is the governing question for our main view simulation.
09:52:51 The both temperature and acidity can be regarded as one the profile that evolves in time. And this as data and asked are based on the fluctuating laws obtained from our recent work, which is also a DNS work as presented by as.
09:53:04 Let's go.
09:53:06 This new term here is the harbor diffusion term, we introduce it to balance the ultra valid catastrophe will not go specifically into this detail but for those of you who are interested, please refer to Tim radicals previous talk.
09:53:21 And this gap. This are not term represented restoring force to balance the vertical diffusion.
09:53:27 Because, now our system is now it's a homogeneous system, if we, if we don't introduce this term we will expect that this is more now stay in the equilibrium state, so that we need to add a restoring force to balance the average diffusion so that assistant
09:53:44 can stay in the equilibrium state. This is just the artificial term we're introducing our people in our work.
09:53:52 OK, so the simulation results, as, as you can see in this finger, we initialize our profile. Okay so firstly, this is the profile of temperature as a function of that.
09:54:05 Initially, it is chosen as the hyperbolic tangent shape. So that's the profile has a high gradient at the center, and low grade and add the boundaries, by assuming it in, you will be able to see that layers.
09:54:21 Initially formed near the boundaries at low grade invasion and and slowly, extend to the central regions. At this time of the simulation, you can see that the, the, the layering structure appears in all of the vertical domain, and they have approximately
09:54:43 the same step sizes, and this is because, as we discovered in our theoretical arguments. The, the, we have the same gamma mode, as in homogeneous case so that is deciding that initially forms that it should be everywhere the same.
09:55:01 And in this bottom column I want to show it bring your attention to this layer merging events where if you if you look at this bad boxes, you can see a one specific layer merging events and this is actually referred to as a bee merger in the literature.
09:55:19 And this merging events, actually happen much faster in the local region region, than Hagrid, which, if you check this time you will find that in local region region there.
09:55:48 remains in his original step sizes, and this is also the case in later times in which you can find that step sizes are always much higher because in the lower region, because it's much, much more efficiently than a staircase in hybridization.
09:55:55 And finally, on to the end of the domain, we can see that the staircase is stop merging and stays at the at the stage where we have the higher step sizes and the local integration, compared with hybrid and reason.
09:56:10 I want to also bring you to another point of view, regarding our simulation which is applause the density ratio, our role as the functional of that. And some versus some brief illustration to how to read this figure.
09:56:27 We're CRO smaller than one. It represents that the system is in unstable state, so that it represents the mix layer, and when you see that, see the pic.
09:56:36 It's represents the position of the interface.
09:56:39 So in this way this finger again captures is the evolution trend of our simulation, like, the. You can see that the staircase and firstly form in the local region region.
09:56:56 And again, from this finger, you can see that in their emerging events firstly happened in the lower region region where the step size is already very big, but the step size is as the hybrid and region remain small, and this continues to the end of simulation.
09:57:14 What I really want to bring your attention to of this. Our role clause is that the are row if you look carefully at those figure, our role as center domain, are always higher than, than the boundaries.
09:57:28 In other words, our role as a hybrid and regions always higher than the intended lower than region. And this is also the case after the layers form, as you can see in these pictures.
09:57:40 Firstly, a bit, so out of just common that this is going to be a very important differences between the staircases forming northern region that hybrid the region.
09:57:52 But firstly why this is the case,
09:57:58 this, this can, this can be actually explained quite easily.
09:58:02 Because this is actually a consequence of the flux balance. And if you look at this, this formula for flux is the multiplication of the diet pickle juice 50 times the background BlueJeans in the low grade integration we have the low by region by going
09:58:19 great and if you want to reach affects flux balance with hybrid and vision. What do you can only do is adjust the type of diversity to a high value. Right.
09:58:28 And because the diagonal diversity as a function of our roles that decreasing function for higher ku must you, this will lead to a lower our role in your simulation.
09:58:44 In other words in lower the lower region region, you must have the lower RL, so that you will you can achieve flux balance.
09:58:54 And in a laser simulation, you can find that the initially higher our role before the formation of layers will lead to a higher test.
09:59:09 Higher interface fluctuate interface density ratio, after the formation of layers. So by our I we're referring to the interface density ratio.
09:59:15 Ok so now we understand that ri is always smaller in the lower than region. We can now start to analyze layer merging the stability of the system.
09:59:26 For layer merging stability it matters how, how, like desire the system, the two layers want to merge together, and we will just directly show our conclusion here without showing you the formulas.
09:59:44 So the growth rate for the emerging stability is only a function of two important variable here, which is one this is the layer depth, the step sizes, and this is the interface density ratio plotting here, as the world roles rates as a function of our
10:00:05 slotting. When the step sizes are constant. And you can see that if you have higher interface dense ratio, you will always have lower layer emergency stability, which means that if you have higher ROI, the system will will not want to merge.
10:00:25 And this explains why, in our their emerging stage in the lower than region, because in lower than vision, our eyes always smaller you from this finger you can expect that the lambda is higher, which means that system will want to merge more well versed
10:00:45 more quickly than the high hybrid and region.
10:00:51 And this can be also seen in this contour plots of lambda, as the function of HLRRI, you can see that the, the instability has the high value, we have lower RNHO but have a very small value, we have the high rri and high HDL.
10:01:11 This several plot this several thoughts here, several points here shows the position of the equilibrium interfaces that is captured in the 1010 time equals 100,000 in our simulation.
10:01:26 As you can see, these are the position of these interfaces.
10:01:30 What do you guys see that the equilibrium stage of our system is achieved. When they reached the when the lambda is around two times 10 to the minus five.
10:01:40 And this is actually a consequence that we introduced that restoring force in our simulation. And essentially, the system is now it's in equilibrium states, it's actually in the closet equilibrium states.
10:01:53 It's just like, for the growth rate that is that is smaller than our restoring force, it's impossible to grow in our system.
10:02:02 This is actually the one of the artificial properties in our system that we want to stress here, but regardless of this, of this question, you can see that as the controller of constant lambda, we have small ri you always lead to a higher HL which explained
10:02:24 why in the equilibrium stage.
10:02:28 The northern region, we will have always have the higher step sizes.
10:02:34 So now we can start to answer your original question like why low grade and region has higher step sizes. The short answer is just because of layers in the lower than region Merced, more sufficiency for that for the long answer as we have demonstrated
10:03:01 there are three steps. The Firstly, the flux balance relationship confines that interface density which are I to higher in the lower than region. And then this higher ROI and lower their region will lead to a stronger layer merging stability as we have
10:03:17 shown in the last slide. And finally, the stronger layer emerging stability ball leads to a more 40 more staircases, which tells you why the local innovation, always have higher step sizes.
10:03:22 And so these are just our explanation on our models. Now we want to compare it with real oceanography data. And this data is actually taken from the Tennessee.
10:03:33 I'm sorry I kind of hide the title here. This is from Tennessee which the food environment is taken for over 20 years from 1973 to 1992 and the profiles for temperatures later uploaded here, you can see that these.
10:03:53 In 1973, there are around 10 steps stops in that is 73, and later in 1979, it's around six steps, and in 1992. It's merged into around three steps.
10:04:10 And you can also check from this finger that's the layers in the hybrid integration is always merge last efficiently than the layers in the logarithm region, which is consistent with our description in the simulation.
10:04:27 And because we have already understood from our simulation that this is a consequence of different interface ratio at different regions of the staircases.
10:04:39 We actually made the calculation of the ROI in these interfaces, showing his curves. And again, we find that it's in the hybrid integration reason you always have higher ROI.
10:04:54 And for for the lower than vision to have lower ROI which is consistent with our prediction.
10:04:59 So, the reason that we, and we want to only use this data to compare it with our observation, with our simulation is that our argument is really tightly related with a layer merging humans.
10:05:18 Right.
10:05:18 With the end. And this is actually the only available candidates wish in an ocean observation which shows the clearly shows the merging of the staircases in time span of over 20 years.
10:05:35 And so this is the available candidates only available kind of for us to make the makes the conversation. And this. Generally, we have this, as we have described this supports our simulation results.
10:05:49 So, conclusions and discussions, the conclusion, again, utilizing the short description, the lower than region has the higher step sizes, because the layers in the lower than rich in mercy, more sufficiency and but there are actually several deficiencies
10:06:07 in our model. The first we, as we have commented before the equilibrium stays in our model is actually a cause equilibrium stage because we artificially for be the system from evolving more along the time due to the introduction now they were storing
10:06:25 for us.
10:06:26 This directly leads to the result of the equilibrium staff sizes are uncertain because it's related with a strength of the restoring force introducing no assimilation.
10:06:38 And another caveats is the interface density ratio in normal days I was four to seven is much larger than the, the interface density ratio that is observed in the ocean.
10:06:49 And this is actually this discussed in a previous work of chemo radical in 2014. This is a consequence of the lack of pump station of turbulent mixing our model.
10:07:01 And finally, in our simulation. If you Transform, transform the time into the dimensional timescale you will find that layers merged authority, two dimensional time Scoble run two years.
10:07:14 If you compare with its with the turn is the day that you will find this find that this is a pretty short time scale. And we haven't yet been able to understand this trend.
10:07:26 Currently, so this is the end of my talks, I want to bring your attention to this open questions like can this systematic and they are emerging events captured the internets the data also be seen in other places.
10:07:41 So far, this is the only data that clearly captures are very well captures these emerging events, but it's a question of whether this will also happen the other staircases because we currently just do not have enough simulation data to tell the difference
10:07:58 between the evolving staircase, or the stable staircase.
10:08:03 So, is this evolution process ubiquitous or unique is an open question. And another point that we want to make here is that, um, should we treat the thermal a nicer case in the ocean observations, as generally cause equilibrium stage or full equilibrium
10:08:22 state. So as you can see from our simulation. The.
10:08:28 This when the layer merging instabilities are as low as 10 to the minus five. In our non dimensional timescale, it's merged so slowly. That's you may expect it to have the next merging events probably waiting for another 10 years or so, the question is,
10:08:47 can we regards to the stage at this stage is where the power system as the course equilibrium stage or a full equilibrium state, and or more fundamentally does fully equal the equilibrium stage, even exist.
10:09:04 So I guess these are just my common questions, and I'm really happy to learn your thoughts on these questions. Thank you.
10:09:17 Thank you very much. You can
10:09:21 questions.
10:09:24 Ask them.
10:09:26 Hi. Thanks to Jen, that was a really nice talk, especially like the comparison with observations, the first time I see it it's excellent.
10:09:35 I wanted to ask you. Earlier in the talk, you actually show a simulation of an evolving large scale profile into a staircase right now in order to the formation of the first layer is you can explain it using the initial government stability profile, but
10:09:53 the later layer merges you presumably have a pyramid transition for how the system evolved, after, after the layers for him because of course that's not captured by the standard GM instability model.
10:10:06 So, what was that what model use after the Leaders Forum.
10:10:11 Okay, so
10:10:13 thank you Pascal for a question. I think I want to ask answer this question from discovering questions we use a similar to our, our model.
10:10:24 As you can see this.
10:10:33 What we are doing in our simulation is just simulates the these 1d profiles and 90 debugging time.
10:10:33 And as you can see that all of these parameters are just
10:10:38 like this new and are not here are just constant, what's it's really of freedom in this model is just a function of form of this success as a function of our role right.
10:10:52 And for our model which shows to use a flex weight and laws obtained from our previous work, but it's actually can be. It's actually have the same flux relationships with one that says previously done by Chancellor in 2011.
10:11:10 And, yeah. Does that answer your question.
10:11:12 Yeah, but that one is only valid before the Leaders Forum. After the Leaders Forum your flexes will depend on other things like the layer height and the interface of flux right okay okay I can see your question.
10:11:25 So what's happening in the in this nation is like this. This relationship is valid at, at every interval.
10:11:44 Right. It's like we're, we're using a finite difference elements methods, find a different sort of finite element methods. So we take this whole domain apart into several elements, and for each elements were applying were calculating the our role as that
10:11:52 that small elements, and we're taking this this relationship into this flux relationship into those elements.
10:12:02 Even though this religions are actually obtained as the, as the as the homogeneous of England fields. It's still valid. I guess it's still valid. After, even after the formation of layers because at this stage, we can always regard the local parts of
10:12:21 these profiles, as a small system of homogeneous gradients. So that's the fluctuate and laws are still valid in this domain.
10:12:30 Does this answer your question. I mean, I agree with your statement in exactly the region you're pointing out, right, but if you lift your cursor a teeny teeny bit, you are now in a region where you fully connected right we are there.
10:12:43 I can see them. Oh yeah, this is a part I'm sorry this is the part I forgot to mention. Um, so yeah.
10:12:53 I will return to this slide. This accommodation is only valid when our role is bigger than one, which, which is the in the south of England region, and what our role is smaller than one this is actually the conventional region.
10:13:04 And once we use here is that we use a permutation scheme, which is the same as the RKPP permission scheme, in which basically if you find that the density ratio.
10:13:17 Sorry if you find the boys ratio and square is smaller than were smaller than zero. This means that compaction will occur, and we will start the diagonal diversity for both heat and assault to a large value to represent some mixing.
10:13:34 Sorry to represent the strong comebacks in this region.
10:13:38 Okay. All right.
10:13:39 Thank you.
10:13:43 Alexis, please.
10:13:45 So, you tend this was a fantastic talk thanks for that. I've got some, I guess sort of thoughts or questions about comparing two observations because I think there's.
10:13:57 I think the fact that your theories really looking for equilibrium states is going to limit how, how many places in the ocean you're actually going to be able to find good observations because there's, there's so much sort of inherent time dependence
10:14:09 and even in in your theory with these merging states. I mean that's kind of a time dependent process until you get to these late stages.
10:14:17 I'm curious about whether your theory, I might be missing something about the sort of double diffuse of nature of of what goes into this.
10:14:24 You've talked about salt fingering states but can your theory also be applied to sort of diffuse of convective states.
10:14:31 Okay, so this is actually a very good question. I will have to return to the original theory of governments the blade theory. The reason that I, the gamma is the political theory works.
10:14:48 Okay. The fact is that the government's to play theory works pretty well in assaulting religion, but it cannot answer your questions in that abusive abusive convection regimes in the oceanography case.
10:14:59 And this happens with.
10:15:03 This is because the fundamental assumptions in the gamma instabilities is that the, the flood fluctuating And last is valid. And this is based on the verify if you go to the very fundamental level this, this is the assumption that the system is subject
10:15:21 to the linear stability of the soft fingering instability. Right. But if you go to the other side of story which is a digital connection region, you will always find that the our role in the observations observational staircases is always rent from 22227.
10:15:38 to seven. But the linear stability is only valid from a from one to 1.16, which is a very tiny window, and this is perhaps the inconsistency with the fundamental defensive connection linear instability and observe the thermal Hey last year cases.
10:15:59 And that's why the whole government instability theoretical framework, cannot be directly migrated from the south England side to diffuse the conduction side okay yeah so that's why because, yeah.
10:16:10 Yeah, so, I mean your your other best bet for like equilibrium staircases and observations would have been the Arctic that that that piece of convective regime.
10:16:19 I think the fact that you're sort of looking for these really stable things and even timing and see it's kind of special in that regard. I think it's going to be hard to find.
10:16:29 Like these kind of really long term. Repeat observations that actually shows staircases that have sort of survived that long, it'd be very cool if if I, if you can find more, but I do think it's going to be tricky to look for equilibrium states just given
10:16:45 the sort of time very nature of all your forces and you're sort of seasonal cycles and all fat.
10:16:50 Yeah, that's right.
10:16:54 Thank you.
10:16:59 I'm pleased.
10:17:01 You turn.
10:17:09 That was a very interesting talk.
10:17:05 Thank you.
10:17:06 So I have a question about your parameter, or not.
10:17:11 So that, you know, that's an inverse timescale. And so, that introduces an additional timescale into the problem. Yeah, I was wondering how you pick are not because you know it could be, you know, timescale that fast compared to the growth rate of the
10:17:28 gamma instability could be slow compared to the growth rate of the ability comparable to it, you know, etc.
10:17:36 Yeah, yeah. And this is a very good question, plus some, I will want to like actually show our this description will related with this parts on.
10:17:49 Yeah, basically, this is the available.
10:18:03 Our knowledge, we, we, we find in in our simulation, and the lower pounds is 10 to the minus six is an.
10:18:06 Is that ok Firstly, I want to show that if you have the equation, which is which is here.
10:18:14 If you have a strong or not. It really means that you have a strong restoring force, and each one, quickly, brings the system back to his original stage without letting the further evolution, right, but you don't want this this this effect to be too strong.
10:18:31 Because if if it's too strong, it's will surprise the.
10:18:37 Even the growth of the gamma is the bridge, because you basically you you don't allow the system to evolve Right, right. And so, you will you will if you if it's too strong, you will you will see no instability emerge from system, but it cannot be too
10:18:50 weak if it's too weak, you will find that the system will will evolve, largely the initial hyperbolic tangent profile will soon guys into a complete meets the layer.
10:19:00 Right, it's going to be just a straight line across the domain. So that's why you do need to find a balance between this profile and this is a value that we find new suitable for the are not.
10:19:14 And eventually, but she was chosen value within this range.
10:19:31 Okay, okay, I have a follow up question so you use the same, our note for both the temperature and the solute equation. Yes.
10:19:30 Why is that, um, yeah this is because this is a good question. This is because one way we can understand this or not, is that, um, if you actually think about our hyperbolic tangent shape profile as the Thermopylae, you will actually find that the main
10:19:46 thermo client in the real ocean is actually balanced by the uploading and downloading of the,
10:19:55 the, like the uploading of the ocean, OCEAN water from the, from the bottom and the downloading of the acronym pumping from the surface and this effect is actually the same on both sides and keys because this is a vaccine process right it doesn't tell
10:20:13 the difference between salt and heat, and that's why when we are applying this we're storing parameter or not.
10:20:20 It has to be the same for temperature and humidity, or otherwise it's it loses original fix that we try to resemble here.
10:20:29 Okay, thank you.
10:20:31 Thank you.
10:20:33 Okay, let's do notes of most of the questions I'm reluctant to cut it off, but as well. So, so I want, let's carry on for a bit. Francesca, please.
10:20:46 Thank you.
10:20:48 Very nice talk. So, but I have just a curiosity about what may be the role of the boundary condition if I understand correctly, you have this realization to this prescribe the profiles, the profiles right.
10:21:03 but then top and bottom I end up being really isn't it.
10:21:05 Oh, ok so this is the part I forgot to mention we didn't apply the periodic boundary condition we apply the basically no flex boundary condition I don't know flux the top and bottom.
10:21:15 Okay, yeah. And so, it's just I'm just wondering if you want to apply that to the ocean. What if you apply say no flux to the bottom and some fixed flux or some prescribed temperatures in the top, would you get without maybe without even renouncing to
10:21:32 the relaxation what would happen would
10:21:38 modern general, what is the role of the boundary conditions, you know, in the long term evolve evolution of this case.
10:21:49 I think the boundary condition, in fact is not so important in this part, which is because, as you have commented we have introduced this restoring force and this is the most important factor that tries to keep the balance of keeps the shape of the original
10:22:06 profile. Right. So, and this works.
10:22:15 As every water column across the vertical domain so that it's the elements as the center domain will not be able to feel the.
10:22:23 Feel the influence of the boundaries.
10:22:25 And it will feel more strongly of this restoring force and that's why the actual boundary conditions might not be that important in this particular situation that the central question.
10:22:38 Yeah.
10:22:39 Just wondering, in the real ocean that are no restoring okay yes but there are yeah right in the real ocean, of course, on it some. The no flex boundary condition will not be a proper because, you know, the yard.
10:22:55 a very well. There is a mix there near the surface and, and the more questions, environments in the interior ocean but this, this complex fitted cannot be were to capture that in this simple Wendy model and that's why we have to just introduce this artificially
10:23:16 introduced our knowledge to balance the whole thing. But, of course, yeah, this is, this can be viewed as the limits of our Wendy simulation because it's cannot capture more abundance of the nominees into the model and because we only try to understand
10:23:32 one simple aspect here that is why the layers, step sizes are higher in the lower than region.
10:23:41 I think that you can resolve a lot of the other physical process might be justified might be just properly nor singles.
10:23:52 Thank you.
10:23:53 Thank you.
10:23:55 Okay, planet to next.
10:23:58 Okay, so you have to, shall we say, we might say in in Latin or English de sx mocking up parameters sort of the mew and br not how so how does the.
10:24:13 How strong is the dependence of the result on those two parameters.
10:24:21 Oh yeah, this is a very a very good question for for the new. This is a hybrid diffusion term. Yeah, um, as we have actually we had this discussion after a skills talk, that's, you know, the gamma instability theory, it's actually suffers from all right.
10:24:40 It's staffers from also valid catastrophe so if you do not have this term introducing the model, what you will have what you will see is that the, the, the growth which will the gamma mode will grows, much higher in the, in the, in the high vertical webinars,
10:24:59 which means that you will never have enough resolution to simulate the domain, right, because you have this also valid catastrophe. That's why we have to introduce them.
10:25:15 Things like the step size and all How do they vary do they scale with you. Yeah, this guy with this new nose.
10:25:22 But scaling. This mute if you, if you look at a unit on mute is actually meet her to the Foursquare, over, over the, over to overstep over time, media, Sorry, it's lost to the Foursquare over time.
10:25:38 So, and the combination of me. And another important parameter which is the diffusion diversity kata, and the Kaaba has a unit of meters per second right combination of these two parameters will give you a non scale and along skill sets sets actually
10:25:54 assess the scale for to initially formed and there's, that's what I would have thought okay so mu is essential to defining the scale which brings brings the question, What do you think the physics of new.
10:26:09 I know rad co mentioned that but i mean what I'm curious, kind of a good physical intuition as welcome, you know what, what's the new, how would you go about going going past the level of a miracle parameter.
10:26:27 Yeah, yeah, this is actually a very good question, I think.
10:26:31 So in the soul fingering part there are actually a lot of critical. Non, non, non scale. The most important sky, I think, is the sky, that's the salinity space and this is actually characterized by the non dimensional parameter which is called, which
10:26:50 is usually referred to D in the literature. And this actually.
10:26:56 I can I'll show that definition directly on my screen but on this basically tells you.
10:27:04 It gives you a scowl on the, on the fingering lead and is tightly related with the scale that's the salinity this patient is the center is dissipated.
10:27:21 And that's the that's fingering was is actually as calm as the school has commented, it's from our director numerical simulation it's always find that this initially form the layers is always four to five times larger than the fingering scale.
10:27:39 And why this is the case is, perhaps, from my understanding is because, for this flux grid in last be valid. You have to have the average to be taken over, largely enough domain you don't want to a domain to contain only one or two fingers.
10:28:02 And so this can be the smallest guy. That's this average is valid is going to tell you when initially formed step sizes is at. So, and from the directing America simulation, we can see that these average is actually valid when the average domain contents,
10:28:25 at least four or five.
10:28:33 The four or five times the fingering words, and that's why the initially formed layers is already at that time so long ago. And this will be my answer.
10:28:42 I'm not pretty sure whether it's good enough but yeah this is my time.
10:28:47 All right, thank you.
10:28:50 Okay.
10:28:50 We'll try and get through these questions, Alan. Please understand why you introduced the restoring force for the specific purpose of the simulation.
10:29:00 But it's possible that if you remove that restoring force, you could use the method to address a different question, you could apply jump periodic boundary conditions, so that you have a notionally homogeneous system.
10:29:17 And then investigate the SM tonic similarity solutions, either numerically or even possibly analytically by introducing some a test functions with adjustable parameters, and then see what the transients, but ultimately self similar evolution of a generic
10:29:41 staircase system looks like.
10:29:46 Yeah. Um, yeah I think this is a miracle proposal. And yeah, this is this is indeed can be done but I do want to bring up one comment on this, which is that in our system.
10:30:00 If you.
10:30:01 I want to plot this non dimensional number of the, of the vertical position but if you transfer it to the, to the actual dimensional number you will find that this is actually around 200 meter.
10:30:15 So, what we're really trying to capture in this model is that we are trying to use this profile to resemble the actual shape of the main thermal client.
10:30:26 And if you, I don't know, I mean if you treat this as the shape of the mandible client. It might be weird if you add another sermon on top of one Thermopylae because, by doing the purity boundary condition.
10:30:41 You are basically you want to say that this this is only part of a much larger system, but if you, if it's already as large as thermal clan I guess it's might be some kind of weird.
10:30:52 Yeah, but I think yeah it does help if you have the pure organic condition to see what's really going on well okay okay wasn't clear enough. I didn't mean you just put the jump periodic periodic boundary condition, but to make each the initial salinity
10:31:09 and temperature profiles strictly linear, rather than this. So, okay, then it's fully homogeneous. And then you're not going to get any systematic very young, Zm simply as a point of principle not to compare to some observation, just to find out what
10:31:32 is the similarity solution that long times because, because once you take away the you know the parameter are not which, which, by EOD introduces the an emergent scale, but you know kind of enough, you know, just
10:31:51 right, it is able to use your mouth to say something about, you know, what is sort of the free solution absent boundary and other effects. Yeah, that's true.
10:32:02 Actually I think there was actually the similar kind of simulation done in radical in 2005.
10:32:09 This is actually the first paper he bring up the idea of the emerging events, in which he actually made a very similar.
10:32:18 Simulation as I showed here, except for, as, as, except for he used up your article boundary condition and linear linear gradient, you can probably check these three viewers simulation.
10:32:33 But I think Yeah, yeah, this is generally on a very good question. That's, this is indeed, giving, giving us self similarities of the some of the simulation, instead of adding this are not turn.
10:32:47 Okay, good.
10:32:49 Thank you. Okay. Yeah.
10:32:52 I'll make it quick.
10:32:54 let me know.
10:32:56 Let me know if I'm incorrect but I'm under the impression that the restoring force that you put in the system is a key to to obtaining your results, meaning that what you do with the foxes is very important to get your step size, as you want.
10:33:13 So, connecting to the first week of the program. We had the the talk of Mary Liza, two moments, and I don't know how generic that is but she was talking about all those intrusions, and I was wondering whether stuff from the law or stuff coming laterally
10:33:35 during those intrusions, how would that mess up with your process and how important that is to actually getting your rights there usually with depth of this, of this tech Titans.
10:33:49 Yeah, thank you. I think this is actually a pretty
10:33:55 complex question, because like so. So I will, I want to start with the first point that the married with the talk is already on the diffuse the connection between mobile story.
10:34:11 And it's actually, as I have a sad previous previously is not really the same story as the, as assaulting ringside because the gamma instability doesn't work there.
10:34:22 But for the second parts. The is that that the acts of those intrusion can be very important in the, in the formation of staircases as as presented by by Mary Louise and he and her students, but for this part, we, we cannot yet be taking this into consideration,
10:34:50 primarily because that this is only a one the model. And we, you know, this describe the yd mode is generated by making the horizontal average of the, of the essentially a 3d domain, and what ever the complex physics physics that happening that's where
10:35:11 the world.
10:35:11 It has to be parameters to introduce this model, right, but four dots for the inclusion it's, we cannot yet find a good way to prioritize it, even in the global model so I guess this is the part that we cannot capture in our when the model.
10:35:34 But the inclusion I agree it's probably totally 3d but it, even in one dish. You have this reversal of the slope. And that's the intrusion so that might well how that might mess up the flux is actually probably boxes from below.
10:35:52 Coming.
10:35:53 And that will probably change quite dramatically your your your role, equilibrium. So, Yeah. I didn't want to.
10:36:03 Yeah, I guess, if we have a good enough understanding on the, on the conclusion we can use the promise tradition for its.
10:36:15 We can do that. Good enough commercialization and we can then take the promise station for other models like maybe like this kind of model which has one deep mean few simulation, or even to the large scale model.
10:36:29 Yeah.
10:36:30 Okay. Final question, Brian.
10:36:33 So I think that the salt finger steps there are a curious epi phenomena, but it seems that nobody is mentioning in the data being shown here prompts me to to want to, to connect to reality is is there a simple answer to the question, What is the effective
10:36:57 nuisance number. That is, if you didn't have these steps, you would have a certain flux given here. Yeah, right. I'm asking, what, what is the amplification, what is the amplification factor in flux that you obtained from the presence of the steps.
10:37:17 Yeah, this is a very good question. Um, so, if we, if, if the system has the only the linear gradient. And in a linear gradient we can assume that there is a homogeneous all figuring for us, which means that there are a lot of fingers generated and gas
10:37:34 these papers.
10:37:49 At ASCO the nozzle number is really a function of our role for our role, which is integration for our role equals to the wish for Rox, it goes to the nosy neighbor Thurman also number is around 50.
10:37:54 To give you a dimensional idea.
10:37:58 And if you form a staircase. What's happening is that the nozzle number is still 50 as the interface, but the difference is that instead of having a linear gradient, which is the same everywhere.
10:38:13 You have a strong gradient at the interface and multiply this stronger gradient by this northern amber will give you a much larger fluxes compared with the linear gradient.
10:38:27 I didn't it's a question, if you want to, like, know the actual Fluxus as the interface you will have to know another important parameter which is the, which is, which is that the depth of the consciously, the depth of the interface is actually you already
10:38:47 have very small value but it's gonna be very important.
10:38:51 If you want to understand the success here.
10:38:55 Okay, so you don't can't calculate them, but they're going to be greater necessarily is that correct, Yes, smooth profiles a lower bound.
10:39:05 Yes.
10:39:06 Thank you.
10:39:07 Thank you.
10:39:08 Okay.
10:39:10 Well, thank you again you chin and thank you everyone for participating that was very interesting discussion, I'm sorry it was long but I'm reluctant to cut discussion with the Thank you very much.
10:39:23 Let's move on to Francesco please.
10:39:29 Hello, good evening.
10:39:33 And I should thank David, for inviting me to
10:39:40 this talk tonight.
10:39:41 I got from him, and he made a
10:39:46 couple of weeks ago, that say something like, July to discuss four feet in 15 minutes what would what are your ideas about the gamma effect.
10:40:00 And I'm honored. On one side, on the other side.
10:40:10 I felt that my designated role was that of rain when somebody says parade, so my apologies for that.
10:40:18 Because, asking me to talk about the gamma effect is about asking a Muslim, to talk about Christianity or something like that.
10:40:30 So, before anything, let me say, however, that whatever else I have to propose that is nowhere near absolutely nowhere near to a complete theory. And in many ways it doesn't address, most of the questions that I've been merged tonight in the discussion.
10:40:52 So, first of all, just to develop some, some, by the way, do you see my screen.
10:41:02 Yes.
10:41:03 Yes. Okay. Okay. So just to develop some intuition and try to bring you all into that, what is my mindset on this issue.
10:41:14 Let me show you some movies from two dimensional simulation tonight I will show you two dimensional simulations.
10:41:23 either non forms that cases for Mr cases.
10:41:27 This would be the first one will be simulations that are not formed inside cases, they have selected any number of 10 to the 13, our users are really number because of my domain, as the top and bottom rigid boundary layers with prescribe temperature and
10:41:46 in saline, and therefore I have a nominal density ratio, which is determined by my prescribed interested in selling top and bottom, in this case is 1.2, and then the diffuser durations one third then they have a plant or.
10:42:01 So, except for the top and bottom battery conditions set the parameters are very similar to the simulation this scene from the sky, before, but those were 3d and desert.
10:42:11 Okay, so this is buoyancy. And normally, that means, you take the horizontal average of buoyancy at every height.
10:42:19 And you, you take buoyancy minus disorders, on average, so what you see is going to see minus, there isn't an average at the height of Twitter you see the thing.
10:42:31 And you clearly see this Wednesday cutting structure, what we would call the sole fingers.
10:42:37 Except that in this regime they don't really look like fingers anymore. Sometimes they look like kind of strange mushrooms, but.
10:42:47 And here you see the whole domain.
10:42:51 In fact, sorry you don't see the whole domain that the actual simulation is actually much larger, because that you don't see a lot the boundaries.
10:43:07 The domain is, is something like 5000 grid points told, and 4000 bullet points wide. With periodic boundary conditions of the side and fixed and fixed by the conditioner, talking about.
10:43:14 Okay, so this is what you get. Once the simulation is essentially an equilibrium.
10:43:19 But now I will show you exactly the same field in exactly the same portion of the domain, except that what I've done is I first completed an Instagram, of the discovery as you normally.
10:43:32 And then I looked at what is the value of the 95th percentile of this histogram. And I only draw the contours of buoyancy anomaly at these 25th percentile, but positive and negative.
10:43:49 So, the red controls are the 95th percentile.
10:43:56 Positive going up, and the blue contours are negative, and going down. Of course this is boosting so it stops top bottom Sumit.
10:44:06 So my first point is that the extreme of buoyancy here are carried by structures that login and.
10:44:18 So we, the fingers here are not something that looks that looks like some sort of stationary convection cell that just switch this around that and does it say is thing.
10:44:29 But it's some relatively long leave the login engine structures that start get into existence.
10:44:39 At some time.
10:44:41 Keep moving. roughly always in the same direction until something destroys them.
10:44:47 And this something, typically, not always, typically is the interaction with some other structure of the same nature, typically another one going in the opposite sets, it just smash on to each other.
10:45:02 And something else.
10:45:06 So this means that.
10:45:09 Okay, this is the movie. Let me start the movie again.
10:45:13 Okay, so now here we have two different lengths fails, one is the horizontal leads the lateral scale of this structures, which essentially is the same length paler as that of the linear instability.
10:45:33 And this is because this structures in order to keep moving. Remember that this structure and moving in a, in a boy and security. So, here you don't see it because these are the anomalies, but these are moving in a boy and security so to keep moving.
10:45:49 They need to exchange the temperature in salinity for in fact they need to exchange temperature and not so much sunny day in order to keep having, and keep building to this level defusing mechanism, a bonus, you know.
10:46:07 Therefore, otherwise they would almost immediately reach an equilibrium hard to find their agreement might and maybe isolate with them by sada, a solution a little bit and then disappear.
10:46:19 So, in order to create this buoyancy anomaly. This game of differential the facilities has to be in place, adjust as in the linear instability, and therefore is not surprising that by and large the ladder of scale of these things is roughly, that of the
10:46:41 linear instability.
10:46:43 On the other hand, the vertical scale.
10:46:46 Well, that's a bit of a mystery in the sense that if you for example if you look at Snapchat the snapshot, like the image that is on the screen right now, is not moving.
10:46:57 You may be mistaken. In considering the vertical the relevant vertical scale, the scale of these blobs.
10:47:05 Our that is not solved because this this really are Lagrangian structure, they move, they go down.
10:47:12 And it's an open problem I would say to characterize, and quantify what is say that the typical path length, or in what determines the typical path length of the steps.
10:47:27 And I suspect that in the future, that may be some surprises in looking at this sort of quantities. So this is something to keep in mind,
10:47:36 another observation that may be worth doing is that even though this is a nonstick is for me regime.
10:47:45 This blobs are severely deformed. These are not rounded around these things so here we are, where's the anomalies that as they move. They typically tend to get in particular when they're not interacting with other similar structure tend to get this machine
10:48:03 like
10:48:05 hot structure.
10:48:07 So this means. Okay, so one exercise that one can do is to try to do these simulations for many different three ready numbers, or if you prefer, different this situation so so as to make the connection more and more active and more and more vigorous and
10:48:29 then measure the rhinos number of these structures so you have a characteristic sighs I got a taste of velocity, and therefore you immediately considering us number you discovered that as you go up the vein, that the rhinos numbers, also go up but and
10:48:57 to a certain point that it's a bold one so you cannot say anymore that these things are essentially still occasion
10:48:56 objects, they are actually things that, for example may have wait.
10:49:06 Alright so
10:49:09 next step. Now, what we do in this simulation, I keep the same salinity really number but I dropped the, the nominal density ratio.
10:49:20 And this is actually a simulation that forms staircases.
10:49:28 And this is the point of scifi all the boys are not just the anomaly. In fact, you see that diversity but gradient, the overall vertical great.
10:49:36 And if I stop this simulation, you might, I don't know.
10:49:42 I well this is going to zoom, but I'm not able to zoom to actually magnify This is image, right now, but you might guess that in this image you can see in the buoyancy field this tiny structures this motion like like an angel structures that move up and
10:50:05 down. And of course the timescale of the movie is different here, here is much faster.
10:50:10 Just because of the choice of the frame rate, and now you start seeing the formation of staircases, you can kind of guess from the shade of the colors that players are formed
10:50:22 yet. So, there is a layer here where there is my cursor, and then another layer here and something may be happening here with a strong interface here and the strong interface here that are banned by internal internal gravity waves essentially.
10:50:49 So, even at this stage, in particular here in the in the interface or region so what we will call the finger songs. You can see this tiny structures that probably appear to you just as little dots, either dark or light or dark of then then there's running
10:51:06 for the lighter than serving food.
10:51:11 And so this is like an unjust structure, then are generating these inefficient regions and they should Clostridium into the, into plumes and move around the
10:51:24 and create vortices in this in this well mix regions.
10:51:31 However, if you just look at buoyancy is not immediately clear.
10:51:38 At least it's just me very clear to me, what is going on.
10:51:41 So let's look at another field is exactly the same simulation. But now we're looking at selling Ed and again this is not selling the anomaly. This is sullied.
10:51:51 And probably the very first thing that it's apparent is that he and his flame like movie.
10:51:59 The flames let me call them this way, seem to have a scale which is larger than those little buoyancy dots.
10:52:08 So now it seems that something has happened to the horizontal skates.
10:52:16 And, of course, because it's the same situation after why you would see now that the layers for me.
10:52:28 Yeah, it should be quite clear where there are the layers.
10:52:33 And in fact, if anything, at this stage here for example in the interface, the scalar salinity seems to be to have gone down again.
10:52:43 Here for example.
10:52:51 And now finally, but this, if anything, looking at the end, at least to me as confused the ideas, even more.
10:52:59 And now let's look at kinetic energy.
10:53:02 Okay, this is an initial state, in fact it's even too early to late sorry.
10:53:08 Okay, let's stop here.
10:53:10 So here, we kind of see this little mushroom structures that move at this like random structure of the move up and down, and they of course because they are a velocity they have a kinetic energy and therefore this is what you see the structures in the
10:53:24 kinetic energy, have a letter of size which is compatible with what you see in the Boise field.
10:53:32 And it's compatible to that of the buoyancy cannon structures, those little like Rams with moving things that I showed you before, but now as we go on.
10:53:44 This thing is some way so for organized and you start seeing things like this little swim.
10:53:51 These other little sphere.
10:53:52 So there are overtones. There is some little overturning circulation here and there.
10:54:00 That is becoming quite a part.
10:54:05 And as the simulation goes on. These are returning to condition that is here and there, it's a bit intermittent.
10:54:12 Now, for example, it's really clear that in fact at this stage.
10:54:18 In this level there is already essentially mix layer so we have these are almost already something like devoted that conviction cells have really been a connection some something vaguely like that.
10:54:36 And now, you see, you start seeing in the layer below the permission of the layer below as well.
10:54:43 So I would say that kinetic energy was a bit more revealing than than the previous two things.
10:54:52 But the most important thing in looking at the kinetic energy, probably to observe is that problem is this fact. Initially, as long as we don't see those overturning circulations
10:55:08 D kinetic energy is relatively low.
10:55:12 But then, when. Whatever happens, that makes those overturns, then it really kicks and, and the kinetic energy becomes strong. Not only that, but okay let me, let me.
10:55:30 Not only that but when you finally have these vortex like structures in the kinetic energy field that it's undeniable that there has to be some sort of mechanical mixing going on in this fluid.
10:55:48 So, when we get to the stage that we see do this, this overturns like here for example.
10:55:56 Now, this stuff is not just double diffusion of course the double diffuse the exchanges are the local scalar still going on.
10:56:06 But this thing is mixing mixing a mechanical way just as if it were to school.
10:56:11 And this is one of my main points.
10:56:16 Alright so let's have a look now.
10:56:20 A closer look at, remember that in the buoyancy. we didn't see large scale structures, we only kept seeing this tiny blogs or fingers are called them mushrooms call them as you wish.
10:56:34 So this is these are snapshots for a tiny region of that simulation, not just before the staircase appear.
10:56:42 And I picked this example but there are many other structures that are just like this.
10:56:48 So, in buoyancy or in fact in buoyancy and normally you see this mushroom like structure disorder, these are these these in these and so on and in fact what it turns out is that this blue stuff this blue created that went down as it went down, it dragged
10:57:11 some fluid down with itself.
10:57:15 And triggered into existence, these other fingers.
10:57:20 Remember that under these conditions when the overall certification is as it is in this situation, if you just push down a portion of fluid of the right size, then immediately you trigger a web fusing stability.
10:57:39 So you're creating a boy and cinema, and of course the same thing in the opposite of course when you go up, because it's the usual cartoonish explanation if you are if you wish.
10:57:51 But, which is very robust actually from a physical point of view. If you bring a portion of fluid down and then that portion of food find itself, you know what he called the region, and therefore he loses he doesn't lose as much salinity, if he does the
10:58:06 right size, and therefore, here you go, you have created some wins, you know.
10:58:11 Now, if you look at the Selenium normally you already see something, it doesn't quite have the same, the same lateral size, so all these structures here are sort of blurred into a single solidity stuff patch in the salinity picture, and in temperature
10:58:35 of course is even more blurred.
10:58:37 And if you look at vertical velocity we're not surprising that the random number here is that the momentum is the most definitive thing in this simulation.
10:58:46 So the vertical velocity is it all the structure all this let me call it cluster. All this cluster is collectively going that.
10:58:56 And, and this is associated with a kinetic energy that is now building up and becoming sizes.
10:59:04 So, This clusters, in my view, the perfect candidates to play the role of the road, in the ballsy mechanism that has been discussed at length that in this in this week's.
10:59:25 Okay, So now.
10:59:28 Well I should have.
10:59:31 Right now, I haven't thought about the government fit at all. So let's, let's discuss about the gamma effect.
10:59:39 So, if you start having a view of this system, which essentially is one dimensional so only the vertical matters and the kind of average over, over the horizontal.
10:59:56 Then you can, which is a view that I actually like because it simplifies things very much and they think its overall correct.
11:00:07 But, what you have is that you can find you can write the two conservation laws for temperature in Silicon Valley essentially in this way. So you have a flux of temperature of flux on Selena silly the and I took this F on top of this Plexus just to recall
11:00:24 that these are the flux is due to the motion of this little like range of structures.
11:00:31 So those are what I would call the finger flexes.
11:00:35 And this is, we learned from a scholar that the starting point of the team or Let's close gamma effective.
11:00:44 Oh well, let me add an ingredient. I hope that my movies have convinced you that at a certain point when you are just going to form the state cases but you're not there yet.
11:00:58 Mechanical mixing starts to be important.
11:01:02 And therefore, let me just modify these conservation lows by adding and editing music flux. So, I still have my finger flux which is there, it is not going away, it will be there all the time, even though I have no clue of what that might be.
11:01:22 I mean I have no more than 40 slides but it is there.
11:01:26 And then, let's add some key, some edit if you see video multiply in the temperature question by the temperature gradient, and the same edit if you see the multiply by the sudden integrated for the salinity.
11:01:41 So collectively, all together, all this stuff is now the actual temperature flux and the saline flux. So now that 10% said it flexes are just this some of the finger flexes, and the differences.
11:02:01 Alright so what is buoyancy buoyancy is just a linear combination of temperatures serenity and the coefficient of this linear combination is the density ratio, and gamma.
11:02:13 Now, it finally appears, is defined as the ratio of the finger flux is a temperature over the finger Fluxus of selenium, and, of course, this gamma may change we, we have no reason our priority to say that this is not going to change.
11:02:32 Okay, so now let's try, let's combine these things, and we just plug these two equations into depression for buoyancy. Let's write some sort of law for the buoyancy forces for the rate of change of points, which is course cannot be closed, but with a
11:03:14 bit of manipulation I can still manage to only make appear only the finger flex is a stuff salinity, and the salinity gradient explicitly and temperature is hidden into the density ratio, and in the gap.
11:03:15 And this down, it's a relatively
11:03:15 simple manipulation.
11:03:16 So now, If we had no, add facility, this first term would be our buoyancy flux do only to the fingers and this buoyancy flux contains gamma and if this gamma does you know is non monotonic and so on and so forth.
11:03:33 who knows what's going to happen.
11:03:36 Even because there is a z that even if down here so that should apply to these if gamma is normal GMC height.
11:03:44 But now we also have these other term, which is an add facility, and which does not contain GM.
11:03:59 The real problem is that this is all together. This is all mixed up, if, if I have a simulation.
11:04:07 It's really difficult to disentangle.
11:04:11 What would be the first term, the buoyancy flags, from what should be the second term.
11:04:19 However, if I say that this first term is roughly always the same or at least it doesn't change in bursts in time.
11:04:29 While this second term may change in bursts, because as we have seen that, you know, it may think in this this case it is those overturns that we see in energy that are initially before your, you have a steady staircase.
11:04:46 There are moments in which you have a burst of kinetic energy and then it subsides and then other births and subsides that when you form clusters of cluster may be passing and then you have a bangle the kinetic energy and then from for some time.
11:05:02 Nothing really.
11:05:03 So, if you just look at this in time and you take the differences of the fluxes in time.
11:05:27 Then you should expect that the difference between two relatively close by.
11:05:27 Time instance in the flux have been the buoyancy flux should be roughly dependent on the density ratio and on the selected flux but should not depend on down.
11:05:35 If there is some substantial mechanical mixing.
11:05:41 And this is what I tried to, to actually ascertain from a simulation. This is also to the attempt to the 14 is even higher.
11:05:50 Really.
11:05:51 And this iteration 1.05 this is really fast really quick, really energetic.
11:05:58 And what I do is I start from the very beginning. So I think the conductive solution, and they took that with a tiny bit of white noise, so that is unstable, and therefore the double diffuse instability is triggered that the fingers, grow and then they
11:06:19 create this Lagrangian structure in this login under structure start to zipping up and down.
11:06:25 And as this, as this goes on, I say that my, my friends, regularly on a regular interval and they compute my flexes were so I because I, I already know what the solution is going to do.
11:06:41 I identify the in the domain, a vertical band.
11:06:46 And I know that in this vertical band the mix layer is going to appear.
11:06:52 However, in this analysis I will stop before it's fully mixed, way before that.
11:06:59 So initially, in this band that is a constant gradient, and in this gradients are going towards.
11:07:15 Really sorry against the ratio, which eventually will become one.
11:07:16 When the step is fully formed.
11:07:19 So this point that this is the 31st point in this time sequence, this this points are a time sequence. So this is initial the initial and then as the linear instability grows, we have this thing.
11:07:33 This to green lines are essentially this formula plotted for two distinct brilliance of gamma.
11:07:41 So, initially I'm just into this wedge. And then, if anything, when, when the fingers are just fingers.
11:07:50 And then they're starting to be a really fast growing really fast. If anything, the gamma diminishes, but if the density ratio were growing is the shoe increase, according to the gamut here, But at the moment.
11:08:24 The growth essentially stops and I have something that keep oscillating back and forth. So if you just look at, you know, casually look at this for as a whole, you may say hey, here is your gamma effect. Because see, now you have something that goes towards
11:08:27 the high gamma, but this set of points can flip flopping back and forth into this wedge. And in fact, it does so if you if you compute the differences.
11:08:39 Exactly.
11:08:41 as this little hand wavy argument about mechanical mixing do.
11:08:48 So my conclusion is that in this particular simulation.
11:08:52 I do not see the gamma effect.
11:08:56 And this leads me to my conclusion,
11:09:00 which is the following the Gamma Rho density ratio so that this, the flux ratio as a function of the density ratio.
11:09:13 In the nonlinear regime, as we heard from a scar is reconstructed from a collection of statistical studies simulations of different density races, you first do a simulation of assessing this ratio, and then you do another simulation a slightly different
11:09:37 ratio and then and, and you measure the fluctuations in these simulations. And from there, then you get to the questions for your gamut here.
11:09:48 Okay.
11:09:55 This procedure assumes a near equilibrium condition between the boys Academy structures and their supplements. There are other programs in this procedure.
11:10:00 One is the ethical the size of the domain, in which one does this is this gamma really independent of the size of the domain but that's actually kind of a detail, the real main point that never really got me to accept this this gamma theory is this is
11:10:23 this is an assumption that there is an equilibrium between the buoyancy carry structures and the surroundings and maybe there is, but they don't see it because.
11:10:33 Okay, let's say that I am one of these boys occurring blobs.
11:10:38 The status has not formed yet, but maybe there is some tiny infinitesimal perturbation in the overall result for the average Canadian sir.
11:10:49 And I'm mentally sifting through this Floyd either upward or downward that, why should I immediately react to and you see this in my perturbation in the in the desecration, I mean by the time I feel that there is any physical perturbation the desperation,
11:11:03 I'm already somewhere else.
11:11:06 So,
11:11:09 in other words, I would argue that the process of formation of staircases cannot be considered something some quasi equilibrium process, which is what the gamma theory of centuries, is, is assuming
11:11:31 of course it could be that I am not seeing how these mechanisms work there there is actually some sort of buzz I believe him that I'm not seeing.
11:11:42 but then please explain it to me. So in this sense, I can see that the gamma effect theory as an incomplete one, and this is something that emerged before already in the discussion, but it does not explain in this specific context of fingering convection,
11:11:58 how this game effect that may arise that not in the linear theory that's okay. But in fully nonlinear staircase in forming regime, when you have this highly nonlinear structure like engine structure they're going up and down and.
11:12:39 And this is essentially where with where I'm at, and the only thing that can say more is that, of course, this idea of mind, this this ballsy idea the alternative idea that there is this mechanical mixing that is what actually triggers the
11:12:37 case for me process is something that I personally have not been able yet to transform into a reality of is something that you can compare with the medical simulations or experiments.
11:12:52 And I think that David the in the in the students are doing a lot more progresses on these still I confess I still have to watch the recordings of David's students talk to see how much further.
11:13:10 This has gone.
11:13:11 So, here is how I stop. Thank you.
11:13:19 Thank you very much Francesco very interesting thought provoking
11:13:26 questions
11:13:34 should be should I look again when you first let's pretend you are ya good.
11:13:38 Well, Jim, thank you Francesca.
11:13:42 It looks like the this killer of your energy flux baltics is the key parameter, right. So, do you have a feel as to what would control that scale is it an on off cannot scale of your system.
11:14:00 What was the emergent scale there because that essentially mixing, this is actually something that we try to nail with with Michael.
11:14:27 Yes.
11:14:20 Several years ago when I was working on this topic and then we never really managed to nail it. And the whole thing is just died out maybe one day I will, I should I should get back to that but anyway there are essentially two candidates one is essentially
11:14:29 the wind sister certification so and I've asked me those kind, kind of scale.
11:14:35 There is another candidate or whatever, which is, you have these clusters. So, the idea is that when you have one of these Lagrangian structures that is fast enough so who's really number is well above one.
11:14:49 This as it moves, it creates awake in in the way it aggregates is it triggers all these other kinds of blob like structures that then start moving as a cluster.
11:15:03 Now, the question is, this cannot go up to any arbitrary scale, a cluster cannot grow to any arbitrary scale.
11:15:13 Because if it does, essentially, it would hamper these very precious double diffused exchanges altogether so imagine that you are displacing as a cluster really why the portion of fluid.
11:15:27 Then you have let's say if you are going up, then you have something which is relatively called the moving into water and water.
11:15:34 Because it's really big.
11:15:38 The World War they cannot get into the cluster.
11:15:43 And the trigger these double diffuse exchanges because it's too big.
11:15:48 So if a cluster grows too big, it's simply one working that it will shut shut down the double diffuse process that it for the class they cannot create the buoyancy anomaly.
11:15:59 So there has to be some sort of equilibrium size, where this cluster mechanism is still viable but is not big enough to kill itself.
11:16:10 And the problem is that we don't have a tear we don't have other than this hand waving argument. We don't have anything to try to decide what should be the equality, the right size for cluster.
11:16:25 But then I'm sure that will chamber chime in in this, because your letter mechanism that's.
11:16:33 That looks awfully similar to things while he has done with us okay, because we go for instance and that looks like in plasma as our beloved France or avalanches.
11:16:47 So nothing I know nothing about about this should I should start studying.
11:16:53 But at least it's very interesting so we keep question is if you have all of that data available.
11:16:59 Do you see a systematic distribution of time delays between the establishment of the flux and the response of the gradients, if you if you were to compute that's because that this is.
11:17:11 Yes, this is actually a very good question. I should probably find the student and the student don't think this on doing exactly this analysis at the are the honest answer is I never actually looked at that that sat down and did exactly that.
11:17:34 Okay, but
11:17:45 I mean, obviously a two, well, three points, but maybe I'll keep it to to listening to your description.
11:18:00 You know there are these this meeting has brought me back to reading old, old papers from Ed seagulls reckless youth on things like, you know, mixing length theory for blobs of stuff with wakes which is what you just said, that's on hand and also non
11:18:21 local mixing link theory based on noon on.
11:18:27 Well non local kernels and so forth, have you have you looked into any of that.
11:18:33 I mean there's forward for the theory, and the.
11:18:41 The honest answer is yes I'm kind of vaguely aware that these things exist because I was very close to the.
11:18:50 But, you know, it's essentially this is 2013 or so that haven't been doing anything on still fingers, really anymore so it's sold somewhere in my throat in my to do list, very down the to do list.
11:19:06 And yes, I should get back to that, that's, that's the only thing that should say and sit down, take all these papers studied think can maybe get something that.
11:19:18 Now the other thing is what GM alluded to you suka suka suka and I developed this idea of looking at jams.
11:19:28 In a sense, it's inspired by you know wisdoms old stories of jamming and traffic flow as a route to coarsening and so forth for various problems and plasmas, but I mean the key thing in that is time delays.
11:19:44 And I notice you are explicitly mentioning time delays and that the something new. There might be double jams right you could imagine. We just had a jam in the heat flux, but you could imagine of a flow for the heat flux for the heat and a flow for the
11:20:03 salt. You know, you could think of it as two different streams of traffic, maybe yes yes yes that that actually sat and play a game like that I mean, or even just the two streams of the two like random team so something like a calculator to do some sort
11:20:22 of calligraphers equation that pops out exactly what you get. Exactly. Yeah, yeah.
11:20:32 something like that.
11:20:30 Yeah.
11:20:32 maybe at the end of the segment this summer.
11:20:36 I could. Yeah, sit down and play with that.
11:20:40 Interesting last, last year, I guess, will be around
11:20:49 boyish, please.
11:20:51 Yeah, it was really interesting, but I also have several questions I'll try to have them related to each other.
11:21:03 crystals equations which you wrote, I don't think they had not only in your account.
11:21:14 And on the other hand on linear terms of velocity, the whole velocities are hidden in those flexes. So those efforts that I Britain those boxes or whatever comes out to the blood of the objectives
11:21:37 not exclusively in somehow hidden under flickers, but you have you have add, add facilities are so clear how you can have it if you, if you wish to Yes, okay, if you wish is to scale problem in the sense that on one side you have the velocities associated
11:21:51 with this like random yc coding structures.
11:21:57 And those are those that go into the, what I call the finger flexes.
11:22:03 All right.
11:22:04 Okay. But, and then you have the velocities at larger scale
11:22:15 here.
11:22:16 So, these are the flex is produced by the range of structures with the motion.
11:22:23 But as these Lagrangian structures move.
11:22:26 I feel, looking at the simulations fairly confident to say that they also engine, there's some larger scale motion is overturns at these.
11:22:37 And they in this case at the in some way models that.
11:22:43 That's another question that was asked a question about it was middle scale.
11:22:52 So, did you try to measure the rate of dissipation, you kind of called the submissions or because add facility, could be related to dissipation of manager dissipation of temperatures of energy, but Zanna took the flow dependent characteristic and it's
11:23:08 possible that this is a little detail which you, you know, messenger theory to take your garments to do.
11:23:19 How would you link the disposition to the gun, I didn't ever look at the dissipation and.
11:23:27 Well one of the ways to do it. You can you can really simple.
11:23:35 It's a way to do it and there are some sophisticated way to put it all together, but jamming the dishes at become the function of this equation. And then you can bring the one sentence number and wasn't a scale, everything will come back to your model
11:23:56 and you will have more skills and more parameters to to play around a little bit happens should really think about that because if what you say is viable, you're pretty much discovered the mechanism for justifying the GM effect in the nonlinear regime.
11:24:16 Because you're saying that depending on the, on the density ratio, you would have a differential diffusion temperature in Sydney.
11:24:26 And the reflecting to the flexes of temperature in Sydney and therefore justify the fact that their clocks ratio as a dependence on the density ratio
11:24:39 on, you know, just from Huawei.
11:24:41 Okay, need to click on Yes, then it's definitely, it's definitely something to look at.
11:24:48 Right, thank you.
11:24:53 Is your theory Francesco intrinsically salt fingers or is it.
11:25:01 I would say so for example I would, I definitely wouldn't know how to apply that to the upside down situation the one in which they say the diffuse effect about the future, because in that case that you don't, you don't really have like Ranjan carriers
11:25:19 and small scale structure that then in some way aggregate in order to form larger structures.
11:25:24 So there's no Won't she think in that case, I don't know the problem of the ball sitting is always finding what is the role. Yeah, that's that's my take, you need to you need to have a row.
11:25:38 And in this case, I hope it and this is probably the only thing on which I'm fairly confident. I'm confident that this aggregation of so fingers of like range of structures, is that all of this in this division and beyond that as you have just discovered
11:25:52 that I no matter, because I didn't God so.
11:25:57 But that's the only thing, on which I'm feeling confident in the opposite in the upside down case.
11:26:03 What's that old I don't know, maybe there is a road but not that I can take off.