08:03:09 Because I'm going to start out with a couple of animations.
08:03:16 That'll play in the background while I just introduce this session, and then I'll be switching to PowerPoint.
08:03:25 So, Okay, I guess I'll get started.
08:03:30 So, this animation corresponds to Arctic staircase that we've been discussing earlier in the program.
08:03:43 So this is 50 meter.
08:03:46 Vertical range.
08:03:49 Now, importantly, and this is ok so it's going to run for a year. So this gives you an idea of.
08:03:59 The top. How far along we are now. So I'll be discussing a modeling approach that differs from most of those we've been discussing in that.
08:04:13 Typically, the goal of reduce modeling is to find a consistent way of removing or eliminating filtering the small filtering out the small scales.
08:04:28 Here is a quite a different strategy the strategy is to create a framework in which it is computationally affordable and efficient to retain all the scales.
08:04:39 So this simulation now It lives on our one the domain of like many of the approaches we've been discussing.
08:04:48 It's now it has five decades of scale resolution. So it is resolving down, fully resolving.
08:04:59 Even the salinity profile with attachment number so it's resolving well below the Commodore of scale.
08:05:06 Now you're only seeing a density plot here, so you're not capturing the full in this image, the full degree of resolution because densities combination of temperature and the relatively smooth.
08:05:21 I mean the relatively smooth temperature as well as the sharper density profile of further along in the.
08:05:28 Okay so this simulation finished. I'm going to start a different one now.
08:05:42 So that's the one you just saw, is a, like I said, it's a physical case the full scale Arctic staircase. This one doesn't correspond to any particular target.
08:05:55 Physical case, it's just an interesting case. That was run for the purpose of showing a lot of a lot of activity it's a lot of low stability.
08:06:07 So, again, I'll let this run, as I continued explaining it.
08:06:12 So, each of these simulations involves profiles of salinity, temperature and the three velocity components, but I'm showing you the density here which is composed of salinity and temperature.
08:06:27 Okay. I'm sorry I'm only seeing the old curve.
08:06:32 Did you share so I have to reach out to stop sharing risky. Yeah I think so, yeah. Oh, I get it, even though it's a map I still have to choose the right curve in the right screen.
08:06:42 Thank you. Thank you very much.
08:06:45 Okay.
08:06:47 Okay so good. So let me start this over, actually it doesn't, it doesn't matter because I'll just let it go from where it is.
08:06:58 So, to see the full, I'll show you some examples with the full resolution just a still frame, but also in the climate.
08:07:09 Working Group next week.
08:07:12 They're going to have a focus group on sea ice and Martin Klein will be showing you the same modeling approach applied to a staircase undersea ice. We're also have the salinity profile alongside the density and temperature.
08:07:29 Temperature profiles. Okay, so now I'm going to just give you the context of what are we trying to accomplish just during this session.
08:07:39 Now, the modeling, I'm going to describe is applicable to a wide range of turbulent flows is one dominant direction of variations that means boundary layers free share layers and so forth.
08:07:53 In that context, the staircases are one of many topics we've addressed in fact, the last time it was any work by me or my co investigators on staircases was at least a decade ago.
08:08:09 So, given that we now have the so many people sit in staircases gathered in this program. We want to offer a code that we have available to those who are interested in, potentially, using code or in coordinating with people who might choose to use the
08:08:27 code and particular Scott Munch who's also listed as a presenter today. He has put together a nice MATLAB package that would be happy to share and help people get up to speed on the method.
08:08:43 So, so what I asked you to focus on today is the question of.
08:08:52 Do you want to know more about this method, and. Okay, now I have to do. Screen Share again switching to my screen share switching to
08:09:08 my PowerPoint sure this all works.
08:09:11 Okay, I hope you know even if you're not see my power, and my PowerPoint.
08:09:17 Now I'm going to go there with you. Good, thank you.
08:09:21 Now I'm going to go to slide shows you see my screen. Okay, see I haven't, not only see full screen.
08:09:30 Good.
08:09:32 So I want, so I'm going to kind of breeze through some of the details just to give you enough of a flavor of the method, so that you can address the question Do I want to know more about this.
08:09:46 Do I want to potentially want to use this method, in my own work. And in that basis. We can follow up on some of these points in some mutually convenient arrangement.
08:09:59 Okay, so, so now I'll go forward with this. Okay.
08:10:08 So, so I'm proposing to resolve turbulent convection on one the vertical domain. I had mentioned velocity profiles and temperature and salinity, but we're going to start with a, a preliminary formulation, that's even more basic than that.
08:10:25 We're just going to try to advance.
08:10:28 We're not even talking about double diffusion yet we just want to advance a density profile.
08:10:37 In time, but we're going to use temperature as a surrogate considered to be losing ask. So what are the middle requirements for resolved space and time resolve troubling convection now on a vertical one the domain resolution means that we're going to
08:10:54 fully resolve the frickin transport of temperature with the true molecular thermal the facility. So, vertical infection well it's going to have to overturn fluid.
08:11:07 And it's also should be non denominational, which is when the analog of being divergence free now infection. That is, especially local and continuous in time in the usual sense cannot satisfy these requirements.
08:11:24 So affection will be implemented by instantaneous for your arrangements of vertical property profiles, which are going to call, which falls under the general a framework called map based direction.
08:11:37 Now I want to reference this to the stochastic ballsy model that Francesco popper Rolla described a couple weeks back in which he introduced stochastic mixing events, in which he said I'm going to pick some interval of the vertical domain, in which to
08:11:59 apply a mixing event. And that.
08:12:04 And so that's very much reminiscent of the key elements and I'm going to describe to you, but the difference here, it's, it's twofold. First of all, we're going to have separate addiction events, which only addict, do not do any reversible mixing.
08:12:25 And second of all, instead of just stochastic Lee sampling these things, irrespective of the state of the system. We are going to be very conscious of the state of the system and its influence on when and where we should be performing these events and
08:12:41 those are the key unique features of this model. Okay, so what does our individual addiction event, look like. Now, in order. It turns out that to obey the basic requirements of addiction is much more constrained than just devising a mixing event because
08:13:02 we want to fulfill some of the requirements I just described, Sophie look first on the right here.
08:13:09 The.
08:13:11 Basically, if you want to be preserving the one the analog and the Solon oil condition you conclude that on some discrete line and here we therefore we suppose you have some step function linear profile, any property.
08:13:24 What we have to do is some kind of permutation of the cells.
08:13:29 And then the limit. Will you take the resolution to the to vanish, the continuing definition formulation of this map is shown as indicated here, you can see that it's reducing limescale is increasing gradients and not introducing any property discontinuity.
08:13:49 And there's not really many ways. Many reasonable ways of doing this and there are specific reasons why this triple map is preferable. and then we'll go into that detail.
08:14:01 And you can see it's much like a diagram that Francesco showed that it qualitatively emulates a turn over now I'm just using the vertical line here because I'm not typically presenting to an audience interested in the physical cases but of course we're
08:14:20 going to go to the vertical line in later slides and use the z coordinate.
08:14:25 Okay.
08:14:28 So, during the first week of our program. Alexis pointed out that it's hard to define a buoyancy frequency for a given configuration because sometimes it's arbitrariness as to what vertical range, you're going to use to evaluate that points the frequency.
08:14:45 Well, now I'm going to have a sequence of events, but each event is parameter prized, and this is also in common with Francesco is approach, you have to pick a location, and a size of the events so here are two possible events.
08:15:00 So, once we have an event we have a well defined interval in which we can define and evaluate the buoyancy frequency. OK, so now once we have a frequency, we can formally construct a mapping right distribution, using that frequency is a function to the
08:15:17 two parameters available.
08:15:21 And I won't go into the detail of.
08:15:24 Why is it alpha minus two here and so forth.
08:15:27 But we do ultimately ended up with a mapping rate, distribution, where if we sample from that distribution, then we will have occurrences in these events at the locations and of the sizes in correct proportion to the Associated buoyancy frequencies, extremely
08:15:45 important is that everything is likewise parameter eyes by time because as soon as any operation, changes the state of this or any other variable that's been taught being time events on this vertical line, then this collection of n values changes.
08:16:06 Well, it's just too costly to continually update this lambda. So instead we use or rejection that that, and that's algorithm and statistics I'm not going to go into it now.
08:16:17 And it does.
08:16:20 But I do show those schematic here of what the basic time investment cycle, looks like motivated, largely by the needs of implementing a rejection method, get us a little too far in that deep to drill for our presence discussion, but I do want to mention
08:16:39 that in this formulation we introduce proud dependence by imposing a lower bound on the buoyancy frequency, recognizing that discuss damping would prevent and turnover from from occurring.
08:16:56 If the forcing is not it doesn't exceed some threshold strength and.
08:17:03 And so, so it's a physically based criterion, but it does it has a consequence of introducing parental parental and even rentals number of dependence.
08:17:18 Okay, so we're not even specialized yet staircases we're just talking about convection in general.
08:17:18 And this alone can reproduce various canonical results for Ghana scaling again a very interesting slide to the Scott, to discuss but I really have to just go right past it but one point is it that the parameters is set by based on some of these canonical
08:17:37 bulk properties. So in order to do any tuning then to apply it to staircases. So, you know, most of everything on this is everything on this slide, this is just to show you that the particular case, I'm going to refer to on the next slide is a staircase
08:17:56 generated from this initial set of temperatures literary profiles that are being heated from below, with particular reference to an experiment that was done by Hubbard in London and so actually some others that there are various run down experiments provided
08:18:12 data on this.
08:18:14 And so when we saw have been discussing temperature so now we add a vertical salinity profile says nothing changed in the basic formulation we can time events any number of
08:18:29 variables profiles on the vertical line along with their micro physics, could be thinking and it could be something else. So, this is now showing a snapshot of a staircase With 10, you can see the sharp jumps and cylinder and a profile.
08:18:48 Now this isn't resolved with four decades of resolution because we're doing the laboratory scale case not a physical case.
08:19:00 In a homogeneous configuration, we get layer merging where you already saw some of that. On the other side, on the animations but again this is just the preliminary model.
08:19:23 And, importantly, what we're, of course very interested in interfaces, the symbols are the results, again with primary set just by the bulk, you know, transfer scale and the Prime Minister was set on that basis.
08:19:25 And so, the measurements have been correlated, this kind of well known correlation theory is kind of down here.
08:19:34 And this is just a part from the point that it captures.
08:19:40 Many the relevant regimes.
08:19:42 The well basically there are three regimes that captures all of them. It says, has a remarkable degree of quantitative accuracy which I cannot fully explain but still it's interesting, but in this deserves more discussion that I have to keep going here.
08:19:59 So far, there's been no velocity field so what's, what does that mean it means that when an overturned occurs, which would, in the physical world. Be can convert potential to kinetic energy.
08:20:11 It's as if it was instantaneously dissipated. And what we really want this potential energy going into kinetic energy that eventually gets dissipated which is why the modeling would include kinetic energy of course, when it does.
08:20:36 Problem is where there's no buoyancy it's only kinetic energy. That's just beyond the need for justification.
08:20:36 So, if we compare to copper Linden, one reason.
08:20:54 One reason why the layers are too persistent here is that it's lacking the mechanism of kinetic energy to be breaking dissolving and merging and migrating, the layers.
08:20:59 Okay, so now we're going to be introducing kinetic energy.
08:21:04 Before I tell you how we do it I'll just indicate that will immediately enable us to do one thing I was mentioning, which is just to, to look at stirred Dave least stratified fluid we are not relying on conductive instability as the source of energy to
08:21:20 produce the turban wins. This is just qualitative, this is not a one to one comparison here it's just to say that's been seen experimentally as you know and Ott does reproduce this.
08:21:33 We have worked where we've done quantitative comparisons, but I'm just going to go past that, as another set of topic for the future.
08:21:41 Okay, so now we still have an Eddie rate distribution but now include, it's going to include here as well as poignancy contributions. Importantly, when we do a triplet map on now, a velocity profile Uz we've increased the sheer therefore increase the
08:21:56 likelihood that there's going to be small Eddie's within, say a region like this, then what have been the likelihood over here. So we haven't Eddie induced cheer inflict amplification driving smaller readies produces an edit cascade.
08:22:22 And it's your dominated flow, we get as an outcome K to the minus five thirds, it wasn't baked into the model as I showed before, if you just doing pure convection, you can get a ball again with spectrum if you only use the density profile.
08:22:27 Okay so, just to give you a sense of what the model can do, if we're going to discuss staircases we should also be aware of the base flows in which they occurs and such as Kimball cabling handholds jets, but most importantly I want to introduce the notion
08:22:42 of a space time diagram. Okay, so this is time advancement on a vertical profile. So if we look at the time cord. Every vertical line segment is the vertical range of some Eddie event that's occurred during this simulation.
08:22:58 Okay. So in some sense, I think of it as like a skeleton of the turbulent development. Now, the fact. Now, I really want to point out this clustering of these events, okay if you have a big event that's bunch of small wins.
08:23:14 Next week, now that's precisely reflecting the fact that the big event generated the highest year and this year, generated.
08:23:23 You know, accelerated the future events. In other words, the velocity profiles are themselves are the underlying mechanism for carrying memory from test events to future events and this is what creates time correlation between cast and future Eddie events.
08:23:42 So, just to make the point about quantitative capability the model.
08:23:50 This is a variable density kept Kelvin handholds, there's no points here, just to see the variable for variable density event so this is not boosting estimates for variable density treatment, just to show for this case, some velocity profiles versus density
08:24:07 ratio mean density profiles and here we have scaled turtleneck genetically energy budgets for due to different density ratios, just to make the point that this model doesn't just capture sort of scaling properties and whatnot but it really has at least
08:24:26 in some instances, I'm not going to say always detailed quantitative capabilities. So let's look under the hood here, just briefly.
08:24:35 The Eddie sampling that I mentioned is going to now incorporate, not only points effects, but shear effects. And so we.
08:24:45 So we can escape. We knew and on an energy basis. We can turn, we can say the unknown Tao, which is going to be a timescale who's in versus the frequency we're looking for, can be expressed in combination with the Eddie low, Eddie sighs l at the given
08:25:03 instant. The Eddie we've chosen as an energy such that we can quantify a kinetic energy contribution. And in a potential energy contribution and like before we had a threshold for energy turnover which we based on which we have a viscous based an effect
08:25:24 threshold rentals number for ready turnover in this three relation that's determining our unknown towel in from that collection of tasks as I said, we can build up our distribution lambda which we then sample from using our rejection methods so that's
08:25:40 the concept, so you could see the mixing life theory, concepts, come into this, but they were applied locally to each Eddie, rather than globally. So it's kind of a conceptual extension of the whole mixing life philosophy, philosophy, but I want to emphasize
08:26:01 that philosophy profiles here are only indirectly influencing the steady occurrences through, through their kinetic energy content velocity does not directly affect fluid, which results in some so instead of counterintuitive, in some sense, formulation.
08:26:22 In addition, I mentioned that we want to be able to convert the potential energy to the, to the equivalent kinetic energy so introducing here's a terminology that Francesco introduced in fact we introduce the colonel, not only to do a triplet map, which
08:26:39 as we indicated is just a completely conservative operation, but then we have to add some chosen function. That's really a wave of like objects and therefore call it a colonel.
08:26:53 That's chosen, so as to make the appropriate change in the kinetic energy within the Eddie, but without modifying momentum so we're doing the correct closure of all conservation laws.
08:27:06 So this slide, next couple points. One is the premise number of applications, which is why.
08:27:14 Scott and I and other of my co investigators and his have not had a big emphasis on layering which is very interesting but it's just one of many things we're doing.
08:27:25 So that's why we're very eager, in terms of our overall strategy to basically
08:27:33 share as much as possible our codes are our method with other people because best way of making progress is by coordinating with subject matter matter experts who can do a deeper dive, then we're able to do in each of these applications, you probably
08:27:48 see a lot of things here that might find interesting even, you know, plasma things so even though I'm emphasizing thermal hairline. I wouldn't rule out other applications potential forces me and so on.
08:28:01 So, now I want to mention some possible applications of the approach to the research that people in this group are doing. So, the ones in blue, I'm going to elaborate in later slides.
08:28:17 And here's a list of others.
08:28:19 I do want to mention though that using can heal your micro physics, with an ODT. It's an interesting analog to NaVi Stokes can help all your, but it's where it's sort of a poor man's navios can heal your that's long been on my to do list for a lot of
08:28:38 reasons that people know above and beyond layering now I'm even more interested so I'm hoping there might be some kind of dialogue on that possibility.
08:28:48 Okay, so in some work.
08:28:53 That was our most recent on layering in 2011. We did very extensive parameters studies to develop some Bob correlations. I mean it's about hundreds of cases I mean, this would not be affordable using DNS to say the least.
08:29:08 And you can see why there is a print parental number and so on. And we you know we got some consistencies, but there were some, some things that were not strictly an accord with pure power law scaling, and I think some of this is well worth investigating
08:29:25 both from a validation viewpoint to see if these correlations are of practical utility. And, you know, to try to understand some physics behind it all.
08:29:37 Additionally, we went into the Richardson number and also developed a correlations there and based on shared interfaces. But here. I want to be on that show you how Eddie spacetime diagrams evolved.
08:29:51 Now this is not all the Eddie's over given time for just a random subset of the edits that were involved in these simulations, showing that as you increase the sheer you do break up the interfaces, but you can see how you have interfaces that are becoming
08:30:13 quite impacted by turbulent intrusions but they still remain somewhat intact. So this makes the point that not only is this approach useful for looking at staircases as a whole but because it's resolving in space of time, or the molecular transport.
08:30:34 It could be an interesting way of probing.
08:30:37 Some of the issues about interface I know in the session is going to follow this can be discussion interface motion. you know interface instabilities what happens during merging, and so forth, your interface to solution, and way this, this could be a
08:31:08 convenient computational laboratory for doing sort of a first look at what's possible as a way to sort of lay the groundwork for optimizing DNS studies that my.
08:31:07 Now, high railing number, their interest remains request interesting question, especially for Astrophysics okay at the say the least imaginable the individual layers and astrophysical configurations, could be have Railey numbers, which they might have
08:31:24 extended beyond the Marcus was regime to the ultimate regime.
08:31:32 Now, the question is, how are you going to reliably do that sort of an extension, unless you have a model that so to speak, can look around corners. In other words, it has enough physics to capture trends regime transitions that you might not have anticipated,
08:31:50 I mentioned Martin Klein will be speaking next week, he did this study, in which she used ODT to simulate a recent experiment.
08:32:02 Experimental investigation of radio, radio radiative lead driven really conviction that showed over a range here in these black open symbols, a transition from Marcus to ultimate.
08:32:18 Well, the Ott with parameters are the set at the lower railing numbers.
08:32:24 So there's no more tuning to go on here, in fact it was set for really convection that rave convection, so no tuning. If the model fits everything for radio convection from over a much wider range, then was possible to observe, using the experiments.
08:32:41 So, so again this is quantitatively predictive and important, and also he took, he got a detailed statistics on it sizes and locations that actually show some of the mechanistic origin of this transition or actually an enlightening us to the physics,
08:33:12 the important point from in our context is if you want to extrapolate DNS, or other methods to, say, an astrophysical regime.
08:33:13 You might need the assistance of a model that can actually carry you through that transition and get the right railway number scaling of your layers. When you're getting into the rail numbers that might apply in that instant instance, I just want to say
08:33:30 that Scott once did a detailed study showing that really convection specifically the model captures detailed fluctuations statistics PDFs.
08:33:48 Really dependence on the midpoint.
08:33:46 temperature RMS.
08:33:49 Again, you don't. The only tuning involved tuning to the bob properties and that basically missile missile number scaling up, in turn, ways.
08:34:01 The simplest way to idealize an internal wave, interacting with an odd simulation of
08:34:12 a staircase is to recognize that an internal wave can just be seen as applying a bulk of vertical translation of the entire staircase and some sign of soil way, and the time derivative of that bulk velocity is an acceleration which by the Kremlin's principle
08:34:36 implies a time dependent generalization of the gravitational acceleration, which can be directly inserted into the potential energy term for that Tao, that determines the Eddie sampling.
08:34:54 So that's all that I want to say and I'm really looking forward to welcoming questions and discussion in the last 10 minutes or so that Bruce designated for us to be together.
08:35:16 So thank you.
08:35:20 And is there any discussion, let's.
08:35:27 ask a quick question. Quick or not quick, I'm happy to have your space, your space time plots seem to look like having some fractal character, did you explore that at all.
08:35:41 This model does, in fact, reproduce fractal scaling.
08:35:48 And in fact, Well, it gets a little bit complicated because I've only told you about to have at least four flavors of the model I took what I call dp and Ott.
08:36:04 The simplest framework in which to explore fractal scaling is instead of the sampling.
08:36:14 Eddie events based on the difference, the instantaneous state of the system to do something simpler, and just do a nominally homogeneous system, and then assign a power law distribution of Eddie sizes, based on what Common Core of scaling with, with give
08:36:31 called the linear it model. And then it's very straightforward to extract the clean fractal scaling, and it does not only exhibit fractal properties, but in fact publication where use of the approach has helped to reconcile some differences opinion of
08:36:57 opinion in the prior prior literature as to how to interpret the fractal dimensions that are extracted from data so yes this is an interesting tool for studying fractal properties,
08:37:20 if I may. Yeah.
08:37:22 Did you ever try to use your approach for finger connection, not the staple center gradient hidden from below the upside down.
08:37:33 You know, we, we haven't haven't tried that.
08:37:36 Of course we knew we couldn't produce the fingers per se in one day, but potentially, we can produce some of the collective effects of fingering such as when you have a fingering layer, you know, over, you know, developing an interface between two, as
08:37:53 far as you know that that program has not been been approached with one dimensional tumors. No, I think it would be a very interesting thing to try. And again, this gives me an opportunity to emphasize.
08:38:07 We haven't been consistent complete or systematic and what we've done because it's just been one topic, among many for Scott, and he and others. And so, absolutely.
08:38:21 Our approach has been like Swiss cheese with more holes than cheese. We've just hit on some promising things to look at here in there and left plenty of topics wide open, very much worthy of investigation, especially considering the fact that in the last
08:38:42 decade, which might call the golden age for DNS of layering and staircases. We haven't been doing anything so.
08:38:49 So there's a wide open opportunity to for synergies and with the DNS that has been done, and will be ongoing.
08:39:04 And I asked something along. Absolutely. I'm sorry. If you do, thermal connection unstable thermal dimension which you have. Yeah, and your model, your model has enough in it to get to get the right temperature profile i mean it's it's very non staircase
08:39:20 in that case right you get a boundary layer and you get a diabetic in the middle, and then a boundary layer at the other end.
08:39:28 And, and your model has enough to pick all that up.
08:39:33 Well, absolutely. I mean I haven't shown you instantaneous plus. But, I mean, you know, this is this. See I haven't even shown. But, but it does it has exactly, it has said as shape in this paper you can see both instantaneous images, and
08:39:56 the paper and just pull out a some images that we have. Yeah, but it but absolutely it captures the boundary layers and it captures, you know, and then you know the basically captures the S shape.
08:40:06 I've had time I would just pull up the pattern of the,
08:40:20 And of course the fancy very interesting fluctuation affects me look at the at the profiles, and I can send you a paper with that and I don't know.
08:40:32 Scott even has any.
08:40:35 Well, I know Martin who's who's online.
08:40:38 who did that, that transition to the, to the ultimate regime, he has plenty of images at his fingertips for you, but it's not only images but we're capturing the fluctuations statistics.
08:40:56 I mean, just by tuning the parameters to the Meet Bob transport.
08:41:01 So the, so the only difference in, in, in your modeling of
08:41:08 straight really Bernard convection and double diffuse the connection is that you have the two gradients but it, but you're, you're mapping everything would just in one case it would just be doing it to the thermal gradient, and in the other it's doing
08:41:21 to the both the thermal and the salt gradient. Right, it's just right. It's just an input difference. Okay, we're just changing our inputs, like a DNS us just set up a different boundary and different initial boundary value problem.
08:41:35 Okay. In fact, I'm going to go back to see
08:41:42 the difference between this and this is only the initial conditions.
08:41:47 I get Kelvin Helmholtz if you put in a step function initial velocity profile planner jet to put in a top hat initial velocity profile.
08:41:55 Now this reproduce the things I haven't discussed like, how does the Reynolds number. There is a function of time, at the time, dancing simulations, and so on.
08:42:03 So, um, yeah I mean, the goal here was to create something where it hasn't a physics. So, to do different cases all you have to do is send initial advantage or conditions in fact.
08:42:30 On my last slide, reference code that up.
08:42:29 At the very nice.
08:42:32 c++ codes, been set up in a very modular form, where we have something called domain cases. So you just set up your favorite initial that thing you know, you define which you define your variables and you set their initial conditions you set that boundary
08:42:49 conditions. You don't even have to know anything else about the model.
08:42:52 And you can run the code, you don't even have to know what what's what's going on under the hood
08:43:02 can relay or jet, whatever you want. Let me interject here Hi I'm sorry I couldn't make the start of the meeting I had something else going on.
08:43:10 It is getting pretty close now to 945 and I think it's important that we wrap up this section, shut everything down so they can start a new with Pascal section which starts at 15 minutes.
08:43:20 So maybe you can just take an opportunity to see some closing words are there some last minute questions but we should wrap up next couple minutes. Okay, I just want to emphasize this slide.
08:43:31 Okay, I just want to emphasize this this slide. More information available. Scott Wunsch, you know, he's, he's a co pilot co lead here I did all the talking.
08:43:43 I think he's probably okay with that he would have said something so but he hasn't, we're very happy to share information code, whatever.
08:43:50 So, please get in touch and let's see if there is some ways that combination what we have with your expertise in the subject matter and so forth, can lead to interesting work and interesting results.
08:44:06 So thank you, especially thank you Bruce.
08:44:19 Great, well thank you very much. It's great to see so many participants I hope you enjoyed that and I hope you'll, you'll have an opportunity to follow up on it.
08:44:16 So for now I'd like to ask to yes indeed clapping all around.