09:04:06 You should probably get going. Good morning, good afternoon to everyone, welcome to mechanisms, three.
09:04:17 So we have an interesting lineup.
09:04:22 Today, Three people that haven't spoken.
09:04:28 Formerly, as it were, in the meeting.
09:04:31 So we're going to kick off with Joe Brown Joe, Joe.
09:04:45 He's worked on various problems in vortices and instabilities mainly in astrophysical desks.
09:04:52 And so we're looking forward to hearing about the zombie vortex instability of you go Joe.
09:04:57 Thank you. So, this is work that I've
09:05:03 been working with my former PhD advisor film markets that many of you already know.
09:05:10 Over the past, I would say 15 or so years. And so what I thought I would do is basically tell the story of the zombie vortex instability because I think the the historical story of how we discover the instability is interesting in its own right.
09:05:29 So of course I'd like to think funding agencies, the National Science Foundation astronomy astrophysics grants as well as computational resources from exceed.
09:05:45 Okay, so a little bit of a quick motivation.
09:05:52 We're interested in the dynamics of the hydro dynamics of protoplanetary discs, these are discs of gas and Dustin orbit around newly formed stars. And it's a remarkable fact that the planet we're standing on at one point started off is literally microscopic
09:06:06 dust grains and in order in you know in a timescale of order about 10 million years you grow from microscopic dust, up to Planet size, which is a remarkably short amount of time.
09:06:23 You know, on a cosmic scale.
09:06:26 And so here is just a menagerie of protoplanetary discs. These were taken in.
09:06:34 This was released in 2018 by the darts collaboration darts s collaboration is four discs around T Tauri stars with sphere.
09:06:45 And you can see this this is scattered light from the dust in these discs, and I remember when these images release people like literally gas.
09:07:08 of these protoplanetary discs of gas and dust. And so the specific motivation that we were interested in is understanding how what hydrodynamic processes.
09:07:17 Allow microscopic dust to form the first generation of planetesimals the building blocks of planets. So basically you want to go from microscopic dust grains.
09:07:29 you know, through millimeters sizes centimeters SIZES All the way up through planetesimals sizes which are kilometer to 10s of kilometers.
09:07:37 And that's that that gap is the hardest gap to go through in terms of understanding from a physics point of view, because you really just dealing with small rocks.
09:07:47 small Dustin rocks and, you know, you go down to Ocean Beach and you pick up to pounds of sand and throw it at each other you don't get bigger, it'll get boulders.
09:07:55 So how do you generate the first generation of planetesimals How do you get to 10 kilometer sizes.
09:08:03 And so that was one of the main motivations for for trying to understand the hydrodynamic processes and so back around in the early 2000s we got interested in vertices studying vertical structures and maybe how those vertical structures input up later
09:08:20 this might trap dust shield the dust from the turbulence within the desk and allow the dust to coalesce into an aggregate into planetesimals. So that's the motivation.
09:08:34 Just a quick by the numbers, I'm not going to go through all of these but if you're a geophysicist or laboratory fluid dynamics is not familiar with the scales or protoplanetary discs, it's a very different environment than then geophysical or laboratory
09:08:52 fluids and plasmas.
09:08:55 So just one of the interesting numbers I find is the density.
09:09:00 We're talking 10 to the minus six kilograms per cubic meter. So, those pictures made it look like oh these are really dense environments. They're not the reason why you can't see through them it's just because there's.
09:09:20 They're big, there's a lot of, you know, there's a lot of stuff there, but it's it's actually pretty diffuse.
09:09:18 So these are these are very diffuse and even though they appear on the screen as if they're a dense environment they're not it's you know the density is 10 to the minus six and chaos.
09:09:30 The other, the other thing I like to just jump down to the bottom.
09:09:36 In a well mix state if you had a, you know, take a cubic football field 100, meters cute.
09:09:42 There were in that volume hundred meters cube there'd be only one kilogram of gas, and a few billion dust grades.
09:09:50 So these are these are very diffuse environments.
09:09:54 I just want to quickly mention that we're interested in hydrodynamic instabilities and there's a whole day you know on a parallel track.
09:10:02 There's a lot of people study protoplanetary discs and are more concerned with MHD, and the role of magnetic fields.
09:10:11 We were specifically looking in the quote unquote planet forming regions which are in what are known as a making the MHD dead zone, and so on this plot it's the middle it's the darkest brown in the middle.
09:10:23 It's in this region where it is believed.
09:10:26 There's not a, there's not sufficient coupling.
09:10:30 There's not enough ionization is not sufficient coupling between the magnetic fields in the gas.
09:10:35 And so, magnetic MHD instabilities like the MRI or sort of quenched in this region. And it just so happens to coincide with a region where, you know, planets form.
09:10:46 So we were interested in hydrodynamic purely academic not MHD, but of course.
09:10:53 Lots of people do focus on the MHD stuff.
09:10:57 And so as I said our motivation was trying to understand.
09:11:02 Looking at vertices three dimensional vertical structures embedded in these protoplanetary discs, because we thought these might be sites were just make what I deleted the eyes of these storms in the desk.
09:11:14 And this might be a great place to form planetesimals. And so we know in nature whenever you have rapid rotation intense shear and strong stratification, you get vertices.
09:11:27 And so our idea was, oh let's look, look, look for these sort of structures in protoplanetary discs.
09:11:37 But of course protoplanetary discs are you know the type of vertices, you might get into protoplanetary just are not going to be the same for what you get on Jupiter for a very important reason on Jupiter there's a sort of in most planetary atmospheres
09:11:52 there's a very strong hierarchy of timescales you've got the turnaround time with the vortex that's one timescale you get the orbital timescale or the rotation time scale up the system itself.
09:12:04 And then you get the brunt vice love frequency or the inverse burn for this price low frequency for a period.
09:12:11 And on in planetary atmospheres these timescales are very well separated right on Jupiter, the turn, you know, think of the time to go ones around afford Texas is a border a week, that type of the plan to spin is 10 hours and the front Vizsla frequency,
09:12:26 you know, and stratified regions of the order of minutes.
09:12:31 And so that hierarchy of timescales allows you to, for example, imagine that planetary atmospheres are very, very well two dimensional.
09:12:43 Because the rapid rotation and the strong stratification allow you to sort of decoupled some of the scales and protoplanetary discs if you, you know, if you do some old back of the envelope calculations, you've got it's a really interesting environment
09:12:57 because you find out all these timescales are basically have ordered the same thing.
09:13:01 Right, the top turnaround time for a vortex the orbital time for a system, the brunt Vizsla free they're all the roughly the same time scale and that what that indicates right off the top of the image right off the top is that the vertical structures
09:13:30 you can expect are going to be inherently three dimensional structures, as opposed to the very flat and things you might get in a, in a planetary atmosphere. And so I'm just a quick note about the computational method we don't actually simulate the hydrogen
09:13:37 and mix in the full disk because you just don't have the resolution to, you know, if you really want to capture the small scale features of the turbulence you can't actually simulate the whole this so we do sharing box simulations where we zoom in on
09:14:02 small box of gas and Dustin orbit, and we go into the rotating reference frame with that box we hop on to that box and rotate around, around the protostar. And in that box there. The gas appears to have a, you know, for especially small box and it appears
09:14:09 to have a linear shear and that linear shares the origin of that is just the fact that when you're close to the star the gas is orbiting slightly faster when you're a little bit further from the star, the gases in orbit a little bit slower when you go
09:14:20 into that rotating reference frame. It makes it look like the gas that's a little bit further ways going backwards. It's just, just the fact that we're in the rotating reference frame with that box of gas on and we do fully 3d spectral methods.
09:14:40 4848 48 in the horizontal plane, and in the vertical direction we use chef he said polynomials.
09:14:47 So we don't. So we allow for the fact that we're not going to have uniform stratification we allow for the backpack. The stratification structure in the vertical direction might be complex.
09:14:58 And so we use ship on your nose to resolve direction.
09:15:04 We get to the side where I just show you the equations and these are familiar to most of you maybe in a slightly different form. But the first equation this is just, you know, the background here is, is linear in our coordinate system we have y, the y
09:15:20 direction is going in the orbital direction the exes are local radio direction.
09:15:26 In the disk, the vertical direction is in hydrostatic balance that's the second equation that's just hydrostatic balance, and the two middle equations just start equations for the vertical temperature structure and the vertical stratification, we can
09:15:46 relate that to a potential temperature and abroad Vizsla frequency for understanding the Vertical, Vertical structure we don't assume that the vertical structure is uniform.
09:15:58 And then if you jump into that rotating reference frame into that box at the very bottom, you just have Euler equations and I should note that we, we work with the inelastic approximation we're not assuming boosted ask.
09:16:14 We, the vertical structure we might be going over, you know 345 scale heights, local scale heights and they did pressure scale heights and the desk.
09:16:24 And as I said, we're motivated by the fact that we're not expecting our vertices to necessarily be to the flat but three dimensional structures and so we didn't want to impose the boost and ask approximation.
09:16:38 Instead we allow we allow for the inelastic approximation.
09:16:42 And so this is just what the equations look like in that rotating reference frame the terms, the rest of the terms. Look should look familiar to you.
09:16:51 You know that data twit all of our data bar that's just a fancy term.
09:16:57 And that's it appears in both the vertical velocity, and in the, the temperature for the potential, the equation for the potential temperature.
09:17:07 And the last equation is the, the inelastic approximation.
09:17:12 So as I said we were originally motivated by, you know, just looking for coherent long lasting three dimensional vertical structures in the mid plane of the desk because we were interested in, and using them as sites for planet testament formation.
09:17:28 And I just want to point out one of the interesting things about three dimensional vertices.
09:17:35 As many of you, everyone here on on on this conference knows that vortex lines vortices it is divergence free and that means vortex lines must, they must want to close loops or they must go off to infinity.
09:17:49 And so, if you're. I should mention the the sheer profile inner protoplanetary disk is anti cyclonic so you only expect anti cyclones and on my figures here anti cyclones out there for custody vectors, the 40 points downwards.
09:18:06 And so you would expect the vortex lines to form closed loops and so
09:18:13 here is if you imagine having a two dimensional life vortex but you know having a go off to infinity, this is what I call them or vortex would look like.
09:18:24 On the right hand side, and then we get some simulations of those things and and plotted the actual vortex lines and this is, this is the sort of mess it looks like this is really a three dimensional.
09:18:42 This is the result of a three dimensional simulation of a vertical structure in a protoplanetary discipline you can see, yeah the, the core of the vortex.
09:18:52 The the vortex lines are all really well aligned but then they they form closed loops and then also go up and format, I believe, turbulent mess.
09:19:04 And so let me just quickly show some simulations this This goes back about 15 years, because this this this will take us a little bit further along in our story, we would put, you know, at the time, so just going back to say the early 2000s 2004.
09:19:22 We, we played God in the sense that we would just put the vortex in by hand we at that time, we weren't sure what the formation mechanism what might cause for disease to form in the first place.
09:19:33 so we would literally just put them in by hand.
09:19:37 So we would literally just put them in by hand, putting some quality equilibrium, by hand, and then just let it rip as an initial value calculation and we would imagine that it would readjust and do whatever, but we hoped it would come to some sort of
09:19:49 long term steady state and then we would eventually throw in some Dustin and do some, some couple Gus gas dynamics.
09:19:56 But we were disappointed at the time to find that all the vertices we put in the middle of the desk by hand.
09:20:06 They, you know, in that, let me just run that while I'm talking again, one second of the simulation is 42 orbits and local years, or if it's on the desk.
09:20:15 This wouldn't last for, you know, a few hundred orbits, which sounds like a long time but not enough time to be relevant for planet formation.
09:20:23 And so we were really disappointed we were like, you know, trying to figure out, well, why did the 40 cs go away and we realized we discovered that there was an actual linear instability in the vortex and anti symmetric mode in the vortex thats related
09:20:38 to an elliptic instability and it would just, you know, you put the vortex in there by hand and it would suffer this instability and would eventually eventually go away.
09:20:50 And at the time we were we were sad.
09:20:53 But, purely by accident, we noticed that, oh yeah but there's this junk up in up off the plane so let me just read you the middle of this simulation is the mid, middle of the desk in the middle of the desert the main claim that this is special because
09:21:08 think about the symmetry in the middle of the desert there can't be any stratification, the local stratification just from geometry must vanish at the middle of the desk right because at the very middle of the desk, you can't have a vertical component
09:21:24 of the gravity.
09:21:26 If there's because the gravity is not due to the gas in the disk itself it's actually the gravity from the protostar. And so just from symmetry in the middle of a disk, there is no gravity from the start the gravity strictly in the radial direction in
09:21:39 at the in the middle of the guests so in the middle of it is there is no vertical component of the part of Stella gravity, which means there is no stratification in the middle of the disk.
09:21:46 And so what we noticed is oh at the middle of the disk where there's no stratification you don't get cut Long live coherent vertices. Oh, that sort of makes sense.
09:21:54 You have no stratification wouldn't expect there to be long live coherent compact more disease.
09:22:00 And there was this mess above and below the midpoint of that is where the gravity, the vertical commander of the gravity was stronger, and at first we just thought it was like remnant turbulent junk, we didn't think there was anything interesting to it.
09:22:12 And then just purely by accident, I decided to take slices through that turbulent junk just to see, you know what's there.
09:22:21 And again it was completely, you know, it was a different day I might have not done it.
09:22:27 And so here's that same simulation.
09:22:30 But doing a slice through the middle, and an A slice way up at the top.
09:22:39 And this blew us away because oh the mid, the vortex at the middle went away. And remember we were playing God we were just putting vertices in by hand and hoping that and we're going to interested in studying them we and we hadn't really thought about
09:23:01 the formation mechanism, you know, that was on our agenda at some point to think of realistic formation mechanisms. But off the me plane to our surprise.
09:23:04 Look at all these coherent vertices, and we didn't put these ones in by hand.
09:23:10 They just sort of appeared.
09:23:13 And, and at first we just thought you know from the side view we thought it's just triggering junk but then when we did these slices we're like, oh my god there's these turbulent vertices, but they're not in the middle of the desk they're off the plane
09:23:23 they're below above and below the midpoint of the desk, and lo and behold, these are coherent and they're long lived, and we didn't have to form that they formed naturally.
09:23:33 And so we were really interested in what what what mechanism formed these off the plane policies.
09:23:39 And originally.
09:23:41 Here's another simulation from the side.
09:23:43 This is actually a boost in simulation because we wanted to simplify the system to see if we could get a better handle on what was going on.
09:23:53 And so, at first, our initial assumption, or initial working hypothesis was that before the vortex we put in the middle of the desk by hand as an act of God, acted as a wave generator it generated internal gravity waves that would propagate away from
09:24:09 the midpoint of the desk, and they would they would go into the stratified regions above and below the disk where they would break deposit their energy that energy would roll up into Florida season, and so we're working hypothesis was.
09:24:21 This is just the vertices that were formed right just be a wave energy produced from the vortex we put it the middle of the disk.
09:24:33 But if you would expected that if you thought it was just internal graduates propagating away from the mid plane as a wave generator.
09:24:42 In the protoplanetary disconnects you might be, you might expect to see refraction of those internal gravity waves and we didn't notice that, and then of our simulations to be noticed that the, we didn't see the internal ground gravity waves refract the
09:24:54 way we expected them. And so, this takes us to around 2013, where when it feels postdocs junction Jang had this eureka moment he said, Wait a minute, this isn't the St Andrews cross with internal gravity waves propagating with MIT plane, these are these
09:25:13 are critical layers of 40 season forming at the location of critical layers and so that set us on a path of looking at forced Barrow clinic critical layers.
09:25:31 And so, this became our new working hypothesis was that you, you know, maybe at some yet in your disc in your in your simulation you have some, some spectrum of turbulence out of that turbulence just, you might get some inverse cascade you might get a
09:25:48 vortex, say, and so that's in panel be you get this some initial vortex.
09:25:54 But associated with that initial vortex on either side of it are critical layers, right where you get a critical layers where you get a, you get a wave, wave speed is associated with the actual matches the local flow speed.
09:26:10 And so, especially with this vortex. There is. There are some critical layers on either side. And what we noticed was that at these critical layers we what we said is they were sort of receptive to receiving forcing.
09:26:27 And so at these critical layers, if you work out the Eigen mode solution for these critical layers. There's region, their regions where there's very strong, up and down willing at these critical errors and, and these strong upwelling and gown willing
09:26:41 at the critical layers can generate vortices it by tapping into the rotation of the system. And you get these die polar vortex bands which would roll up into new vortices.
09:26:55 They would merge. And now, in panel, he now you have a new vortex that was produced in it would it would excite its neighboring critical layers and as it excited its neighboring critical layers, it will generate another generation of vertices and so we
09:27:08 had this idea that Oh, one vortex infects the neat infects the neighboring critical layers turns them into vertices which would, in fact the neighboring critical errors and turn them into four pieces that we imagine a dead zone rising from the dead and
09:27:25 one one zombie vortex becomes too zombie for disease become for zombie for disease as they infect the neighboring critical layers and and just for fun we decided to release this paper on the archive on Halloween, as well.
09:27:40 So, that's so that's.
09:27:42 So that was the dawn of the zombie vortex instability.
09:27:48 And so again, here we're just looking at this is the Eigen mode associated with the critical layer and this is the vertical velocity inside that critical.
09:27:58 This is really stretched out so you can see the x axis is really stretched out, but this is really actually a very narrow localized region, and you get very strong upwelling and down welling and tighten that up willing you down well and you can tap into
09:28:11 the rotation of the system and generate for TriCity.
09:28:14 And I should highlight a, you know, so we sort of hand wave this hand wave our understanding of this.
09:28:23 Neil bone forth and his collaborator. In, 20 2020 released a really interesting paper where they really dived into the theoretical mathematics, you know, using match das and tonic expansions to really study forced nonlinear the nonlinear dynamics of forced
09:28:40 Barrett political leaders to sort of get at what is going on inside these critical layers that allow them to, because normally when you think of critical layers in the non dark clinic context, right interesting, they don't really do anything.
09:29:06 know they form a cute cats, you know Calvin's Catseye but they don't actually dynamically do anything dramatic, but here these forced to bear clinic layers really seem to do something dramatic. And so, here, here, I showed you that picture of the vertices
09:29:09 marching across the domain. Well here's an actual simulation where you see it.
09:29:14 You can see the, you can see the generation of the die polar bands, critical layers being excited and and and new voices being for me you see them literally just marched across the domain.
09:29:27 Critical layers getting excited, rolling up and afford to seize triggering infecting the neighbor and critical layers, generating new vertices and, etc.
09:29:41 I should comment that a lot of folks were skeptical when we first released this work in the mid, 2010s, because people have been studying protoplanetary discs with numerical simulations for decades and net no one had ever found the zombie vortex instability
09:29:58 so why did we find it Why did other people miss it.
09:30:03 And part of the reason was, and I think part of it is we were using. You know, very high resolution spectral methods which you need to resolve those critical layers.
09:30:13 And you also need the right spectrum I needed additional turbulence, you need you need a broad spectrum of initial turbulence.
09:30:22 To to trigger and interesting enough a lot of the initial you know when people were doing simulations of proto player discs, through the 1990s and 2000s, a lot of times they just ignored the vertical stratification because they didn't think it was very
09:30:36 relevant for what they were doing. And so, Obviously the vertical application is very, very important.
09:30:45 Um, I just want to mention in terms of layering so I want you probably saying how does this, how does this talk connect to layering and so I want to show you the layering phenomenon that we observed.
09:30:54 So, in 2018 we started to investigate different types of verticals. Again, we didn't want to just focus on simple models of stratification so we wanted to really see if this does this work and non uniform stratification if there is there's something special
09:31:07 about the stratification. And so here's a very high resolution simulation of the zombie vortex instability that I want to show you. and this will make a connection to layering.
09:31:21 So I'm just going to run this simulation you see it go.
09:31:24 White means no vortices it, you can see the zombie vortex instability, generate more disease.
09:31:35 And you see the creation of these zones these red zones. And here's where it gets really interesting this blew us away.
09:31:43 We're getting bursting phenomena.
09:31:46 So we're getting so there was the. So, the, you know, an early simulations we never let it run out for thousands of orbit so the idea on this paper in 2018 we decided let's run this thing out for literally thousands and thousands of orbits and see what
09:32:04 happens because we hadn't done that yet. And that's where we noticed that, oh wow we get this zonal flow right so I'm plotting participate, there's a vertical component of the vortex city, this is a full 3d simulation, obviously.
09:32:16 But you, you get this. The zonal flow creation, embedded in the protoplanetary disk.
09:32:23 But then we also get this bursting phenomena where it goes through phases where it gets it gets very limited.
09:32:30 And then it bursts, with more vertices and then lemon arises and it bursts, etc.
09:32:37 And we didn't, we still exploring that we just we that bursting phenomena we just discovered in 2018 and we're, we're investing which still trying to wrap our brains around, around that we see is one more example of that I'm going to finish up in the
09:32:55 next couple of minutes.
09:32:57 So no amount of time, but here I'm also plotting. It's the same simulation and plotting vertical component unfortunately but now you're seeing the almost applauding the vertical velocity field, as well as the fractional potential temperature anomaly.
09:33:12 So you can get, you can see some how some of these other these other parameters, very.
09:33:19 But yeah, you can you can clearly see that that bursting phenomenon.
09:33:24 And then here's some still stills of that movie. And so here I'm plotting the kinetic energy. And you can really see you know If so, you know, initially, the kinetic energy is there, other than this, this is the connectors you can talk with this year.
09:33:38 So, initially you had no kinetic energy or very little kinetic energy it grows as the zombie for disease takeover. And you get your initial zombie apocalypse.
09:33:48 And that but then here's where you get that real amateur ization spike of new vertices real m&r ization spiking and forces, and it really seems to be like.
09:34:01 It's a really
09:34:05 has a real well defined period here, which which really surprised us and so then I'm also plotting Mach number. You can see this is this really is a because you know in production and cares People always ask oh is this supersonic flow or sub sock and
09:34:20 can see this is this is definitely the background she is obviously very supersonic but this is substances sub Sonic structures embedded on top of the background here and then I'm also plotting Rossby number because, you know, we can from geophysicists
09:34:35 you might want to say, Well, how does how does the vortices virtuosity in these vortices compared to the rotation the system and. And so you can see the, the, the, the Rossby numbers not tiny, but it is, you know, less like less than one.
09:34:51 And then you can see here's the this turbulent bursting some still shots
09:34:58 of that of that process.
09:35:02 I'll skip that I'll skip that. Um, oh, as a one of my final slides, I'll just mention that.
09:35:12 Can I just say Joe you're just watch the time a bit, just one more, I'll finish in the moment my final minute. So this is just the background, this is just you can see the layering the zonal flow structure.
09:35:25 And, Oh, I won't even get into this.
09:35:30 We did we, you know, one thing you may wonder about is, these are very, very thin small critical areas a very thin small structures, you may wonder well what about dissipation does that kill off the phenomenon.
09:35:41 And so we spent a lot of time in our 2018 paper going through the radiative transfer to the cooling mechanisms and to see to make see if the, if the Zvi survives if you add cooling mechanisms to dissipate the temperature in the critical errors, and in
09:36:01 for all realistic scenarios we still get CBI, but I'm happy to talk about that offline.
09:36:06 So I will go in there.
09:36:10 Thank you. Great. Thank you gentlemen so interesting talk.
09:36:15 We have time for some questions.
09:36:22 Pat.
09:36:24 Thank you.
09:36:27 I was curious, two points. I mean, all this is great but you have a disc, and the first thing you got to worry about in the desk is the creation process.
09:36:38 So what the bait is the base accretion process here basically this idea of the MRI working in the streams at the at the the surface of the disk.
09:36:52 Yeah, so that's the that's candies layered model, or the MRI and so yeah you may have.
09:37:01 You may be at the surfaces, you might have some accretion, but within the dead zone, the question is for sure.
09:37:13 I'm saying what what do you think the baseline accretion mechanism is.
09:37:15 Um, so that was one of the things we we did investigate is we thought perhaps Zvi can be contributing to the accretion
09:37:26 by.
09:37:29 And I, that was one of my last slides, we looked at how angular momentum was redistributed due to the CBI because that equation is due to angular momentum redistribution.
09:37:42 Yes.
09:37:42 And so
09:37:45 what's the balance of value you sort of anticipated my question I mean it. In other words, your zombie thing will I think it will, it will transport angular momentum.
09:37:59 And now the question is in the sense is it going to be inward or outward right so and are we showed in our 2005 paper that that we do get outward transport of angular momentum so that you do get some mass flow in, but there's some sad from the zombie,
09:38:19 yes. Okay, you It could you explain how that comes about, please, um, we're not 100% sure it has to do with the fact that you have the, you basically have this.
09:38:31 You know the domain fills up with these interacting vertices that just happened to have the right correlations, understanding why the correlations point the way they do is, it's not at all clear because it could go one way or the other and you don't necessarily
09:38:46 a priori know we don't actually have an a priori theory for why the correlations lined up the way they do due to CBI.
09:38:56 Um.
09:38:56 The other big caveat to this whole issue is, and I don't think anyone has a really good handle on this but whenever you're doing local simulations local sharing box simulations because you need, you want to really have all your resolution on a very small
09:39:11 box to do the implemented transport connecting that local simulation to the global angular momentum is is not really well understood how you do that correctly.
09:39:21 So my answer my question was the phase correlation and seems you don't know yet.
09:39:27 Correct. Okay, there is there definitely does seem to be a correlation that leads to our transport of England mental but why why that is it's not clear yet.
09:39:43 Hey, good.
09:39:43 Thanks. This is a really cool Talk. Thanks for sharing this right, I sure just had a clarifying question.
09:39:49 When you were talking about these critical layers for the the you know waves interacting and critical errors to generate these before to seize the thing that I didn't quite follow was when you know when I think about critical areas I think about, you're
09:40:02 saying that it's a matching between the face speed of a wave, and the local flow speed.
09:40:09 And I was confused about what that wave was and like what equilibrium, it's an Eigen mode of.
09:40:16 And I could just, could you clarify that as the question makes sense or let me just quickly jump back to that slide.
09:40:25 Here it is.
09:40:26 So this is delta em. I mean it's little Delta sub mm is wave number.
09:40:32 Little delta is the separation of the, of the critical layers.
09:40:38 And so these are these are not your usual bear tropic tropic critical layers, these are these are bear clinic clinic critical layers.
09:40:47 And so, you actually have, it's not just a local wave speed but it's also a coupling to the brand facelift frequency the local brand violet frequency as well.
09:41:02 And I see Neil has his hand up so maybe I'll let Neal's the expert on the critical layer stuff and go let Neil jump in.
09:41:21 You have to answer a question first Neil It's the new rule, many are allowed to ask.
09:41:21 Yeah, the, I guess, with the question was what is the critical error, and what's the Eigen function. That was the question.
09:41:32 Or I can function is defined about what background state.
09:41:38 And why does what is the critical error move around, sort of, what, what I'm wondering.
09:41:52 So it's this is not a classical critical way, right. So it's not where the mean flow speed equals the face speed. It's where the mean flow speed equals the face speed plus or minus a characteristic internal way speed, which is related to the horizontal
09:42:14 wave number and the local buoyancy frequency. So, and then you have you have to have both a vertical wave number and the horizontal wave number. So you get this funny displacement of the critical as in both the horizontal space in the vertical space.
09:42:22 And so that that's sort of why you get this, this curious pattern where the things as Joe was saying, you know, you see these vortices displaced sideways and then after the mid flight.
09:42:36 It's very important. But otherwise, in terms of of a linear stability problem if you look at the linear way equation for this configuration, you've got a linear shear in the horizontal and vertical stratification.
09:42:48 And there's a singularity there that comes up and and it's at these positions and and you know if you look at the local structure of a linear solution then it looks similar to the kind of singular most that you get with classical critical as but that
09:43:02 that offset by this this characteristic gravity wave spirit. I don't know whether that helps. It does. Thank you. Okay. And can I can I guess my question, you know.
09:43:16 So, you know, you your final simulation that you showed quite a few details on that that was specific to this case where you're really thinking about it being an accretion desk.
09:43:29 But, you know the the original one that you showed from what 2013 that that was where you had something quite generic right you've got a linear share flow.
09:43:41 And so it's it's horizontal flow it's horizontally shared and then you've got vertical stratification and that's all it is. Yes, which makes it sound as though it ought to be much more prolifically observed than it seems to have been so I guess my question
09:44:08 or somewhat provocative question is, is you know Why is this not seen anywhere else.
09:44:06 Well, alright so that's a that's a fantastic question and so one of the things, you know, one of the questions we we got was that is this just all some weird artifact of our numerical algorithms is, you know, could have Has anyone else observed it in
09:44:25 a completely different orthogonal set of computer simulations.
09:44:30 And so we worked with folks who weren't using spectral methods using a find a good enough finite volume method. Basically the Athena code. And we gave them the prescriptions we said, you know, do this set up your simulation with this horizontal share
09:44:50 with this stratification with these, you know, we gave them the parameters we said, use your completely different methods and but we'll give you their prescriptions.
09:45:00 And lo and behold they found, they found CDI right so it isn't it isn't an artifact of all of the different things we did with our code, and our methods and our setup, you know if you give someone the prescriptions for it, they can do it and so you what
09:45:15 you do need though, and in I think when we're working in an interesting set of parameters space where you're, you have a horizontal shear and a vertical stratification, and those two, but those two were working in a, in a regime where the horizontal sheer
09:45:32 the weight the times that the timescale associate with that horizontal shear is of the same order as the timescale for the vertical oscillations and I think that's a that's a range of parameter space that doesn't show up saying planetary atmosphere so
09:45:46 that's why people in say planetary atmospheres weren't looking for it because you generally as I mentioned earlier, generally in the geophysical context you have this very well separation of timescales and here we're working with in a regime where all
09:46:07 timescales are aren't the same water and that's just not a premise that a parameter space that had been previously really explored.
09:46:13 And I think they were there is work, there is work in the geophysical context in, you know, exactly the generic setup that you said.
09:46:17 Actually, but some.
09:46:30 Okay, Phil I see you've got your hand up, we're kind of out of time. If you're going to say, can you be super quick, just so that can. I'm sure I'm with respect to the critical layers that the other thing that's interesting about the critical layers and
09:46:36 upon rent price low frequency and other things is that whole thing can be awesome interpreted as a resonance. If I have like a protoplanetary disk, where I've got everybody orbiting at different speeds so there's a sheer.
09:46:49 If you go look at the separation between two orbits is a certain distance where the the two, two lumps of something going with the flow would would be it's the same as neutral angle going around the disk.
09:47:03 if that if that separation time is the same as the brunt Vizsla frequency that is that is when you get this critical layers to be energized so you can interpret the whole thing also in terms of resonances.
09:47:17 That's all I wanted to say.
09:47:20 Okay.
09:47:20 Thanks, Phil.
09:47:21 Thank you Joe we better move on. that was very interesting presentation.
09:47:27 Okay, so next we have one cell phone. It was also not given a talk in this meeting so Antoine did his PhD at MIT. He's now a professor in the current Institute, he's an expert on numerical methods for equilibrium in plasma dynamics of fusion devices but
09:48:00 he's also got wider interest in plasma turbulence, and he's going to tell us about some of those today so please and thank you very much, David for this introduction, because it's exactly what I wanted to say that I'm in some ways that don't belong to
09:48:07 this conference at all.
09:48:10 conference at all. But I'm very glad I got invited to this session or whatever we call it staircase, because I recognize many names I know. And because there are indeed topics that are now dear to me.
09:48:22 So thank you very much for the invitation.
09:48:33 And I want to emphasize that this is what I'm going to talk about today is very much the, the work of Nietzsche and the Maya Quran and the way it happened is I contacted nd, to see if some of the methods for into the quantification he had developed would
09:48:39 be applicable to turbulence and plasmas, and as I introduced him to model has I, how's it gonna work a tiny model that I thought we could try his methods on, he discovered what I might call flaws of this model, and that's what I'm going to talk about
09:48:54 today in link with staircases with avalanches, and a coherent structures which also maybe belong more to the session of Neil and co tomorrow.
09:49:06 So, our setup today will be what you've heard from several people I just want to make sure everyone's on the same page so you imagine plasma and a Taurus and a notorious, it happens that there are gradients of the magnetic field strength, and curvature
09:49:23 of the magnetic field, and particles and such field drifts, and they don't drift in the same direction for positive and negatively charged particles so the, let's say in this picture, the positively charged particles go downwards the negatively charged
09:49:40 particles drift upwards.
09:49:42 And so when you create a temperature perturbation hot on the left school and the rights the difference and velocity, creates a chart separation.
09:49:52 And the result of the charts oppression is the creation of an electric field, and a particle in the electric field and the magnetic field also feel adrift, but this time the drift in the same direction for both positively charged particles and negatively
09:50:05 charged particles.
09:50:07 And so the drift is such that the instability the the initial perturbation is reinforced and creates what we would call drift waves and drift instability.
09:50:18 And so that's exactly what you see in a complicated kinetic simulation here's the top of my can use a life in real life there's no hole here, but for simulation reason there's a whole.
09:50:30 And so you see those streamers that corresponds with those drift instability. And on the left hand side. The, the, there's no instability because the drift arrange so that there's no instability but on the output side there's this instability.
09:50:46 And that's why we will call the primary instability.
09:50:49 So that's the primary drift and stability, but then what we're going to be interested in this talk is a turbulence on top of it, and the behavior of the nonlinear behavior.
09:51:01 So people found out a long time ago that just collision on transport couldn't explain the level of transport measured in fusion devices and instead, you might want to understand the level of transport by random walk by Eddie deck relation time.
09:51:16 And so, then you're diffusion coefficient goes that goes like the anti correlation length squared to the engine correlation time.
09:51:27 And what's this ending correlation time. Well, it's roughly the time to go around the, the Eddie. And so that's determined by this so called he Crosby philosophy that talks about just the slide before.
09:51:42 If you have a potential distribution that is concentric like this electric field will be radio, the magnetic field is towards you and so the equals be velocity is around, and so that determines your scaling and your velocity for the anti correlation time
09:52:01 time and when you work out everything.
09:52:04 You found that the diffusion coefficient in this random walk by any direct correlation goes like proportional with a temperature and inversely proportional with the magnetic field.
09:52:16 So the D is inversely proportional with the magnetic field strength is unfortunate because it means that for good fusion we're going to have to pay more with magnetic field based expected right we're doing magnetic fusion, you have to pay more magnetic
09:52:28 field to transport less.
09:52:31 However, these proportional with T. That's definitely unfortunate but also unexpected it's as the plasma gets hotter in some sense better actually confinement degrades.
09:52:42 And so, as a result of this mechanism in the most modern fusion devices they found the transport is dominated by this type of turbulent driven transport.
09:52:52 And what I just described may be considered as the low confinement mode, which is experimentally verified and the scaling is such that is not favorable for fusion.
09:53:04 Fortunately, people made the lucky discovery now 40 years ago, as they put more heat into the fusion device, they noticed this bifurcation to a better confinement mode called the high confinement mode, which is characterized so the high confinement mode
09:53:20 or the red curves here, characterized with a higher temperature and especially a temperature pedestal, near the edge for both the electrons on the ions, and it definitely pedestal.
09:53:31 So seems like there's a region, as compared to the low confinement mode with transport is improved.
09:53:37 And when they didn't measurement of the electric field they found that the low confinement moon has nothing particularly interesting near this region happening, whereas the high confinement moon has very strong regional electric field which means, both
09:53:53 in terms of magnitude and variation which means. A equals B velocity, and which means here. Okay, so there's strong evidence that the better confinement better transport is associated with Crossfield rotation and cheer.
09:54:09 And indeed, when you run that in simulations. Again kinetic simulations, where you use start to understand what's happening.
09:54:18 If you force the absence of velocity here in your simulations you get those very long streamers link with a primary instability that I talked about in the first slide, but once you allow yourself consistent flows, you see that the sheer flows, cut the
09:54:34 streamers and reduce the essentially the end decoration. So, there seems like to be and I'm going to play a movie. Yeah, there we go so you see what I'm going to call zonal flows.
09:54:45 So, the tremor instability is almost too fast to be seen.
09:54:51 Sorry. Yeah, It's too fast to be seen, and quickly are set up those workflows if you look hard enough, you see the zone. Well, let me move to the other simulations them easier.
09:55:06 So here's a primary streamers primary instability and suddenly the zone, zone will flows, which here the turbulence, and we use a transport.
09:55:15 So what we're going to be interested in of course and we have been for the last month in this conference is the role of zonal flows on this transport.
09:55:28 And very soon I'm going to take this slab approximation, and for many geophysical people you're going to be disturbed because the radial direction is going to be the x direction where the, the direction of the zonal flows change, and the direction of
09:55:43 the zone flows in the y direction which is kind of flipped from your physical dynamic and fluid dynamics site, I apologize for that. But it's a convention we adopted 40 years ago.
09:55:55 The other links phenomenon that I want to talk about today is the existence of nonlinear upshift of the critical ingredient for the existence of the regime with strong turbulence and strong transport.
09:56:08 So when you run kinetic simulations you notice that when, whereas the linear theory would tell you that the critical gradient let's say for this parameter is this gradient scaling for the temperature.
09:56:23 In fact, in the simulation they observe strong transport on the for much higher gradient of the temperature for whatever the parameters are, it's always a verified, and that is called now in our community the demands shift.
09:56:39 And those two things we're going to study today.
09:56:42 Um, and the last thing that I want to discuss is so so sorry I want to explain something here.
09:56:49 So in the demonstration we're going to see that as normal flow as a dominant and there's an interaction between the turbulence of the zone and flows, and it's a mostly nominal regime with a turbulent behavior, interacting with his own workflows.
09:57:03 Whereas, in this regime is turbulent dominated and I want to talk about that the regime too much. It's also fairly well understood base of regime that slightly less interesting from a fusion point of view, we'd like to operate more here of course.
09:57:17 And so I'll discuss more mostly this regime with strong zonal flows and interacting turbulence and not the high determine regime.
09:57:26 So why, what does it look like in that regime with strong zonal flows. When you still do a sea transport, of course Bates, a regime of transport characterized by non tissues, none diffuse of transport, which appears to be scaled free an avalanche mediated,
09:57:43 so I'll talk about that all this is a very interesting paper that I'm citing here, which involves quite a few people on this call and in this conference.
09:57:53 I only get to mention GM because I didn't have room here.
09:57:57 But there's pads, there's a year, and other people, I think.
09:58:02 And so what you see is here I'm plotting a simulation by Toby has grown or kinetic simulation. Here's a radial direction so there's no force will be going up and down, up and down.
09:58:15 And here's a regional direction here's time and you see the heat flux here is plotted.
09:58:20 And you see that the heat flux goes in bursts, and it goes into a, it appears in terms of regularly coherent structures, which made me and people have started to call avalanches.
09:58:33 So we're going to discuss this avalanche and behavior. And this burst of behavior today.
09:58:39 Another view of this is now this paper by long the author and CO, which was another kinetic code. And they also see now is the heat diffusion of it but you can use this as a proxy for the, the heat flux just like the slide before.
09:58:56 Now the regional direction is here, the time is here and again you see those bursts. In the heat flux, and you see those regularly propagating structures that are quite coherent and the novelty I would say compared to this plot, it's the same sort of
09:59:09 behavior, but what I'm plotting at the same time, is that he Crosby sharing rate so the velocity sharing rates, and you see a very strong correlation, both in space spatially and temporarily between the sharing rates, and those regularly propagating structures.
09:59:27 And the final picture of the same story is, to me, the most understandable because it's with a reduced fluid model by event off and co an Oxford group here aren't you see the, the temperature perturbations as a function of radius here's your y direction,
09:59:45 and here's the heat flux, as a function of time and radius.
09:59:51 And so you see the normal structures so you have very strong zonal structures of the temperature perturbations of the interface where there's normal flow is small right because it has to transition from, let's say negative to positive, you have turbulence,
10:00:04 so you have flattening of the temperature profile year, creating those staircases so you have steep temperature flat temperature in the transition region of zonal flow, a strong zone off here again which allows for a strong temperature gradient.
10:00:20 And what they observe is at a certain moment of time of times those normal flows weaken a bit, and solitary structures are created, which run into this more turbulent layer where there's no flows of the weakest create this burst, which corresponds with
10:00:36 this burst until a new zonal structure is reestablished later on. And again you see those coherent propagating structures. and this burst corresponding to the
10:00:49 birth of those solitary propagating structure and their collision with a turban layer.
10:00:56 Okay. And these group has called those soldiers structures Ferdinand's for reasons that I could discuss, just in case I mentioned that in the rest of my talk, you've heard the word for the non.
10:01:09 So here's a right point to ask, maybe one or two quick questions before I I talked about the new model that I want to discuss in case you're confused.
10:01:25 Otherwise, maybe David you're, you're telling me if I should just move on. maybe it's the best.
10:01:30 No one is put anything in there. Okay, great.
10:01:35 Just carry on. Okay. So, ideally, I'm going to be straightforward. Ideally, if you want to be accurate to simulate all this you need to solve it so Dr kinetic equation the very big Connecticut equation very expensive, but people like to have very simple,
10:01:48 when models how's it going me my husband dr Walker attorney.
10:01:52 Because if they're simple to code and it's simple to add or subtract physics, and understand the basic phenomena.
10:01:58 And so there's quite attractive. So nowadays, to understand the systems instabilities on the floor mechanism, and the fundamental properties of the workflows and here's an example of Parker and promise you start with a random turbulence and no one behold
10:02:13 in the hustle me my model zonal jets I created this is radius as a function of timing says only average flow.
10:02:20 Okay. And so, That's the type of model I showed you, Andy and then Andy notice maybe a, an issue with one of these models, and I'm going to show it's key to understanding.
10:02:32 Many of the phenomena just discussed with a more complicated models.
10:02:35 So let me just show. There's going to be to sort of physics heavy slides so I understand, so we understand the modifications and then it's going to be mostly phenomenology that I'll show you.
10:02:47 So it's very important to understand where those classic army mind Marconi models come from is the so called. So let's focus on how's it going Mima, so they take the so called a diabetic limit where there's no resistance to it.
10:03:01 And you assume that the electrons have such a small massive there's no inertia. So in that case in plasma physics, the electron pressure is balanced by the, the electron pressure gradient is balanced by the electric field.
10:03:13 And so if you assume a uniform temperature, your electric field and the temperature can be brought on here into the electric field.
10:03:23 The pressure, the only thing that's left with the pressure gradient is density gradient you can integrate this, and you have that your density is function of space, but only in the x direction because it's a parallel gradient, and is an exponential with
10:03:38 a potential.
10:03:40 Now you're right, everything in terms of the zone mean and the zoom of fluctuation.
10:03:45 And you find the density so it was all mean, and a potential fluctuation.
10:03:53 Okay, which I'm going to tailor expand like this.
10:03:57 So why is there only until the here, I need to explain that.
10:04:01 So if I write mass conservation for the electrons mass conservation right d by the end by dt plus v dot read n equals zero.
10:04:10 It takes this form.
10:04:12 And I'm going to take the zonal average and integrate my parts, I can write it like this, and know the the wider team of the density, following the previous, the previous slide here the wider range of of the density is just some of this form so when you
10:04:30 plug in here you see that I get a perfect derivative. And so, according to mask observation you cannot have a net radio electronic flux.
10:04:39 The zonal mean of the density cannot depend on time.
10:04:44 And so I conclude that my density can always in general be written that way.
10:04:48 And so what we're going to solve for as done the sofa the density of fluctuation.
10:04:53 ie, only this zone only varying until the, which is also fighting in this idiomatic approximation.
10:05:01 So that's very important note net regional and electron flex.
10:05:06 And so there are two has gone me my models, both so for the potential vortex city, but in different form.
10:05:13 The original Hasegawa me my model didn't understand this. Well didn't notice this issue of the net regional transport of electrons, and so it has the full potential here.
10:05:25 In contrast, the so called modified has it got me my model has only the only varying potential here.
10:05:32 And this small modifications has a lot of implications.
10:05:39 So the first thing to notice is this model is not the origin of one is not governing and variant whereas the modified model is Galanin and variant, as we should.
10:05:53 In plasma physics, it's not the same in in a quasi geocentric model but here we need galleon invariance on the the modified model has it.
10:05:58 This model now naturally has no net radio transport of electrons, like I showed before.
10:06:03 And when you do this model you notice that the zoom of flows that are observed a much stronger than in the original model. So, this moment indications has a big impact.
10:06:14 The one downside of the hustle me my models is actually the model has no drift instability.
10:06:20 So, I'm going to show you what that means but first let me show you in practice. So you do, you start with initial states here on the left, whatever the initial states in the modified for the Gumby my models, you're going to get zonal flows out of that.
10:06:34 in the so called Tony how's it going Mima or the orange animal has given me my model, you don't get some workflows, you have to work hard to get them.
10:06:48 So, because there was no drift instability in the house economy model that's built in, you have to if you want to study is overflowing actually have to preserve the system you have to add a stochastic term by hand and people don't always like that.
10:07:02 So the next easiest model on the hustler walk autonomy models where you add a bit of electronic resistance.
10:07:14 Along the lines which has this extra turn as compared to have cigar minima. And this resist diversity is enough to create this rift instability.
10:07:19 So the hustle work attorney models are good because they have no because they do have a built in drift and stability.
10:07:27 And again, that was two models the original hustling or what could be considered roses only average part and the only fluctuating part for both fine n.
10:07:50 And the modified has to go walk a tiny model, notice by new Mata and Dewar notice that this term comes from a parallel gradient.
10:07:47 That would kill, there's only average so the only thing that would survive is a five children.
10:07:53 And so the original hustle go work at any model is not going Galanin in variant, and it's hard to generate all flows, just for the same reason as the origin will husband on me my model, the modified hustle got waka tiny model has the good advantages of
10:08:08 being Galanin new variant, and of creating strong zonal flows, as desired.
10:08:13 Now, however, let us look at the limit of no resistance at so no resistance is equivalent to taking alpha to infinity. If I had explained what alpha was in detail.
10:08:26 So when you take alpha goes to infinity.
10:08:30 For this equation to be well posed you need the density to equal five. And so that means that your potential warp city becomes the same as the original hustler Mima potential participate.
10:08:43 So the original has a dog walker attorney model does converse with the original has to go on me my model and the right limits.
10:08:50 Let's look at the modified half second and work at animal law, however, so their alpha goes to infinity means until that goes to fight Tilda.
10:08:59 But that means that your potential for TriCity still has the zonal average density, which is not the same as the modified Hassan NEMA model. And so the modified has to go work at any model may not converge to the modified how's it going my model.
10:09:17 Let me skip this, I can explain in more detail this issue of convergence, but and you might have noticed an issue, and then they might have realized that maybe you can define a new model that he called the flux balance, how scott walker tiny model, where
10:09:29 you defacto defined that you're going to solve for this potential fortress t involving only until the and not inborn like the modified it has a guy Walker team model.
10:09:42 And when you do that by construction, the model converges to the right model and the appropriate limits.
10:09:49 By comparison, if I sold for the same quantity is the modified has a guy walk up any model, I would have extra terms extra nonlinear terms.
10:09:58 And so the two models and modified Hassan, has a gala waka waka Tony in our new balanced Hassan What can you have the same linear instability properties that are very different from a nonlinear dynamics point of view.
10:10:14 And so I'm going to finish with that and then show you how that impacts avalanches and solitary structures.
10:10:20 So first thing to notice is in our new model. It's a bit hard to see on the slide but you always have zonal flows present, whether you are more turbulent or more organized, whereas in the modified Hasegawa Tony model, when you are very resistant very
10:10:36 unstable unstable you have no zonal flows whatsoever. You don't get zonal flows in the high
10:10:44 conductivity limit and sort of the house ago me my model. So here's the difference, and other difference is that there's no flows while being more robust in the balance costs that go What can you model.
10:11:06 If you compare with the husband Ah, the modifying how's it going to walk a tiny model. So you see that's all model, the old model. There's no flows are more robust as I showed in the previous picture but they're more variable.
10:11:12 Now is the important piece that connects with the first part of the talk is, we wanted to see if we have a nonlinear upshift of the critical gradient for strong turbulence in our model.
10:11:25 And so this may be seen on the left, this is the
10:11:30 transport the particle transports the particle flux, as a function of the density gradient.
10:11:37 And let's see, let's say for instance the red curve has a critical gradient for the linear instability which is point oh one so way below, not even shown in this plot.
10:11:46 And we see that the transition to the strong transport is much, much higher spots. The same for a different viscosity, you see that whatever the parameter we have a strong shift between two types of transport, sort of a demonstration and a strong turbulence
10:12:03 regime.
10:12:04 This is, this is another part that shows the same thing, except now we compare with the old modified how's it going to walk a tiny model, which has has agreed you'll increase of the transport there's no bifurcation between the demands regime, and a strong
10:12:18 preference for him, like in our model.
10:12:20 So the electron dynamics and now fixed that we provide is the condition in these models to achieve a demonstrate the true dimension.
10:12:29 Another thing that we observe is with our electron dynamics as compared to the modified Hasegawa Qahtani non plotting the political flex as a function of time.
10:12:38 This is radius.
10:12:40 In the middle you see those bursts of avalanches, but near the edge of the domain you see that in our model we have those solitary coherent propagating structures, which is a definite sign which is correlated with the sign of the, of the sheer.
10:12:56 I'll talk about that shortly, but you see that in our model whatever the parameters we have those solitary structures radio coherent structures, appearing like in the more complicated simulations and are completely absent in that model that does have
10:13:10 a different electron dynamics.
10:13:14 The last thing that I want to talk about is within our model, we can also lose those solitary propagating structures, those coherent structures, instead of if we apply periodic boundary conditions so let me explain.
10:13:28 So here in those situations we in this situation I showed before we had this so called channel geometry, which you can you can imagine we have was on either side of the simulation.
10:13:40 And because of this channel geometry if you look at the zonal profiles the function of radius here's the zone of velocity, near the walls, we create very strong sheer has you can observe here.
10:13:52 And this exactly and those regions.
10:13:54 And the regions of strong here that we have the coherent structures peering.
10:14:00 And in the periodic boundary conditions I'm not pulling plotting the velocity profile but we wouldn't have the strong cheer.
10:14:06 And that's the reason why we do not see the solitary structures.
10:14:12 So I'm done with my talk, so I said that transport and topamax can be separated into regime, so to say. A more organized chemistry team, and a strong determinant regime, that's less favorable for fusion.
10:14:26 In the dermis regime many, many people have observed non diffusers transport with periodic bursts, and the particle and heat flux is avalanches and coherence only three structures.
10:14:38 And I've presented the only known Hasegawa walkathon model, which has a demonstrates, which has avalanches which has caught here in solitary structures.
10:14:55 treatment of the parallel electron dynamics. But that's not enough, even with the correct treatment of the parallel electron dynamics, strong velocity fear is absolutely required for the existence of those coherence auditory structures.
10:15:07 And that's it for my talk. Thank you for listening,
10:15:14 and keep on trying. Interesting. Cool.
10:15:21 This is really cool to see this shift is always been kind of a really cool phenomenon it's exciting to see that there's all of a sudden, seems like at least three different groups, trying to come up with good models for it and this one's really convincing.
10:15:35 I, I've never worked directly with Tesco and even will continue equation so forgive me if this is a dumb question.
10:15:42 I know that I was talking to, what's his name, Darren Ernst.
10:15:47 What is has been really interested in how the ion mass.
10:15:54 You know this isotope of fact how that affects the demonstration, I think, and is there an eye on mass that you can vary in this model of yours.
10:16:03 And if so, is that something that you've looked at to see how the damage shift varies with the ion mass in this model.
10:16:11 Um, so, in the model itself know. So in those simple equations whichever you take.
10:16:18 Oops.
10:16:18 Whichever you take whichever hustle go workout tiny model, you take some hard stuck. Okay.
10:16:31 Yeah, whichever model that I in my us is not really a parameter. Um, what I could imagine is.
10:16:45 this equation from first principles I could maybe see where this comes in. But I think unfortunately those models are too simplistic for that. Sorry about that.
10:16:50 Thank you. Yeah, okay, the plasma guys are lining up.
10:16:58 I sign it, I signed it each and every one of you, I send you to each and every one of you
10:17:04 surprised when you mentioned the beginning, the word by, even though, right, which is, which is also two dimensional model. Now if I remember where the Oxford group claims that the image shift is hard to say related to the so called finite glamorous effects
10:17:24 which is, which in your mother would be, you know, considered consistently replacing cute about history by troopers plus density. So it's bit more complicated, so So did you try to look into the differences and commonalities between those two models.
10:17:40 Yeah so so that's an excellent question so I really like this word by a by even if I have to say and so we're actually computing so they claim that the damage shift is determined by the competition between the Reno stress and the magnetic stress, if you
10:17:57 will. And so, I wonder if there's the same yeah we're looking into whether there's a same competition between.
10:18:04 For us it's not a magnetic but the grad n stress if you will create enough density stress and the renewal stress. So we're looking into that but I don't know the answer yet.
10:18:12 Yeah.
10:18:14 Okay, thanks.
10:18:17 I'm not sure who was next I think it was gear anyway let's go with him.
10:18:25 I think that was no.
10:18:25 Okay, I know Pat was just on top of the list always just because he's on top of the list. Okay, go on pants off you go.
10:18:33 I have a couple of clarification then a question. I mean, your points, as I understood them was that wells, the drift waves and the zone flows obey different equations.
10:18:51 You should be careful not to bring the congregation along in the fluctuation P.
10:19:05 Yes. So, the first point and the second one, I would rephrase just to say that.
10:19:13 I think structure, it doesn't have to be do with channel forces, I think, strong shear is necessary for those coherent structures, but you may mean demanding, depending on the models you may get strong shear with periodic boundary conditions, right I'm
10:19:25 not making your point there. Yeah, we made a difference. I mean I my first comment is I'm sorry. All of those are well known. Okay. People don't write papers about these things they just, you know, you can find papers where it's done right, and there's
10:19:40 no shortage of papers like that. Yeah.
10:19:46 So I mean, I I'm curious what the point here is second point is you talked at length about sheer but since you go to has a gala waka Tani. There were, you didn't tell us anything about the congregation, and in particular the zonal congregation is quite
10:20:05 is quite an interesting business.
10:20:08 And even in the large alpha, or a diabetic regime, you can have significant zonal congregation but of course if you run alpha at the edge. Alpha may be above one but not always by much and then the zonal congregation is significant, you have you have
10:20:27 any comments on that.
10:20:31 I'm, I'm not trying to understand the.
10:20:34 If you have a zonal mode zonal flow that we babble about you can also have a zonal density perturbation with the same symmetry that's a congregation. That's right, and all you need to have an avalanches of propagating correlation when you think about
10:20:50 it so propagating congregation. So it's have some interest to understand the congregation so I was curious if you had any comments on that. Yeah, well, so we do observe that I didn't put it here, we do observes the coordination of the identity profile,
10:21:08 that's very much stronger in the channel geometry than than the periodic geometry, surprise, yeah I mean, Yeah, and I agree with you that nothing here is a revolutionary My only point is that you may not have to do more complicated model than half ago
10:21:26 or continue to study all this, but that was my, my main point.
10:21:30 I don't think has ever shown in the literature. Well I think every guy shame not everyone writes the obvious right.
10:21:40 I understand the obvious thing, but final comment I just have to make the first papers on avalanches in this business where there's sort of two one from a spreading perspective and one from an avalanche ng perspective where was that VA in 94 and diamond
10:21:59 and harm and 95.
10:22:01 Okay, so you're, you know, the these ideas have been around a long time and I think were derived before the age of proof by color picture in simulations just a remark.
10:22:14 Okay. Yeah, no, I appreciate all this, all this I agree with you. I remember my, my point is eventually to use those models to do uncertainty quantification i'm not i'm not saying that I'm going to teach you about avalanches I read about all this, or
10:22:29 one of the simplest, I just want the simplest model for which I can do answer any qualification that's relevant. And I haven't found any other model than this one, that combined simplicity and those properties.
10:22:46 This is my only comment. Whatever. Thank you.
10:22:52 Yeah.
10:22:53 I also had a question in the paper that you were showing. Is there a in the way you walk Easter they describe or you understand thing between the, the generation of these birds and the onset of the inflation of the layer, because, looking at the first,
10:23:18 second and the fourth layer there's also in very static almost enter term, and yet you have this very big events occurring in the middle so what's the causality of all of that is that the diversity is necessary to create the layer or what role does it
10:23:41 play yeah so what they argue, what they are you is that the zonal flows are weakened by viscosity, and the weakening allows the formation of the solitary structures that then collide with the turbulence, which, again, regenerates the solo flows so I think
10:24:02 that's a cycle, if that makes sense.
10:24:05 And so the burst is a consequence of the fact the burst happens because as normal flows have weakened. Allow the formation of those so called Ferdinand's if they want to call that that.
10:24:17 And then the burst happens, and once the turbulence has regenerating the normal flow you're back in your general cycle.
10:24:26 Okay, so I guess. What year is it.
10:24:37 I was guessing it was almost four or five average or here or whatever that's maybe not what the show here.
10:24:55 What is it sorry I don't hear you so well.
10:24:48 The, the, the, the color.
10:24:51 The for one on top.
10:24:57 perturbation of the temperature, delta t.
10:25:00 So, Redis hot blue is cold.
10:25:05 And this is the heat flux in the bottom.
10:25:11 Okay.
10:25:13 Okay, well thank you very much.
10:25:18 A low time.
10:25:19 Yes.
10:25:21 My question was regarding magnetic shield, and I just sent these up, Sheila simulations, but do you have already or. Do you have plans to look at adding magnetic share because that's of course known in the context for example of internal barriers to condense
10:25:35 the flows and also possibly generate vocal avalanches.
10:25:41 Yeah, exactly. Yeah, I we agree with, with you and Tony and we have a plan to to generalize the model to include magnetic here so we're looking into that now.
10:25:53 At first we want to do the Z pinch geometry like given us, but then we realized this would be missing precisely. So, I'm. We're looking into sort of a school bench geometry.
10:26:06 right and then especially in the edge of course to make it took them are relevant but you have very closely spaced rationales by then of course the, the radio skills for both the staircases and the, the avalanche absorption of our generation would be
10:26:20 likely very much dominated by the boundary conditions from the magnetic geometry. Right.
10:26:25 Yeah, but then I would agree with Pat, you know, Pat diamond what he just said that. I don't want to make a career inventing a little models that people know.
10:26:34 The idea was more to find the simplest model that has interesting dynamics. And now to do a, see if we can do on turn the quantification on top of that.
10:26:43 So I agree with Pat that I'm not sure how much promise there is to, you know, add little terms here in there, because people have done that for a while so since, since my name was taken in vain.
10:26:55 Just a quick comment here and a million people lo thar have done has to go and walk the tiny with magnetic here. So again, there's, there's nothing really new.
10:27:06 I can send you some references if you're interested. Okay.
10:27:13 Right.
10:27:14 One final question for Petros, please. Yes.
10:27:21 Can we go to the that bifurcation diagram that you showed showed with a cup.
10:27:27 I was wondering why so slow loading. Yeah, there we go. Yes, here it is. So this happens at the cup Baikal's one, how does this depend on the on the on the irresistibility on the on the on the on your on your dissipation parameter.
10:27:44 Yes, the threshold depends on alpha right so it's company called one alpha alpha alpha is changing or alpha is the same for these two simulations. So here are 5.5 on the left.
10:27:57 Your.
10:27:59 I'm talking about this graph here, where are there's one here from top up.
10:28:03 No, no, this one. No, the left one.
10:28:06 Yeah. Your eyes fixed here alpha is fixed so cup is changing
10:28:14 transition happens for lower resisted. It happens a lot slower.
10:28:21 It's as you reduce some you knew is a viscosity is a dissipation based on resistive at that's why I was gone. Okay, the viscosity so the more the more viscous it is essentially the later.
10:28:36 So, this type of transition sort of gets slower. That's correct. Now, how does the, how does the flow change here at when this transition takes place I mean.
10:28:46 On the left you have zonal flows and on the right, what happens. Yeah, so on the right you start to have a much more war to city, and let's see if I could plot.
10:28:58 If I could show you an example. It's, it's akin to this transition here.
10:29:02 Why is it low quality, not sure it's loading, I think.
10:29:25 So I.
10:29:25 The zonal flows are still there we go there's all flows I still present, but there are much weaker and slowly but surely you would reach a region where regime where you only have a vertices.
10:29:24 Okay.
10:29:25 Okay. Yeah. So, this is the transition, if you will, you could view it like this, right. So, here you have Strong's overflows here you have a region with fewer zones.
10:29:38 And then you get to recurs overflows. Yes, but by the way, but as as as as with the World Cup as a parameter here I see the parameters, alpha.
10:29:48 That's true, but Kappa Alpha really a symmetric rule if you think of this here.
10:29:53 If I keep Kappa fixed and achieved alpha i get the same type of bifurcation.
10:29:58 Okay.
10:29:59 Yeah.
10:30:00 So, by the way, something that maybe pat me comment on. I apologize for mentioning his name. I think one thing that's not known is a predictive theory for determining the scale of this all flows.
10:30:13 And I got to think of this because here across the transition here I have several, several jets here have two dominant jets several jets again, that's a question that I think is still open and for which we're happy to have small, reduce models as well.
10:30:29 Okay I yeah thank you. I didn't want to provoke packed into into.
10:30:38 Just because out of fairness to Adrian thank you I'm trying to get Yeah, thank you very much.
10:30:44 So I'm sorry Adrian that somebody's late, but anyway, so it's a pleasure to have us our final speaker.
10:30:51 Adrian praise.
10:30:53 Adrian has been an enthusiastic participant in this but he hasn't actually spoken to give a talk yet so Adrian did his PhD in medicine.
10:31:05 He's now a postdoc fairly recently in Pascal's group and he's going to tell us something about his PhD work I think on sheer Sheer Glow instabilities.
10:31:16 Yeah, let's see if I can figure out how to share screen. Everybody see my title slide here. Yep.
10:31:22 Great.
10:31:23 So, thanks David. Appreciate the opportunity to talk about this stuff here.
10:31:28 It's not directly related to, it's not directly staircase related, but it does relate to, quite a few more directly circuits related talks that we've seen in the past few weeks so hopefully I can tailor this to be relevant to the audience here, like David
10:31:49 said this is basically my PhD work they did back in Wisconsin.
10:31:52 So wouldn't have been possible without my advisor is Paul Terry Ellen's Bible, as well as close collaborator empty official and a couple students now jack Schroeder and industry path he worked on this as well Ben dash is basically carrying this forward
10:32:07 as his PC project now so if you've got a problem with the work, maybe take it up with been dash Just kidding.
10:32:21 The title is capturing negative turbulence viscosity and reduce models of unstable sheer flow so what we're after is reduced models that can sort of capture some of the negative turbulent viscosity or the Canterbury was and transport that we see in sheer
10:32:29 flows, occasionally.
10:32:32 So, I think I don't need to motivate reduced models of turbulent systems to this audience too much but let's let's talk about a little bit anyway so.
10:32:43 Here I'm showing some simulations of a common holds unstable share layer and an HD.
10:32:51 The color is just a die to visualize mixing but the initial condition is top half the domain flow goes to the right bottom half the domain flow goes to the left.
10:33:00 And so we have this earlier in between it's a hyperbolic tangent velocity profile.
10:33:04 And this is unstable couple of Hubble's instability, we can see these characteristic kh pillows forming. This is stratified by the way.
10:33:12 So you see these, you know, typical cage bills forming and eventually you get this turbulent system the magnetic field disrupts the vertices that form.
10:33:23 And we get a bunch of small arrange terminals across the range of skills that enhances transport so the turbulent transport in the system goes well above the, you know, diffuse of transport the molecular infusions and and viscosity.
10:33:42 And this is why we care about.
10:33:46 And this is why we care about, shear flow turbulence because this turbulence can enhance mixing beyond molecular diffusion and all sorts of systems that we care about.
10:33:57 And it's difficult to predict, of course, we we try to use direct numerical simulations to figure out what the turbulent transport is in some systems that we can simulate.
10:34:07 But of course, many of the natural systems that we care about such as say stars or other astrophysical systems. The there too complex, or the rentals numbers are just way too large for us to be able to simulate them directly and accurately.
10:34:21 So, we need reduced models that we can you know use to predict trivial and transport in the systems that hopefully we can test the models and numerically tractable regimes you know with with moderate rentals numbers and then we can extrapolate those models
10:34:36 to higher rentals numbers. So that's the overall motivation for this work.
10:34:41 We're not claiming to show you know we're, we're, we're pursuing tools for this models.
10:34:49 And we're looking at systems that other people have looked at before and we're not disagreeing with with other people's views of the systems but we're trying to take a new approach in pursuit of reduced models.
10:35:01 So one motivating example that I find quite interesting.
10:35:06 If you have the same system. So in the previous slide one of these unstable sheer layers, and you add a horizontal magnetic field, let me go back to the previous slide.
10:35:17 If you have this sort of unstuck agency will share later and have a horizontal magnetic field that magnetic field, even if it's week, it'll it'll stabilize the instability just a little bit, it will reduce the growth rate and instability, just astronomer
10:35:32 magnetic field it will suppress the instability altogether but if it's a week magnetic field it'll reduce the growth rate.
10:35:38 And if you have instability different turbulence, you might think that, in effect that reduces the growth rate of the instability will then reduce the subsequent turbulence.
10:35:50 And that's not what seeing here so on the left, this is simulation from plot at all. I think it's Astrophysical Journal 2008.
10:35:59 This isn't the only people that have shown this but it's, I guess the first one that I was aware of in the and magnetized case, if you have no magnetic field, just to decompose instability, you end up with a coherent vortex in the middle grayscale here
10:36:15 is density. This was a compressible simulation. And these little arrows are the velocity field.
10:36:21 If you add a week magnetic field, this vortex gets disrupted and you end up with a whole bunch of small scale turbulence and much more turbulent transport as well and this is, you know, arguably a little bit counterintuitive, from a certain viewpoint
10:36:35 just because you're reducing the growth rate with this magnetic field, you're increasing the insulator turbulence you're enhancing the turbulence.
10:36:43 And this is something that we like to get a better handle on.
10:36:47 This has also been explored by in this paper by Matt Griffis and Hughes and 2017 I think AFM where they looked at a very similar system, it was incompressible.
10:36:58 And they plotted it here that sort of similar result, where they're plotting the East different panels a different magnetic field strength so each different panels a different simulation.
10:37:12 And what they're plotting is the main flow profile a bunch of different times so the different lines are different times in their simulation, the black line is the, is t equals zero and this is just showing the flow profile for the sheer layer so bottom
10:37:26 half the domain we have flow going to the left, top half the domain we have flow going to the right, and we've got this sharp share layer that broadens as time goes on, the different fields that the different parameters here and this is basically the
10:37:39 magnetic energy divided by the kinetic energy of the equilibrium, roughly, so very very weak magnetic field here still a week magnetic field here but not quite as week.
10:37:51 And you can see that in this stronger field case, as time goes on the shear layer gets broad and much more dramatically.
10:37:58 So there's much more turbulent winds and transport in this simulation.
10:38:04 Even though it's it's the field as a stabilizing influence and really a negligible stabilizing influence on the instability, this is not something that you could predict with, say, a quasi linear model.
10:38:16 This this sort of scaling wouldn't emerge from that. So, we want to figure out, you know, you have these trends in case you're from turbulence that can't really be captured by linear arguments alone.
10:38:29 Can we use other tools that have found success elsewhere to understand sheer flow turbulence. In particular, we're going to be exploring the role of stable modes in the system, I'll explain what that is on the next slide.
10:38:41 But this is something that we're borrowing from some work in the fusion community where, where they've used the sort of a model to study terminal transport in fusion plasmas, and we're going to try to apply that framework to share flows.
10:38:57 So, what do I mean by stable modes.
10:39:00 This ties in a little bit on refreshments talk, a week or two ago.
10:39:06 So a result of that's been known for a while, I think I first read about it in. In, Drazen and reads books from like the 80s.
10:39:13 If you haven't invested sheer flow, it can be stratified or stratified.
10:39:17 If you have an unstable envisage share flow, then for every unstable mode for every exponentially growing solution to your linear scalability problem.
10:39:26 There's a conjugate stable mode.
10:39:29 So, a mode at the same wave number, but different flow structure that is to King exponentially time so you have an unstable mode and you have a stable mode.
10:39:40 And this is tied to PT symmetry of the Euler equations. And that's described a bit in this archive prevent that I came out recently, presumably it's going to be published soon.
10:39:53 What I'm talking about is shown here, this is an equilibrium, you know some unsafe kh unstable flow profile that we studied this is our first paper on the subject.
10:40:07 This is an unbounded sheer flow top half the domain again flow goes to the right bottom half of the domain flow goes to the left and we have this sort of finite transition region in between.
10:40:14 and. This is a while steady flow profile, it's got unstable eigen modes for a range of wave numbers.
10:40:23 Here I'm plotting at cables point for the unstable motor the system.
10:40:28 fi is the stream function and planning streamlines the flow every time you use fine this talk I'm talking about a stream function.
10:40:36 And the Eigen mode of the system looks like sort of two aligned vertices like this, sort of, you know, co rotating vertices I guess that have a well defined orientation.
10:40:52 At that same wave number so at the same large scale.
10:40:56 k is the wave number in the direction of flow so kx at that same ku also have a stable mode.
10:41:03 This is a perturbation that decays exponentially in time in the linear regime.
10:41:08 So for every unstable mode like this you can sort of do a reflection of it and you'll get a stable mode the decay is exponentially in time.
10:41:16 Now, this is these are invested modes, so this decay is not a viscous dissipation.
10:41:22 Instead, what's going on is that this unstable mode is driven by, you know, the energy production term is basically the rental stress and it's drawing energy from the sheer flow from the background shear and consequently the stable mode is the reason
10:41:38 to traditionally time is because it puts energy back to the background share flow. So, one mode of corresponds to energy transfer from the main flow to the fluctuations.
10:41:49 The other mode corresponds to energy transfer from the fluctuations back to the main flow.
10:41:55 And they have equal and opposite growth rates, and they're the same wave number. If you write this is important if, you know, if you do one of these typical simulations where you have some unstable equilibrium and you seed instability with some random
10:42:10 perturbation some white noise fluctuations. Then, in general, both of these will be present in your initial condition, but presumably the Stephen mode is decaying exponentially in time, if you're fluctuations are small and you can assume that they evolve
10:42:24 linearly. And so you might ask who cares about these stable modes right if they're decaying exponentially from a small amplitude, then then why do we care about them.
10:42:34 Well, for one thing, there's evidence of them in experiments.
10:42:38 I'll show leader and simulations as well but before I get into this figure here I want to note something that may be familiar to many people in the audience but if you have the sort of energy transfer from the main flow to the fluctuations or from fluctuations
10:42:52 to the main flow. That's going to be equivalent in the system to avoid some transport or rental stress so if you're taking energy from the main flow to the perturbations that's going to correspond to some positive turbulence at viscosity that's going
10:43:07 to be a downgraded Wellington transport. And that's going to correspond to a broadening share layer.
10:43:13 On the other hand if you have energy going from perturbations to the main flow serve and negative energy production.
10:43:18 That's going to correspond to a counter gradient wants and transport or negative viscosity, or different opposite, I don't stress, that's going to shrink the shear layer.
10:43:28 And this has been seen in experiments of the sort of specialty developing share flows where they have, they inject two different flows at two different speeds, and then you'd have sort of a spatially evolving shear layer, away from the injection point,
10:43:41 it's showing the picture here is from review paper from the 80s.
10:43:47 Here they're showing. This is not bad it's sort of a schematic of the with the sheer layer as it evolves, away from the injection point so it broadens and then stagnates, and here they're showing the rental stress and you can see that initially you have
10:44:00 rental stress of one sign, corresponding to energy transfer from the main flow to the fluctuations. We're broadening the sheer flow, but you have the sort of transition phases, these are, these are temporary where the rental stress changes sign, which
10:44:16 means that instead of energy transfer from the main flow to the fluctuations, you now have energy transfer from the fluctuations to the main flow.
10:44:25 And if you actually read the paper that they're citing you see that indeed the flow profile shrinks during that time.
10:44:43 So you know there's some evidence that this stuff is relevant.
10:44:37 We wanted to dig into this with simulations to look more closely at this so we did some, I'm going to go in the interest of time lightning fast through, through this stuff but it's it's in this paper 2018.
10:44:49 We did some essentially 2d comma or a flow simulation so we have periodic boundary conditions in both directions with a sign the soil body forcing terms so we have flow in the y direction that various scientists oily and the x direction.
10:45:04 That's maintained by forcing term. And we have large scale linear and fracturing term, and a small scale hyper viscosity as well. And I'm just showing here to sort of orient yourself to the domain, what that looks like.
10:45:19 So sandy soil.
10:45:22 Background flow, that's in the y direction very science literally in the extraction.
10:45:27 So we want to look at what are stable modes doing in the system.
10:45:32 And how can they get excited despite their linear decay.
10:45:37 Sorry.
10:45:38 So the way we do this.
10:45:44 The way that we look at this is some post processing analyses on these track numerical simulations. So from the simulations that I was describing the previous slide these common workflow simulations.
10:45:57 At each time we take the the the stream function the state of the system.
10:46:04 And first, we do a 40 transform in the y direction that's the direction of flow it's the direction of homogeneity, and we take my hat here, which is our for you transformed stream function.
10:46:16 And we expand it in a basis of these linear modes.
10:46:21 So Vijay here is the different modes of the system those unstable modes of stable modes and these marginally Sabre Sabre modes correspond to those kind of critical errors that we were talking about before.
10:46:33 And then beta, here is the amplitude of that mode at each time.
10:46:37 So, with this expansion we can track how the amplitude of these modes changed in time here on the left hand side I'm showing on the top it's a log scale on the bottom it's a linear scale and two different time domains in blue, beta one here is the unstable
10:46:53 mode.
10:46:55 That's exponentially growing in the linear regime here right before saturating and finite amplitude.
10:47:00 And what you can see is that the stable mode beta two is decaying, as you expect before being non linearly driven to a comparable amplitude ends up being at a similar amplitude, as the unstable mode.
10:47:15 And you can show I didn't think I would have time to include it in the sides but from this sort of behavior this linear decay and the nonlinear growth, and our previous work on the subject, both in sheer flows and other people's work in other instabilities
10:47:31 this sort of nonlinear excitation of stable mode, you can get it from.
10:47:37 It comes from nonlinear interactions with just unstable mode so to unstable modes at two different way of numbers, interact to drive the stable mode to significant amplitude.
10:47:47 So this is sort of reminiscent of the generalized quality linear approximation that we've seen great talks on so far at this workshop by both Laura cope and Steve Tobias, the nonlinear interactions that are responsible for this table mode reaching a significant
10:48:03 amplitude, or the same nonlinear interactions that you get in the generalized closet linear approximation.
10:48:10 So I think that's key, you wouldn't see this in a closet linear approximation those nonlinear actions are abandoned. and you don't need nonlinear interactions with with small scales it's this is all large scale stuff on the right here.
10:48:25 This is just showing that in addition to this, unstable mode positive growth rate right in the stable mode negative growth rate. There's this continuum of marginally stable modes, and they get excited as well.
10:48:36 Now, this stable mode being a comparable amplitude to the unstable mode.
10:48:44 That tells you that you know it's not quite equal right there's there's more unstable more than saving mode. But this tells you that there is you know for every parcel of energy transfer from the background to the fluctuations, you do have a lot of that
10:48:58 energy going back from the fluctuations back to the main flow.
10:49:02 going back from the fluctuations back to the main flow. Not all of it, but there is a significant amount of energy going from the fluctuations back to the main flow.
10:49:08 And this means that beta two are the stable mode here can be thought of is playing a role in saturating the instability so you know you have exponential growth that has to stop.
10:49:21 Eventually, part of why it stops is presumably transferred to small scales, but here we can see that also a mechanism for saturating the instability for halting exponential growth can be transferred to steal modes and then back to the main flow.
10:49:35 Something else that that came out of this work that I think is pretty cool, is we tested. A lot of times people try to come up with produce models that one of the assumptions in this model, the inner models might be that the large scale flow fluctuations
10:50:05 driven by this instability so maybe we can assume that they bear the same shape as the unstable mode so you can assume that at large skills, your turbulent flow is just going to look like whatever your linearly unstable mode is. And that makes sense because that's what's driving
10:50:08 that's what's driving instability and driving the turbulence rather so so it's a it's a reasonable assumption to make. We wanted to test that in the system, and we want to compare it to what you get if you additionally include a stable mode so you say.
10:50:22 So, when we take our trivial fluctuations and we're just trying to describe the large scales and the fluctuation so we take out the mean we take up a small scales, and we're left with this sort of filtered stream function at a given point in time from
10:50:37 our noses post processing, this is just a snapshot from a simulation and then we do have filtering.
10:50:43 You can express it as a linear combination of all the modes like I was saying on the previous slide. So this is just something over all the modes, confirming that we correctly, we derive the exact state.
10:50:55 You can say, okay, instead of sending over all the modes. I'm going to just sum over the unstable mode at each wave number there's like, you know, 10 different unstable wave numbers in this in this domain, and you get roughly, you know, a similar structure
10:51:09 but it's not exact, compare that to what happens if you take you assume that your system state is a combination of unsealed modes NCO modes, and you do a much better job.
10:51:20 It's not perfect but you get a pretty good, you know, description of the flow profile, we quantify that in terms of an error so we just took the difference between these two panels and to fail to normal bit.
10:51:32 So this is just the, You know integrated error in fi as a function of time. And you can see that in the saturated state.
10:51:41 Using the unstable mode alone at each wave number and each of these large scale wave numbers.
10:51:48 You get about a 60% error if you assume that the top left panel is basically just bottom left panel let's say 60% air. Whereas, if you add in disabled mode so just adding in one additional free parameter.
10:52:02 You get 20% error so you get a much better description of the large scale free features with just a little bit more work, you know, adding in one additional mode to your characterization.
10:52:12 Additionally we showed in this paper that you can capture the parental stress very well. So the rental stress as well described as you might expect if the flows well described.
10:52:21 And so we wanted to see. Does that work in MHG.
10:52:26 I'm going to breeze through this stuff in the interest of time.
10:52:30 I'm sorry everyone. We.
10:52:33 But, let's see. So, so what's what what I want to focus on, I mean we did 2d incompressible simulations of uncertified sheer flow in HD.
10:52:44 This is basically the physics of what we simulated for those who aren't familiar, this is just not really stoked equation with incompatibilities, and you get your j cross before us here.
10:52:55 So, for those who are not familiar with MHD.
10:52:59 The month of equation looks the same, but you haven't straight across before us here and you have to solve for the magnetic field, the system that we simulated is the same as that Matt Griffis use paper.
10:53:07 We have an unstable flow profile with flow going to the right and the top level domain flow going to the left and the bottom half the domain. A initially uniform magnetic field.
10:53:19 I don't have time for that.
10:53:21 Here, I'm plotting at different field strength so ma here is basically flow velocity divided by field strength. So, a large ma is a week magnetic field and a small ma is a strong magnetic field downplaying the kinetic energy of the main flow over time.
10:53:38 So the blue line here is the week magnetic field case, you can see that the energy in the flow does not decay as much as in the stronger magnetic field case so this is saying that the layer broadens faster with a stronger magnetic field, this this y axis
10:53:54 here can be taken as analogous to a layer with.
10:53:58 So the layer broadens fashion is more turbulent viscosity, the stronger your magnetic field is so this is consistent with previous work we're just making sure that our simulation works.
10:54:09 And we can do the same thing here that we did in the coma Gora flow problem.
10:54:15 We have an unstable mode at each wave number and this country at civil mode as well. Here I'm showing the streamlines of these two modes, it looks just like in the similar case I showed before.
10:54:25 And these are the field lines of those two modes. So, even though we have a field now that contradicts symmetry that PT symmetry is preserved.
10:54:33 So again this unseen world is going to put energy from fluctuations to the main flow. This did.
10:54:38 Sorry, this until mode is going to take energy from the main flow put them into fluctuations and the same mode transfer energy from fluctuations, to the main flow.
10:54:47 And now that'll. Additionally, presumably happen with a maximal stress in addition to the Reynolds stress.
10:54:53 We can do the same mode decomposition, as we did before.
10:54:57 And I think that I don't have time to go into it but the key feature is that the same sort of reduction in rental stress that I think Samantha give a talk on a couple weeks ago, that's captured by this sort of.
10:55:13 I can move the composition and plotting here.
10:55:16 You know the mental stress is a function of z and, but it peaks as equals zero, so I'm just for the sake of simple plot and focusing on the mental stress equals zero.
10:55:25 And that's just shown in the dashed lines here for different often mock numbers of different field strengths.
10:55:31 And the point of this plot is just that, hey, the solid line, which is an expression based on the stable and unstable modes is right on top of the dash line which is actually the mental stress at each time.
10:55:42 And this tells you that models in terms of the stable and unstable modes.
10:55:48 Can this this reduction in their own stress with magnetic field strength so the green line is stronger fielding see that there's less mental stress on the green line there's less mental stress as you increase the field strength, that's captured by the
10:55:59 sort of stable and unstable mode model.
10:56:03 There's other implications for that.
10:56:07 I'll just zoom ahead I mean that in this me she case, if you, if we, the conclusion that we do in this paper was that magnetic fields suppress the stable modes, which means that their category and modes of transport is removed you no longer have energy
10:56:21 transfer from the main flow from the fluctuations of the main flow. And so the energy that would have gone back to the main flow cascades too small skills unimpeded, there's nothing preventing it from Cascadia small scales.
10:56:33 And that means that you're going to have more small scale fluctuations that's going to be more dissipation and that's what we showed in the system as well.
10:56:40 So, the main result really is, you know, big picture, the stable modes transfer energy back to the main flow in a way that gives you kind of great ones and password.
10:56:51 You can show that they're nonlinear Lee driven using with the sort of nonlinear interactions that are captured in a generalized closet linear model.
10:56:59 And the large scale fluctuations are well described by stable on civil modes alone.
10:57:06 So, it's conducive to reduce models, basically. And in this case, we had this.
10:57:13 We published this paper just last week actually, we have this nice picture where this magnetic field suppresses the stable modes, so you don't get their category and wants to transfer you just have downgrading modes of transport, and you don't have any
10:57:26 anything stopping the energy from cascading to small skills, which causes you to have more maximal stress and more small scale dissipation.
10:57:37 And you can read more about it here.
10:57:38 Thank you.
10:57:43 Thank you very much, Adrian.
10:57:47 Nice talk
10:57:50 time for a few questions.
10:57:55 Alexis, please.
10:57:58 I Hadrian I that was really nice talk thanks for sharing.
10:58:01 I am curious about the sort of the sheer instability in the MHD system that you're thinking about, because you said that your simulations were all in 2d and at least for like a stratified cage.
10:58:14 When you're getting into discussions of turbulence and mixing there's big differences between 2d and 3d systems. So I was wondering if you could, if you could comment on that for MHD is that the MHDKH is that going to be the same sort of issue going from
10:58:29 2d to 3d or do you think these results are going to apply in a fully three dimensional system.
10:58:35 It's an important question, I appreciate you bringing that up.
10:58:39 I think that that's certainly going to be an issue in the hydrodynamic work that we did you know remains a question of, in particular in 2d hydro, you have an inverse cascade of energy, whereas in 3d it before cascaded energy and surely that must be reflected
10:58:54 in this kind of analysis in HD, both 2d and 3d, the energy tends to cascade to small skills.
10:59:04 So, I suspect that you know the sort of the, that helps my argument basically in MHG that we still have a small scale cascade in 2d.
10:59:14 But that said, you know, it is certainly a leap to go from what I've shown here to the 3d case.
10:59:21 And I do hope that the 3d case gets investigated, you know, I, because there still is a small scale cascade before cascade in 2d and 3d I suspect that this is still playing a role and also I should mention that symmetry between the for every unstable
10:59:37 mode there's a stable mode that holds in 3d as well so that's a kind of helpful point.
10:59:42 But, you know, until it's shown you know until someone does this simulations, I can't give you much more besides hopeful optimism, unfortunately.
10:59:50 Okay, thanks.
10:59:53 OK, now go ahead of pack of you go VM.
10:59:59 I'm sure you thought about that. I didn't use in your presentation.
11:00:06 It's over. Well, I didn't see that the the key points with those table modes, is that a significant amount of energy or entropy or inflammation whatever should be channeled through the stable modes, before going back to the dynamically relevant votes.
11:00:25 Otherwise you don't really care, right, especially as well, and they are them.
11:00:32 and to to really see that.
11:00:35 Have you tried to looking at, I don't know cross by coherence or transfer entropy or one of those methods that that actually gives you a causal chain and gives you the flow of information and.
11:00:50 And the second question related is, if now you were to cut the interaction with those people nodes, would you see a significant difference in the relevant dynamic piece relevant modes.
11:01:05 So, It's I understand your correction, your question.
11:01:11 Just make sure you're asking if we've sort of looked at the sort of mode mode energy transfer from, say, the unstable modes to the standard modes to, like, really traced from which mode to which mode energy is being transferred, is that right.
11:01:26 Oh yeah.
11:01:28 So, I agree that that's something that should be done.
11:01:33 That's that's something that has been done in the fusion cases were sort of this, you know I'm borrowing this framework from work that was done an infusion plasmas.
11:01:41 And if you look at work by Garth Whelan, in particular, who's my former officemate.
11:01:46 He has done such diagnostics infusion plasmas.
11:01:51 It has not been applied yes to the sheer flow case, I agree that it's an enticing idea, and at one point I wanted to do that and then I graduated instead in the interest of time.
11:02:02 So, I agree with the question but hasn't been done.
11:02:05 But is it just an interesting idea or is it the only relevant idea because you don't really care much if you're too excited stable modes are going to be down to way or that don't really participate into the dynamic, so you really interested and you can
11:02:24 send how show that stable loads, give back to directly relevant or some kind of information. Right.
11:02:35 So I may missing some.
11:02:38 I.
11:02:40 It could be the name missing something with your question, I mean, what we show is that, let's see, these modes, end up at a large amplitude right you can show that without doing sort of a by coherence analysis, and you can show either analytically or
11:02:57 or we did show this with simulations are post processing diagnostics, that if you know the amplitude of the seal mode that you know how much energy is transferring back to me flow and how it's affecting the Reynolds stress in particular.
11:03:11 So, you know, as far as dynamical relevance goes, I'd argue that it is that we did show that it's dynamically relevant in that the stable mode, you know if you compare system where you have a lot of student mode amplitude versus none.
11:03:24 When you have a lot of similar amplitude, that's going to really affect the round stress it's going to, it's strong enough it's going to give you a counter great mental stress, or if not it's just going to weaken throughout stress give you sort of across
11:03:35 face effect on the rental stress.
11:03:38 So that's I guess the dynamical relevance that I'm thinking about does that address your question or my still missing something.
11:03:45 Yeah, it's both the addresses that you have that you you have page long question but it.
11:03:53 Thank you,
11:03:57 Pat.
11:03:59 So you mentioned the magnetic field business which brings to where you mentioned Samantha but it also brings to mind our, our paper with our glorious chairman and other such things.
11:04:13 Did you quantify any other. In other words, a scan of be not and what happens when etc etc.
11:04:21 You made a statement about capturing that I didn't quite see what was captured.
11:04:27 So, I just have a small sample of the figures here and it's a it's a group of papers along paper, but I guess it depends on what you're asking if we captured and we did, asking the, that you know, the, the, the loss or degradation of the negative viscosity
11:04:48 phenomena right that's sort of the main point.
11:05:07 is why and what's happening when you know with these issues of defacing versus Reynolds magnetic competition vs change in the structure of the turbulence etc etc.
11:05:20 Right.
11:05:21 Yeah.
11:05:23 So I certainly agree that that we're not, you know, it's been known this business of what the benefits of doing the Maxwell streets are doing and I don't mean to present statements about rentals and axial stresses that are new.
11:05:35 Again, we're just sort of after maybe we can look at it in this different light, and that is conducive to a new kind of model it's sort of the motivation here.
11:05:43 Yeah, but I mean I'm saying what the way to learn one way to learn something new, is to compare what you get for the, you know what, for the mechanism and the critical field strength in this model to what was been advertised or other approach.
11:06:05 Yeah.
11:06:03 Yeah. I'm asking. Yeah, so, so in terms of alpha and Mach number, so the equilibrium flow speed, alpha and velocity.
11:06:12 We went as high as like 60 or 100 which is comparable to that Mac versus newspaper. We went down to, I've got really low often Mockingbird simulations, but we ended up in the paper focusing on down to maybe often Mach number 7.5.
11:06:26 So that's gives you a feel for the breadth of the range of off the market really simulated. We did a our all of our rentals numbers were 500 and we vary the magnetic Reynolds number from like 250 to 2000 or so I forget exactly.
11:06:40 So we very the magnetic rentals number by a factor of two or four.
11:06:45 So, you know, we, that that's about the range of primary regimes that we varied and the the general picture that we saw is, you know, in terms of just the stresses it's, I think consistent with
11:06:59 what you, what Samantha show we see that, maybe, I don't know if I showed the right plot here show it. No, it's unfortunately not quite getting at it.
11:07:10 We made a lot of plots like what I'm showing here where I'm looking at in solid lines as rentals stress and bash lines is an actual stress and I'm breaking it up into different scale contributions.
11:07:20 And we see that, you know, at the very weak magnetic fields, the maximal stress is totally irrelevant right this is even scaled up by a factor of 10 the maximal stress is lower than real stress by lots and the magnetic field is acting to reduce the rental
11:07:35 stress you get less of a mental stress and so it's not an increase of the maximal stress rather is a decrease to across phase effect in the mental stress, and then it stronger fields consistent with previous work, we see that the maximum stress also gets
11:07:51 enhanced, I think, am I right this is getting at your question, I think there's getting added but I mean I think again mgh, and really qatanani of Einstein I mean and really Zelda which before any of us were born, gives a very specific prediction on you
11:08:10 know product of magnetic Reynolds number time without a Mach number and stuff. And that might be interesting to look at, you know, a scan of that and what's happening when.
11:08:25 Yeah. And then of course the stuff we is Samantha and I was doing was a somewhat different model, which perhaps you could translate to your model that I'd have to think about a little bit but I was just, but in particular the first comment about let's
11:08:36 call it the Zelda which number. I mean that's the that's really the thing to look at. Right, so yeah and I appreciate the comment, and I tried to look at that one of the things that makes this tough, is that it, I think, Alexis, commented on this during
11:08:52 her talk Actually, this is really sensitive to initial conditions, we've got a big enough box so we get multiple coherent vertices that then merge at a time, rate that depends on your initial condition, and that affects this sort of Zelda which scaling
11:09:05 significantly, it's not totally independent the initial conditions. And so that means that the system that we set up is maybe not best suited for answering your question, because of the sensitive initial condition dependence.
11:09:18 But I agree that it's the right question to ask.
11:09:23 Two more so.
11:09:31 Do we have a sensor, Adrian how magnetic fields affect PD symmetry and how would that would change you know this.
11:09:41 The nature of the moves that are created in Paris.
11:09:45 Yeah, that's good question. I guess I haven't looked at it explicitly in the framework of PT symmetry, like like shins paper.
11:09:55 But I can tell you that this business of for every unstable mode there exists a stable mode you can show just from looking at the equations, we wrote them down in the paper, but you can show from in the ABC equations for the linear mindset, not I agree,
11:10:10 I'm a bit.
11:10:11 Thank you very much I agree with that and the bits of shoes. Peter cemetery was so sort of.
11:10:16 Where should I say a deep principle, I'm a bit bothered that magnetic fields, you know, break the principle, that's all I mean, but I agree with you that from the equations is is more transparent.
11:10:27 Yeah. But, I don't know, I'm not ready to say that they do break it. I hope I didn't say that they do.
11:10:39 No no I therefore I don't understand where the argument breaks I guess so. Shin by need to reach into paper like you. Yep.
11:10:44 Okay. Final question, Brian.
11:10:48 So, after the first time step you don't have a mode defined anymore.
11:10:54 In order to have a mode, obviously you have to have homogeneity and time and space.
11:11:00 So how could what structure you using after your first time step can be the mode that you've had in the original profile.
11:11:10 I'm very glad that somebody asked that. I agree with, where the questions coming from, and I spent quite a bit. One of the reasons is paper took so long because I spent a lot of time worried about that.
11:11:21 So there's two different things that we did when I show this plot, where the mental stress as well described by these episodes, and this plot in the bottom right here where I'm looking at mode amplitude.
11:11:34 These are obtained by taking the original equilibrium. And, you know, finding all the modes of the original equilibrium is equal zero equilibrium, and using those as a basis to try to describe the fluctuations at later times.
11:11:49 And I agree that, you know, why would you do that, the the flow profile has changed.
11:11:55 You know, one should instead use the the main flow, perhaps, and I did that as well.
11:12:01 And it's interesting that there's a whole section, it's hard. It's a hard business for a lot of reasons, but it works with caveats. And the short answer is it gets you the same answer.
11:12:14 Whether you use that and there's a lot of sensitivity because there's a lot of non normality in the system, which means that you know the Eigen modes everything's really sensitive to small perturbations of linear operator.
11:12:24 And so whether using the t equals zero equilibrium or the mean flow, you actually you know it can have significant effects on, for instance this plot that I'm showing here, I can go to amplitude.
11:12:35 But in terms of this plot, you know, looking at the rental stress in terms of the mode amplitude and really all the quantities that we care about the system.
11:12:41 We showed that, whether you're working with the expansions in terms of the t equals zero equilibrium modes versus the modes later on, which we get by doing a horizontal average with the flow.
11:12:56 You get consistent answers. And so, shortcuts are a bit justified, hopefully.