09:10:30 So our next speaker is Michael Lebars from CNRS, and he's going to talk about layering generated by McIntyre's if you use of instability around the lens vortex.
09:10:45 Okay, thank you.
09:10:53 Yes, yes. Okay, good. So I'm going to talk to you about layering we can see in the lab around vertices in rotating stratified fluids and you have two examples here this is our experiments.
09:11:06 And this is the old work from the Griffith and Nathan, which basically said the problem. 40 years ago.
09:11:17 So the, the main geophysical motivation for this work, are the so called medis so maybe stand for meditation and water ad.
09:11:28 If you take a vertical cross section the Strait of Gibraltar you say you have the Mediterranean Sea on the right here which is hot, and the soul chair, and the Atlantic Ocean on the left and the Mediterranean Sea flows into the Atlantic Ocean because
09:11:44 it is denser it close to a gravity current to some deep. And then, it spreads and because of the coin is false itself organized into an anti Cyclone which is called a MIDI.
09:11:56 So these anti cyclones have a typical pain take shape with a height of about one kilometer. The diameter of about 100 kilometer. And they past is the green very long duration of time so if you make a vertical cross section inside the Atlantic Ocean you
09:12:19 So typical rose be number of the Santee cyclones is below minus four point 15, and they can, some of them disappear rapidity but some of them also are capable of crossing the Atlantic Ocean, and they finally crash on the Brazilian coast on the other side.
09:12:34 after three to four years.
09:12:37 So using our lab experiments what we want to explain is, why do they have this pancake shape so it will shape. Why are they so lonely, and also have direct interest for this meeting.
09:12:51 Why are they surrounded by layers so this is an illustration the Seesmic image. So what you see here are strong density interface is highlighted by the seismic profile, and the layering that you see here but also on the side of a typical verticals can
09:13:08 have 10 to 100 meter.
09:13:11 Okay, so to make your Mehdi in the lab you need two basic physical ingredients you need stratification with a broom by South African cn, and you need rotation was a query parameter F.
09:13:24 So if you make an injection into a rotating fluid. This is just on the record player, you will build a tailor column, if you make your injection in the state the stratified salty layer has here, you will make a gravity current.
09:13:39 So, stratification limits motion along the density along the axis of vertical axis rotational measuring a lies the flow along the rotation axis, and if you combine the two, then you end up with a pancake vortex that is floating at some athletes around
09:13:58 its natural level of buoyancy.
09:14:02 natural level of buoyancy. So what happens here is very simple, you have a linear stratification some background rotation. And then you come and inject Mercury News fried it goes down to its natural level of buoyancy where it's spread, because it's spreading,
09:14:16 and you have rotations or query sports, make turns it into an anti cyclones and then you end up with a deployed on the cycle, which is sub stratified meaning it has a internal stratification of zero in that case compared to the background specification
09:14:33 that is fixed by your souls.
09:14:37 is fixed by your souls. So before it was just educational movies that I've done with the gentleman who at UCLA is a real experiment is here. So the tank is about 50 times 50 times 730 centimeter, it's rotating add up to one rotation per second.
09:14:54 And here you see to view on bargains a rotating frame of reference. This is a side view. So we have a vertical laser sheet and you can measure the aspect ratio of your pancake vortex and this is a top view, using, or reason to laser sheet as above here
09:15:11 where we can do piv.
09:15:26 So that I bought Xi just show you an example here of the velocity measurements, it has a very good shape regarding it says he moved to velocity.
09:15:25 So our system is correct terrorized by four dimensional parameters. So Rusby number which is a ratio of the radio velocity at the center of the vortex divided by the query parameter, the ratio between the stratification I, which is in in salt water in
09:15:58 tank invaded by Sequoia this parameter, then you as the x men number, and very important Schmidt number which is a ratio of viscosity divided by So DPZVT so it's equal to seven readings a lab experiment. So what I want to explain is, why is this shape
09:16:02 while this alone, leave on why as I surrounded by lanes. So let's start with the first question.
09:16:07 So to predict the shape we can just look at the logistics equation written in the rotating frame of reference. That's why we have a call us term here.
09:16:18 And in terms of the density perturbation real crime compared to an imposed background, which has a constant ruined by Sarah frequency.
09:16:27 And if I want to look at a stationary solution, which is linear and actually symmetric, I just end with the most simplest balance that I can get is the job Sophie, and it was 30 minutes so address the pressure gradient, the buoyancy and the corner is
09:16:54 So if I ride the equation, I have on the radio direction, so Joseph speaking Caribbean and in the vertical direction zero static equilibrium. So, the pancake floating vortex works because you have a density and how many weeks make, which makes a pressure
09:17:03 gradient which translate because of the core use them to some as you recall velocity.
09:17:08 So knowing given volume for your vortex you can look for an exact solution of those to the equation, looking for instance for sorry buddy rotate. Sorry buddy rotation, and arena interval certification.
09:17:23 And in that case, you find a pressure solution for the pressures that is problematic in error credit ticking z, with a factor in front of them that are written here, so in case of a floating on the cyclone you have a negative words be number, and you
09:17:37 have an internal specification that is smaller than the background stratification as I explained to you before so you end up with an editorial equation.
09:17:47 P prime he is he is a pressure at the center of the vortex and you can just determine it by computing the total volume of your vortex.
09:17:57 This equation is just really the solution of this just considering that the pressure has to be continuous between the inside of the vortex and the amniotic fluid.
09:18:09 So, with this very simple model we find that the equilibrium solution must have native soil shape, and we can determine the aspect ratio so the height divided by the radius, and it goes like the square root of the rows be divided by the density I know
09:18:24 many times a co this term. Notice that if you have an anti psycho so it was these negative you must have a sub stratification so the internal part of the vortex must be less stratified.
09:18:39 So that was just for the Sony body rotation you can do exactly the same solution for and find the same aspect ratio low for the motion vortex. And, which fits well with our lab experiments.
09:18:54 So, we have compared this very simple model with our lab results, and we have done two types of experiment in this first one we just inject in our tank it comes to a given volume of salty water, and then we follow the evolutions through time of the floating
09:19:12 on the cycles so the blue line is a theory, the black line is the art of measurements and you see that over time, the volume stays constant but the rugby number decrease, and we have a very good agreement between our theory, our model and the measurements
09:19:29 with no adjustment parameter.
09:19:37 The other experiments we can do is to have a constant injection over time so in that case the volume of our floating on two cycles, increased through time.
09:19:52 And the rugby number remains larger, so we can maintain we can sustain the voltage during longer that again you see you have a very good agreement between the modal in blue and the measurements in black.
09:19:59 So we have done lots of experiments like this you have also lab experiments with constant volume in blue here is a flow constant flux here. This is a theoretical low we have a very good agreement between our lab experiments and theory, but we also have
09:20:16 a good agreement with the measurements we can get from medis in the Atlantic Ocean. Those are all those points, and also with all the types of floating vertices that we can find in nature, for instance, to Jovian vertices, and this point is a great read
09:20:33 books.
09:20:34 So you can see for the graduate but it's a good order of magnitude but it started off center and we think this is because, read, read sport. It's a floating vortex in the atmosphere but it is also shared by the vertices jet so we have re return our experiment,
09:20:53 but also including sheer, and this, I don't have time to show you that but this is published recently in a paper by the masking.
09:21:04 So that was just for the shape so now why is this along leaps to explain why you say, this is the remain for very long time I come back to my name is Lucy nice equation and this time I will keep the time evolution and the diffusion.
09:21:23 Still, assuming linear dynamics and neglecting the soul diffusion which is very good at least for my lab experiment.
09:21:31 So in that case I recover the.
09:21:44 Just perfect balance in the radio directions. I will set the balance in the vertical direction. And in addition to that I as a time evolution characterized by z balance in ZSE to direction plus the density equation, and the divergence free solution in
09:21:55 here. So usually if you have a naughty cycle and so if you have some as he told us he here, you would balance diffusion, the viscous diffusion of this velocity to predict its time evolution, but the dynamics are our floating vertices is not the same actually
09:22:14 the discuss diffusion of the as the veto velocity, give rise Souza Korea is supposed to radio velocity. This radio veracity because of impressive it gives you a vertical velocity and then these vertical velocity at next the background stratification to
09:22:32 give you the time evolution. So it's not at all the classical viscose evolution of the system, you go through the code is supposed to make recirculation to finally have a time evolution of your a creative state was equilibrium state, because I've had
09:22:51 direction of its density no man. So if you further assume from the set of equations that the aspect ratio of your vortex is a small meanings that is it as a small high compared to its radius, then this system can simplify into a single diffusion equation
09:23:06 diffusion type equation for your pressure, where you have the time derivative of the pressure which is given by the lesson of the pressure but in the radio direction times the viscosity which is corrected by this animal f ratio.
09:23:21 So it's kind of a contract intuitive behavior because you have a small aspect ratio vortex.
09:23:26 But viscosity doesn't make diffusion in the small direction if you want it makes diffusion is a large direction, and the viscous diffusion is corrected by this factor, another square.
09:23:40 So, this simple model again works well with our experiments, those are measurements of the velocity as me to velocity as a function of the radius, or values time from the initial vortex to the one after 726 rotation of our table.
09:24:05 And if I just re scale, those low by using the classical viscous diffusion but with this viscous term, I can impose all the calculations that we really haven't set similar behavior corresponding to radio diffusion but corrected by this factor.
09:24:14 So just to give illustrating numbers the usual balance will give you a typical times California vortex of age 12 those of us because at this specific balance which involve stratification and rotation gives you elsewhere in terms of is because at times
09:24:30 square and square. And if I apply this to an ocean where I take a turbulent viscosity for instance of 10 to minus two meter square foot per second for just look at the middle school dynamics of the ocean.
09:24:52 you a lifetime of three months, and this balance, give you a lifetime for three years. Okay, so now I come to my last point, which is the most relevant for this meeting so why are what are those later is that we can see around those memories.
09:25:05 So, the layering around maybe this 10 and up and questions out some very interesting papers so incensed by a hardcore into 2017 at shows, it might they might they might be due to the diffusion instability, because again, the inside of the melody is hotter
09:25:24 and the soul chills and the Atlantic Ocean, but also our model also says that the layering might be due to for instance on the side to some critical layer in study to some relative instead each year around some critical layer that appears on the side.
09:25:41 So for instance, you can see the world by mean wha.
09:25:45 The point is that none of those two mechanism point stability is valid for our work, because we cannot be a devotee is even stability because we have only one stratified agent, we just have sold low temperature effect in our system and our experiments
09:26:04 are very, let me know so we have chicken we have really no critical point critical layer in our system.
09:26:12 So brief is an indent in 1983 show this nice picture where you see the layers appearing by shadow graffiti and suggested it could be you. To so called McIntyre visco adding instability.
09:26:28 So, this instability is a very strange one which is driven by viscosity.
09:26:35 And I will just give you how it works. So if you take some Joseph the balance flow. Well, I hope he knows I'm flying like that, and you take one through a particular and you move it regularly upward.
09:26:48 If you don't consider any diffusion defects, then you would just take thrive excite rabbit hole inertial waves.
09:26:55 But now if you consider diffusion effect and if you assume for instance the viscosity is larger than Sol diffusion. Then, if you move in that direction slowly you might come to this point, but by having your mental maturity Brady but not the buoyancy.
09:27:13 And then in that case you end up with a football tickle here which is dancers and the ambience. And so the displacement is reinforced.
09:27:22 So it's the same type of mechanism as the classical defense even studied it, except that it's not diffusion diffusion diffusion of salt and temperature for instance, but it's differential diffusion of viscosity and just ratifying agent, and he can work
09:27:37 out for larger viscosity of a smaller viscosity.
09:27:45 So my entire I studied the linear stability of this system, by using your local approach. We start from a bass pro and fellow in balance and he looks for plan way calculation around this Bass Pro.
09:28:00 He makes some approximation of course he keeps the fusion because it's the engine for this instability but you neglect your visual effect for simplicity, and he assumes case evaluation he makes a WKD type approach, between the wavelengths of the perturbation
09:28:15 is looking for and the typical scale of the base.
09:28:20 So then, so this work is Reba McEntire but if you want to, easier to read version of it you can also you can see nice paper by Monroe. If you use this typical length scale which goes like where's the viscosity ups here, this typical times care and you
09:28:48 treasures two angles, the angle gamma, that is made by the line of constant circulation and the anger made by the eyes of Pecknold, you end up with this discussion relation in here. And the threshold for the instability that is written here.
09:29:14 I just show you that for mentioning one important point. The Ackman number does not appear in this equation, so you have the two angles that appears on gamma, meaning that this is those two numbers, those two angles would depend on the Rosebud number
09:29:06 on the ratio. End of RFU as a sweet number also but you don't see any dependence on the segment number. So it means that once you reach scale by the typical large scale of your system, you will find a threshold, and the growth rate that does not depend
09:29:24 on the admin number, but on these length scale will depend on the admin number and you will scale with an x men number with a boil, one half.
09:29:33 So from an experimental point of view, being only very few experimental validation of this instability mechanism. Most work had been done in this 25 minute configuration.
09:29:45 So it says rotating studies 2540 layer of fluid. When you put in rotation the disks, and you see the layering appearing above and below.
09:29:57 And for the medic configuration, well there's been only this picture by Griffith and Indian. And then we have tried to make some more measurements, but it's not very easy to study this problem experimentally.
09:30:09 This is because the layering on your peers in the continuous injection experiment in which case we don't have a very good control on the rugby number and we are also looking for an instability that grows on the system that is also growing through time.
09:30:26 So we cannot make a systematic approach, and also in the lab we cannot change the Smith number which is the main control parameter of the instability.
09:30:37 So, I have switched to a numerical study instead and I use a very simple numerical simulation actually symmetric numerical simulation, looking for perturbation around a background balance cushion vortex.
09:30:53 And just to show you here so those are the two same regime except I change the Schmidt number. So in one case, you see the radio velocity in here. In one case, nothing happened.
09:31:04 And in the other case you see the nice during appear.
09:31:09 So in the last two minutes I was just show you a try to answer the question does the McIntyre instead at explaining what we see here and what we say in the lab.
09:31:17 I want those are local energizes ISIS to explain the growth rate on wavelengths and could this mechanism has something to do with maybe layering.
09:31:26 So I have done some systematic study change English meet number and what you see here is the evolution of the wavelengths and the growth rate in red is a numerical simulation, excuse me pick simulation again and in black into the theory.
09:31:39 You see we have a very good agreement for the wavelength. And we also have a similar behavior because of gross right except that numeric is slightly shifted.
09:31:48 And I think this shift is due to a limited scale separation between the size of the MIDI in my system and the layers, so they appeal with a 10% scale of the, of the, maybe so you don't have this nice cat separation.
09:32:08 That's why the theory is not completely quantitative EEG correct, and to better show that if I know decrease the x men number or regarding the wavelengths I have an accident agreement again.
09:32:21 And this one half slope that was predicted by the theory. And when my wavelength goes towards smaller and smaller values and so gross rate also first increase and then saturate to evaluate that is much closer some Siri.
09:32:36 And so the last one for the Rosebud number is systematic in terms of Rusby number it also works very well and the last one that does not work where is the systematic study in terms of the ratio and over F again that we're looking at the growth rate we
09:32:50 we stopped well but then we have a decrease in in with no way that is not predicted by the theory, and this again is probably due to the limited scale separation in my system, at least for the segment number that I can easily access, access with my numerical
09:33:05 simulation, because when f increases. It also means that the height of the voltage decreases and so I don't have these guests separation that is necessary to applies a local therapy.
09:33:17 So I. Nevertheless, I think that the McIntyre except instead he indeed explains what Rufus and Linden showed and what we've seen our experiment and what I see my numerical simulation.
09:33:29 The local theory, those were provided that you have a good sketch separation, and then to finish with a good that mechanism do something for explaining maybe layering.
09:33:40 So, for many years I have two possible McIntyre instead eg one using salt and because it end user using temperature and stability for salts, I predict using the molecular value wavelengths of 4.8 meter and a growth time of 70 days.
09:33:59 So these bros time is not that big but it's still very kind of long because the layering or may disappear very rapidly after that formation for temperatures and mechanisms instability is stable, but the last point I would be happy to discuss with is that
09:34:16 I think there is no reason on this typical scale to use molecular value so I have just explore the range in terms of discuss it from 10 to the minus six to 10, minus two, in terms of sweet number also, and in that case you can have a large range here
09:34:33 when you can predict the good, the relevant link scale for mateys and the rig written revenue growth rate for middies have less than one day.
09:34:44 But my question now is, would it be reasonable to take a typical middle scale that you have the number of larger than 100 of smiles and one of the 100 and viscosity between 10 to the minus four and 10 to the minus two meters per person.
09:35:03 And those are my conclusion. So from a medical point of view I would like now to look at the nominee now Long Term Evolution and mixing. I would also like to look at instead he wins a sprint number is larger than smaller than one up to now only managed
09:35:17 to look at to see they are in Windsor St. Mary's much larger than one. I don't know why.
09:35:24 And from an experimental point of view I told you that the free vortex is not a good way because we don't very well controlled experiments so one possibility would be to have a solid Goshen shape inside the experiments and to make it rotate as in the
09:35:41 spinner experiment by Monroe, for instance, and in that case a numerical simulation also predicts the appearance of layering so that would be a good way to look at this day or any in the experimental context with some control of the BIOS parameter.
09:35:57 And I stopped there. Thank you.
09:36:00 All right, thank you very much for an interesting talk questions please raise your hand in the reactions Bruce
09:36:12 Bolger myself. Thank you very much for an excellent talk.
09:36:16 You made a comment that in the lab experiments, you did not see leering if you just injected a constant volume you only saw it. If you had a constant flux source.
09:36:26 But with your numeric I assumed it was a constant volume initial condition and of course the body is constant volume after it's created. So can you explain why you don't see layering if it's constant volume in the lab experiments in the lab experiment,
09:36:43 the recipe number is decreasing through time.
09:36:46 And I think it's decreasing too rapidly, so that the layering as no time to grow into new metrics, I'm, I'm, I'm cheating Actually, I'm imposing this background various version vortex with a constant Crosby numbers for time and they just look at perturbation
09:37:04 about this impose flow.
09:37:10 That's very clear and I assume with, with a melody because it's so large, you don't have this problem with the rapid decay of the recipe number is a timescale separation between the growth of the layer is typically one day for growth as well layer growing.
09:37:23 wise the life of the media is four years so at this time the could being and length decoupling that is much better than what I can do in my experiment.
09:37:32 I should say is that in CO this platform.
09:37:36 They should be able to see it I know that drew so Marissa has done also some floating vertices experiment.
09:37:44 And he published a paper recently about that. And I assume that it was a one meter experiment, one meter maybe a lab maybe would be possible to see the layering growing.
09:37:57 Maybe.
09:37:59 Thanks again, Monica.
09:38:13 It wasn't clear to me who's next so we'll try Pascal, and then David Rachel,
09:38:10 great talk and my question might be kind of stupid because I, I, this is the first time I hear of these instability and then trying to understand the physics.
09:38:19 The ingredients appear to be kind of similar to the Gold Rush sugar figure instability in the sense that you have some kind of temperature stratification or an angular velocity stratification, are they related.
09:38:31 So that would be my first question and then the second question is the layers that appear, or the ISO pigment layers in the sense that they're following ISO picnic contour so they're inclined compared with the ISO pixels.
09:38:43 So the second is easy is inclined competitor eyes are big nose.
09:38:47 So, consistent with the Medi observations. Yes.
09:38:52 Oh, wow, lots of discussion where people try to look for the angle of those structures.
09:39:00 And that would be one way to explain weathers are related to double diffusion, or to McIntyre instead eg and there is a great paper by this one Riddick on this, looking at the NGO, and the trying to, to know what is the mechanism points stability, depending
09:39:18 on the NGO that the layers make with the eyes of ignore for your first question, which and instability, are you mentioning, it's an ass for stability Yeah, well that's the thing it's an Astro instability that's typically at low a very low parental number
09:39:35 instability which is called a goal right sugar trigger instability.
09:39:40 Here you have a high parental number limit obviously from exactly the, the McIntyre theory works for both.
09:39:48 I printed on a small printer so maybe the same type of instability in the low content number, but in this paper McIntyre discuss both limits.
09:39:58 and maybe Steve to bias if he's here knows the answer to my question since he's worked on the GS F.
09:40:04 It wouldn't be the first time that the same instability discovered in two different.
09:40:11 Okay.
09:40:15 I'm not hearing Steve so David Rachel.
09:40:20 I, I know that's difficult in in the lab to reach larger values of no breath because of Reynolds number difficulties.
09:40:30 But do you expect to see significantly different results I mean the results you reported have no breath, even less than one or order one. I think the biggest value add is 2.4.
09:40:42 But the reality is that most ocean values have no for fr on the order of 10 or more in ocean in the atmosphere it's worse it's like 100 or so. So what what do you think would change in those cases.
09:40:59 So from a theoretical point of view local point of view, nothing much better, you're writing the lab we cannot go and check this limit, because n is limited by the by some salt stratification, and the other way to increase animal life would be to decrease
09:41:17 F and then you are, you have a large segment number and you cannot do much.
09:41:22 So, in the lab, I think we cannot do much more than those by you yes from let's say oh point two to five or something like that, but from the theoretical point of view, you can go.
09:41:37 You don't have a big dependence on NOVRF and you actually know I don't have yet but you have your region assented limit in NOVRF, you need the F, to have the instability because you need some, some,
09:41:55 you need to start from assemble when equation, so you can do this instead he does not exist if you only have stratified, you need some rotation, but then there is not seems specific in the loud and aware fd me from a theoretical local point of view.
09:42:17 Tobias has appeared So Steve, a brief comment or question brief, please.
09:42:25 I was just going to answer Pascal's question but Adrian has answered it in the chat. So that's probably the way to go. Just have a look at agents answer.
09:42:35 All right, great. Thank you. All right, thank you very much. Let's thank Michael again.