 So we move to something completely different now, right? Stratosphere. We have an example. This advances your slides, also. This advances your slides. Before I open this up, I just wanted to announce people that, at the end, I included more slides than I know I'm going to get through, and I didn't do that because I was trying to pack things in. I did that because there are some things that I won't touch on, but I thought were important for you guys to all know about. And what I did was, at the end of my talk, I added all of the references. So if you see them, you can, because I think these are being made. The slides remain public and available, yeah? So then you can go and find the relevant papers and go to the direct sources. Just be aware of that. So show of hands. How many people have worked on the stratosphere, or have even read about stratosphere? OK, like three, right? OK, so I work at this place called the National Oceanic and Atmospheric Administration. And when I first came there as a postdoc, it took a lot of abuse, ongoing. I endure a lot of jokes about that these are names for the stratosphere, the so what a sphere, the ignorosphere, the sponge layer, that it's the rigid lid for Matsuno modes. I think largely just telling me that my work is irrelevant. But that turns out to not be the case. So why should you care about stratosphere? Well, this is just one example. So this is by a paper by Hansen that just came out recently. And they're exploiting what's called a nudging experiment. And essentially what you do is you take the European Center climate model, or S2S model, and you nudge particular portions of the atmosphere to perfect conditions. So it's like saying, if we had a perfect tropics or a perfect stratosphere, what would the effect be on our predictability? And I show three panels here. The black line is the interim and the other is forecast model for 30 plus some years of data. And this is the NAO index for two types of nudging experiments. One is where they've taken observed SSTs and nudged tropics. So that's essentially, I don't know, 15 South to 15 North. And you take winds, temperature, all those sorts of things. And you say, OK, we got that perfect. What's going to be the extra tropical response? The other is a nudged stratosphere. So they pick some layer above which the model is getting perfect stratospheric variability. And what you'll see here is this is the correlation between the NAO for when you get perfect tropics. The correlation is significant, and it's 0.51. However, if you get perfect stratosphere, you get a correlation of 0.72. So the point being there, that despite being a sponge layer, you can get, you know, harness a lot of predictability out of knowing things out of the stratosphere. So I sort of assumed that most people don't know a lot about the stratosphere. So I'm giving a lecture on Wednesday that's going to really go through the particulars of teleconnections and stratrope coupling. But today, I wanted to just kind of go through some basic dynamics of the stratosphere and what's called a sudden warming, sudden stratospheric warming, because that's probably the most impactful type of event whereby you get communication between the stratosphere and the troposphere and enhance predictability. And I'm going to go through just kind of some basics of the triggering mechanisms for how these happen and some basic kind of deterministic predictability. Because when you have one of these warmings, you get maybe 60 days of enhanced predictability. So in my second talk, I'll talk about predictability of sudden warmings in a more probabilistic seasonal sense. But today, when I go over the basic dynamics of warmings, I'm going to talk at the end, I'll briefly touch a bit on the deterministic predictability. All right, so the stratosphere in some sense is maybe a little bit more simple of a, you know, I tease the people that I work with that make fun of me that the troposphere is kind of like this noise maker and you can't figure out what's going on. In the stratosphere, we have, I can distill it down to, in some sense, these two pictures. So essentially the interesting things that go on in the stratosphere is during the winter. We have what's called the Charney-Dresen condition, which essentially says if there's Easterlies, planetary waves can't propagate up into the stratosphere from the troposphere. But in the winter, we have westerlies and the window opens up and waves are allowed to propagate up into the stratosphere. So in the absence of those waves on the left panel here is what a temperature in January might look like for what's called radiative conductive equilibrium. So this is just the temperature that would occur if we didn't have any waves and essentially you get a very cold, cold and, you know, warm, more warm over the tropics. But in reality, this is what the actual temperature structure looks like. This is just from error interim. And really what you see is that actually over the equator, things are colder than they would be in radiative conductive equilibrium over the equator and warmer over the poles. And this is essentially due to the effects of waves propagating up, breaking and driving this equator to pole over turning circulation, which causes adiabatic cooling over the equator and adiabatic warming over the poles. So the balance essentially that we have here in the stratosphere is the atmosphere trying to relax to this very cold pole that sets up this circumpolar vortex. And then we have waves propagating up from the troposphere and breaking in the stratosphere and warming the stratosphere and slowing down that circumpolar vortex. So what does that look like? So what I've done here is I've taken a video and this is potential vorticity on an isentropic surface, maybe in the mid stratosphere. And this is what as the fall progresses from say October to December, this is what the spin up of the vortex looks like. And where the warm colors are is, you can just think of as warmer and where it's blue, it's cold. And the black line kind of gives you an idea of where the edge of the polar vortex is in the stratosphere. And kind of one of the things that you can notice right away is there's a lot of deformity to the vortex. And that's the effect of these Rossby waves that are propagating up from the stratosphere or from the troposphere and breaking on the periphery of the vortex. So these are the waves that cause kind of slowing to throw up the vortex and cause it to not be quite as cold and strong as it would be in the absence of those waves. Those waves turn out to be the portion of the dynamic that causes sudden stratosphere of warmings that are a part of the evolution of the sudden warming. And that is really the, this is the system that we wanna understand if we wanna understand predictability in terms of the stratosphere. Very pretty, right? You just like stare at this for like hours. It's like looking at the cream in your coffee or something like that that my wife teases me about. So I should point out that when I talk about the polar vortex, I'm talking about stratospheric portion. And Darren Wall kind of made this clear because at least in the US that the polar vortex has been talked about a lot in the news. And the point they were trying to make is that there's two really certain polar vortices, one in the troposphere and one in the stratosphere. And you can kind of see what these two look like. And the way that I kind of schematically think about this in my head is you can kind of think about these two vortices somewhat independent, but maybe they're coupled by some sort of string and as they both wobble around, they kind of interact with one another in sort of a loose manner. So if we perturb one, it's gonna do something to the other, though the character of that perturbation is an area of ongoing research and debate. But just know that when I'm talking about the polar vortex today, I'm largely talking about the stratospheric one and its impact of perturbations to that on the troposcuro cortex. So as I mentioned, the most impactful types of events in the stratosphere what are called sudden stratospheric warmings. And when one of these events happens, essentially it's defined by the following criteria. We take a latitude circle where I've drawn the red line around 60 degrees north at about 30 kilometers in height. What we need to see is the pole to equator temperature gradient switch from west release to east release. And we are, I'm sorry, the zonal wind needs to switch from west release to east release and the pole to equator temperature gradient needs to reverse. And I'll show exactly what that means. It's just so you're getting an idea of how big of an event this is. We're talking about over the course of a week, you're talking about the whole column inside this circle flipping by like say 40 degrees Kelvin. So we're talking about a huge dynamical disruption to the stratosphere. And they come in two flavors. On the left here I've showed kind of what a typical vortex looks like. Largely centered over the pole, it's pretty circular. And then on the left, the center and the right are two examples of what a sudden warming look like. And we call them two kinds. One is a displacement and that's because we essentially just take vortex and we just push it off the pole, okay? And the reason you gotta think when you take a zonal average, you're sampling it's not that the vortex is completely obliterated necessarily but when you take that zonal average around that circle, the wind is easterly in that averaged sense. It's not that the vortex itself has become easterly. The other kind of sudden warming is what's called a split and during one of those events which one could argue are a bit more violent is the vortex gets split into two pieces and then they start counter rotating. So, because I like seeing these things I think it's good to see what they look like visually. This is a PV map of a displacement. And essentially you can see it just gets, you see these waves breaking on the periphery of the vortex and then you see the vortex getting pushed off the pole and that's essentially a huge disruption over in the extra tropics throughout the column. The other type of event as I said is the split and this is in 2009 and for this one we're gonna see the vortex kind of get squeezed into a kind of a peanut shape and then be ripped apart and essentially completely obliterated. So, you could imagine that dynamically the difference between having this beautiful vortex centered over the pole like this and then having absolutely nothing throughout the whole column and the stratosphere, you can imagine that that has a, is somehow gonna translate some sort of large impact to the troposphere. Just so everyone knows, there's also some, those were on horizontal surfaces, there's vertical structure to these types of events and they are particularly dependent on one other's displacement or whether it's a split. For displacement, these on the right here, you see these are, the colors are two different levels and what you see is when that vortex gets pushed off the pole, that happens more strongly at upper levels than it does at lower levels. So it's really got like a first bear clinic vertical structure. Splits on the other hand, when the vortex rips apart like that and starts counter-rotating around itself, those structures, those two daughter vortices have largely bearer tropic structure. So they, that what happens at one level is happening through the whole depth of the atmosphere. So just to kind of summarize, there's two types of warmings, the displacement that's pushing it off the pole and the split where you rip it apart. They have a distinct vertical structure and from a kind of a predictive standpoint, one of the conditions we sort of wouldn't wanna know well how predictable are sudden warmings and when I say that in today's talk, I'm gonna be talking in a deterministic sense. So are there conditions that enhance the wave forcing? So this is the forcing that causes one of these events or might there be stratospheric basic states that are conducive to one of these types of events happening? So kind of the traditional theory goes back to Matsuno in 1971 and the idea here is that all we need to do is one of two things. Essentially you need to generate enough wave activity in the troposphere so that we get this huge wave that propagates into the stratosphere and that disrupts vortex and two, there's this notion of what's called preconditioning. So might there be basic states in the stratosphere that are more receptive to huge pulses of wave activity? So how does this, how do they kinda envision this working? So as I mentioned before, there's what's the Charney-Dresen condition and it just says that there's a wave propagation window such that winds have to be westerly greater than zero but less than some critical wave speed. So essentially we have waves propagating up and if that pulse is large enough, it gets into, it hits a critical line in the upper stratosphere say and it breaks and it reverses the winds from westerly to easterly and when that happens, you're now negative so there's no, the waves can't propagate any higher now so your window is closed and this just keeps happening at lower and lower levels and the cascade moves downward. So this is the critical layer cascade that Matt Simmel was getting after. So that critical layer, wave absorption is nonlinear but the propagation of those waves to that critical level is pretty much a quasi linear or linear phenomenon. So then as I mentioned, how do we trigger this critical layer cascade and the two questions are generating enough activity or focusing that this is the preconditioning idea. So evidence that supports this anomalous wave forcing type of idea, this paper by Lorenzo and Darren, Lorenzo Pavani and Darren Waugh in 2004 it gets cited a lot and you can stare at this plot if you like in your free time but I can just tell you what the message is here. They picked 100 hectopascals and for the North War heat flux that's essentially the waves coming through the 100 hectopascale surface in the lower stratosphere and they correlated that with the 10 hectopascale NAM and what they found is that that correlation peaks for about 40 days of integrated wave flux. So the idea there was is if we have periods of anomalously high wave flux integrated over a 40 day period that's the most strongly correlated with variations in the NAM. Okay, so that's. You want to explain what the NAM is? Oh, I'm sorry. So the Northern, thank you, the Northern annual mode. So depending on who you ask, that's the Arctic Oscillation, North Atlantic Oscillation, essentially the NAM is kind of a measure of the strength. You can just think of it as the strength of the vortex, right? So strong or when the NAM is in one phase of the other, at least in the stratosphere you can think of as a strong or weak vortex event. So the idea here being that if you have a lot of wave activity the vortex is very weak and vice versa. So in that traditional kind of theory preconditioning what happened in the following sense. So on this upper left panel here I've shown what kind of say the meridional PV gradient looks like under climatological conditions and you can think of that PV gradient as your wave guide. And I've drawn a red line here showing where you might expect your waves to propagate given that basic state and kind of the tendency is the waves propagate up and they due to curvature effects amongst other things they propagate equator word and they break somewhere not near the pole. And that's not conducive to breaking vortex down. So the idea is that if you can somehow change the PV gradient and then focus the wave activity pole word you're gonna, for a given amount of wave flux coming up from the troposphere you're going to disrupt the vortex more strongly. And kind of the idea here is is if you just take the absolute vorticity gradient the way that you might accomplish this is a precursor wave that propagates up and it breaks. And if we look at the dashed line as an initial absolute vorticity or PV gradient structure and then we add a perturbation that's a wave breaking that mixes low PV air pole word and high PV air equator word. In the end if you then recalculate what the gradient looks like the vorticity gradient you get a sharpened edge just like this. So here's kind of a weak PV gradient and here we have a sharpened PV gradient. And that's so if we have a wave that comes up breaks it reshapes the PV gradient and so then subsequent waves that come up now are experiencing a sharper PV gradient that's helping to focus things forwards. You can see this in simple shallow water models. Essentially if we start an upper left hand panel with kind of uniform PV field and then we go left to right across the screen and you start driving the model and you have a wave breaking event on the periphery of the vortex and by the time that's all said and done we see that the PV field has been reorganized and we have this nice sharp PV gradient that's tight up against the pole compared to what we had at the beginning. Okay, now I'll just leave this quote up here. This is from that Pobani wallpaper and it's just making the key point therefore that they were trying to make which is that these events are driven by anomalous wave activity generated in the troposphere. So that's kind of the traditional view. So that's not the end all story for how warmings are generated. So there are alternative views on this. Picking up on the resonance ideas I've given you a snapshot of the literature here. I would encourage everyone to go out and read it. I won't lie it's thick and it will take some time to, it's like chewing cement or something like that. I've tried to condense it down to kind of its essence so you can just get a kind of a flavor for it because I sort of feel like it's not appreciated and one of the reasons why it's not appreciated is it requires a non-trivial amount of mathematical sophistication and sometimes that doesn't translate in those papers to physical insight or kind of the general populists like us. So hopefully you can walk away from this knowing a little something about these other types of ideas. So there's two types of resonance. There's internal mode resonance and external mode resonance. The big thing to take away here is that there would be different notions of how the warming is triggered. In particular in a resonant framework you do not need an anomalous amount of wave activity. This is an internal nonlinear dynamic to the stratosphere, troposphere coupled system but the big story is you don't need to generate, you don't need a huge blocking event in the troposphere or something like that to drive wave activity. Likewise preconditioning can mean something very different or at least the interpretation of what the PB gradient looks like can mean something very different. Another set of papers that I'm not gonna go over are essentially Eleanor Neal's set of papers and these papers are very underappreciated. They don't get cited much. He's got some work going back to the 80s but if anybody's interested in sudden warming I would really encourage you to read those papers because even if it turns out that they're not right his papers are full of wonderful physical insight and they're very good to read even if you don't dig deeply into the details. So I included those series of references to be complete. So these resonant ideas while they don't need anomalous forcing you do need some amount of forcing, right? You need a wave present there to begin with. Just wanna make that clear that the stratosphere does not generate this wave activity initially on its own. You need some amount of wave activity coming out from the stratosphere. That's certainly key. And I'm gonna also discuss the notions of precondition how they differ in these types of scenarios. So as I mentioned there's two types of resonance. One is what's called an internal resonance and the way to think about this is you somehow build a high latitude wave cavity. So this would be on the right would be the north pole and the left would be the equator. And if we get conditions right essentially what you do is you stop the waves from being able to propagate the equator word. They have a reflecting surface or some such type of barrier in the upper stratosphere. And essentially you can think about this as reflecting the wave back down. And if the upward and downward components of that wave become in a phase they resonate with one another and you get this nonlinear interaction that causes a warming. So there the anomalous wave flexes that you see in the stratosphere are a function of that resonant interaction between the upward and downward components and not just a huge pulse from the troposphere. So that's internal mode resonance. External mode resonance is slightly different. So the way to think about this is if on the left panel we start with fixed topographic forcing that for this case I've shown a wave number two forcing we just generate a stationary Rossby wave that has a wave number two structure. Now on top of that if we have a free barotropic mode that is a traveling wave that's traveling around the periphery of the vortex their kind of idea is that the PV gradient and the width speed of the vortex itself determines the phase speed of that traveling wave. So that determines how quickly this traveling wave is winging around the vortex. So you could think about it like this if you look at the middle panel and you see this structure that's kind of oval shaped you can imagine that kind of whipping around vortex. And if we tune, we change the PV gradient and the wind speed such that we affect the phase speed of that traveling wave to the point where in the frame of reference of that stationary forcing you can see how we have plus plus minus minus if that traveling wave's wave number two pattern lines up it becomes stationary in the frame of reference of the fixed topography those two things line up and it's no longer traveling those two waves start resonating with one another and you get an explosion that triggers one of these warrants. As I mentioned in that sense of time I'm not gonna go through Elin O'Neill's work on vortex interactions but it's a very cool piece of literature and I don't wanna downplay it I just don't have time to go through today but as I said if people are interested in some warrants I would encourage you to read some of his work. So to summarize we have two kinds of ideas sort of three rather. We have kind of the traditional idea which is anomalous forcing and then we have the resonance ideas and then we have vortex interactions. So these are all kind of theoretical notions. What does the data tell us? Well there are some things that we can look at in the data to give us a clue. So if warrants are triggered by anomalous forcing then we should be able to trace large pulses of wave activity from the troposphere to the stratosphere, right? And this should happen at linear group velocity timescales to generate a wave it propagates up to the stratosphere and that should happen at some sort of to the critical layer where it triggers that downward critical layer cascade and we should be able to see this. So what do we see? So this is for the 2009 split sudden warming that was the second video that I showed you and what I'm plotting here is December through January and these are the 45 to 75 degree north averaged vertical wave fluxes in height and what I've superimposed here so you could think of each of these shoots going up that's like a pulse of wave activity. And what I've superimposed on here but with the arrows are standard group velocity timescales. So if we're thinking that things are propagating up at the linear group velocity timescale then we should get a nice strong match between the slope of that arrow and the slope of those wave fluxes going into the stratosphere. What we see for the early December late December events is that it works really beautifully. During those early events we see wave pulse coming up from the troposphere and propagating to the stratosphere at very beautifully what we would expect for standard group velocity timescales that are about five to seven kilometers per day. However, these right to flux events and I hope you can see the contours here these are the events that were associated with the warming and what we see is if you look at the contours is they don't follow the group velocity timescale at all. When something starts occurring here at 100 to 200 hectopascales it shows up almost instantaneously at all heights. Okay, so that doesn't look like linear propagation of an anomalous pulse. This is more when we have one of these resonant interactions we expect the wave energy to kind of explode all at once throughout the depth of the stratosphere. So this kind of gives us an idea that maybe the traditional ideas are not quite right. So essentially how could we look at, I showed you before these ideas of preconditioning of wave focusing, I showed you the PV gradient that was very sharp and indeed for the 2009 warming we did have this very sharp PV gradient. So what that means is in the traditional sense is instead of waves propagating a equator where it here I've shown what's called the refractive index for the DJF climatology in the republic, refractive index essentially just tells you where waves are gonna propagate. So for, you know, this could be, this is that idea of wave focusing in the traditional sense. Whoops. However, we also see, we can interpret this in terms of what it might mean for resonance. This very strong sharp PV gradient can be related to, as memory I told you about the external mode resonance. The phase speed of that traveling wave is a function of the vortex edge PV gradient. So as I changed that vortex edge PV gradient I bring that traveling wave, I slow it down so it becomes stationary in the frame of reference of that topographic wave. And it turns out as you, in the theoretical sense as you sharpen the PV gradient you're slowing down, it's been shown that you slow down the wave, the traveling wave speed and you bring it more closely towards that resonant point. Unfortunately, when you start looking at refractive index, I mentioned a wave cavity and I don't know if you can see it here, but this is the region of wave propagation kind of like in that diagram I showed you. And this looks very beautifully like a wave cavity, i.e. the wave would come up and reflect back down and the regions of white it can't propagate. So when it comes up reflected back down it can't go a equator word and we kind of hold it in. So it also looks like conditions are right for internal mode resonance, right? When I first started looking at this I went to a gentleman named Alan Plum and I was, he's a very sharp dynamicist and I was hoping that he could look at this and they're like, oh no, you need to look at this piece of information to help you differentiate between those two types of resonance. And also you could tell me it was, oh no, it looks like resonance but I don't have a way to tell you which one it is. So that's still an open question. However, all of these things are consistent with the ideas of resonance. Okay, so that's for one event. Can we extrapolate this to a larger series of events? So we're gonna do the following. We're gonna take error interim data for 30 plus years, 35, 36 years. And we're gonna do two things. One is we're gonna define an anomalous wave event as the de-seasonalized 10 day average vertical EP flux which is just, that's just the amount of wave activity vertically propagating in the stratosphere average between 45 and 75 north at 700 hectopascals. And that's, it gives you a time series, right? So that's a time series of essentially wave generation at 700 hectopascals. It's how much are we generating in the lower troposphere that's gonna then propagate in the stratosphere? And we define anomalous meant by two standard deviations. We do the flip side for anomalous winds. I'm not gonna go through that, but know that that result, you get a consistent result and I'm about to show you no matter how you look at this. Okay, so I'm gonna step you from left to right. Just the pictures consistent between the top and the bottom. The top is just for wave number one events and the bottom is for wave number two events. So that's essentially not necessarily displacement but more or less you can think about it like that. And what it's showing here is the black contours are the 10 hectopascal upward wave flux and the colors are the 10 day average zonal, not 10 hectopascal, the zonal wind tendency for all heights. And the one thing you need to know here is that we have taken these quantities and we've divided them by their standard deviation that's level specific, okay, right? And we've done that because one standard deviation of wave flux in the troposphere looks very different than one standard deviation of the wave flux in stratosphere. So you really wanna do this level by level. In essence, what that does is where you see contours, they're significant. So this little hatch mark here is we do correlations with respect to wave events as I define. So that's 700 hectopascals and these are forward and positive and negative lag. And essentially, as I said, the contours are the wave flux and the colors are the change in the wind, right? So during that time period for all the wave number, one large flux events, this is, we get this structure, you see wave activity and the zonal wind co-locating with one another, which is not surprising because the waves are driving the change in the wind. However, if we look, if we separate by wave events that started in the troposphere that did not produce strong decelerations in the stratosphere, what you see is that for wave events where there was no strong deceleration in the stratosphere, you look at this as the tilt here and essentially it's like looking at the plot I showed with the group velocity vectors. For those events, those wave events that started 700 hectopascals and propagate up, you get a nice clean linear group velocity propagation time scale into the stratosphere, but they don't cause a strong change in the forcing. However, if we look at the difference between events that do cause strong decelerations versus those that don't, the difference is that the wave fluxes are anomalous in the stratosphere, but not in the troposphere at all. Not only that, you'll notice that whatever happens in the lower stratosphere, it immediately happens throughout the depth of the stratosphere. And so this is offering another view that it's not just that 2009 warming that I looked at, it's essentially the bulk of all events is that you generate some amount of wave activity and then something nonlinear like resonance happens in the stratosphere and boom, it explodes and it happens at all levels. Why are these differences, are the mean states different or why is this different? Why are the waves different or is the mean state different? That is a open question. As I showed some of the mean state conditions, this is just for one event, you do see a strong evidence of the mean state being different, like the formation of a cavity or something like that. That's an open line of research right now. In fact, this paper we just published it this year. So this is just, I guess these ideas, these resonant ideas have been around for a long time, going back to the early 80s, but they've actually just started to be kind of brought up more recently. And we, in this paper, the whole idea was to just look at the data and see if the data supports one or the other. But as far as taking it beyond that, that's an open area of research. So the one thing I would like to point out here is that that's kind of the composite view, right? So only, it's a kind of takeaways from that, is only 11 of the 53 wave events in the data record, two standard deviation wave events in the lower troposphere are associated with a wind event. If you flip the coin and you define wave events and you get a similar view. So you say, well, only 11 of the 32 strong wave events in the stratosphere are preceded in the wind events in the stratosphere are preceded by a strong wave event in the troposphere. And in terms of sudden warmings, it's only seven of the 28 warmings in the data record are associated with anomalous wave events. So what does that mean? That means that kind of the on aggregate, most sudden warmings are not caused by anomalous generation of wave activity in the troposphere. But that doesn't mean that not all of them are. So if you look through every event, there are certainly events where you've been able to, there's a block or something like that, that generates a ton of wave activity and you can trigger a warming like that. But in general, that's not the case. And that has implications for both the deterministic and probabilistic predictability on seasonal time scales, right? How are we doing on time? What are we? How long have I been yammering? It's 120, so yeah. What time did I? It's just 10 minutes. Okay, yeah, no problem. Because I can, I mean, I don't know what time I started, but I can, I can. I think, 10 minutes? I can make it short. So this is a paper by, that just came out by a co-author of mine and essentially this is one of these nudging experiments and it just hits home kind of some of the topic, some of the ideas that I just put out. And essentially the idea behind the paper was can you nudge the troposphere to be identical and then run ensembles to see if changes to the stratospheric basic state like you're asking, do they matter? So you nudge from essentially 10 hectopascals or I mean, 10 kilometers down. So the whole troposphere is identical in all of these experiments. The difference is, is that 20 days before a warming, you perturb the stratosphere to wind a little bit. And essentially what you see here are our time series of the wind evolution at 60 north of 10 hectopascals. And this is for essentially, they took the model and did a free run, found some warmings, you pick a single warming and then you run ensembles of that warming where the troposphere is nudged. The red lines show ensembles that with the perturbed stratosphere where you got a warming and the blue lines show where the perturbation in the stratosphere to the basic state meant that you didn't get a warming. And essentially when you look at the wave injection, so this is the wave flux between 45, 75 north that set a latitude on a wave flux coming up. And what you see here is for the red lines where you had a warming versus the blue lines where you did not. And this is up at 100 hectopascals or let's see, that's 10 hectopascals, 100 hectopascals and 300 hectopascals. Essentially what you're seeing here is that despite the fact that what's coming in from the troposphere or 300 hectopascals is identical, all you gotta do is bump the stratosphere and you can get a very different set of occurrences. And that tells you something about the fact that the nonlinear dynamics in the stratosphere and the basic state are important. It's not necessarily the injection from below. Another way to look at this is to look at the coherence. So essentially these are plots of height and frequency and essentially what Alvaro did was he said, well let's pick two levels and let's look at the coherence of wave activity with that level with all other levels. And the top panel, the left panels are the 300 hectopascals and the right panel is 100 hectopascals and therefore, whoops, did I, oh these are for the two different types of warmings. And essentially what you see here is that the coherence at 300 hectopascals, so that's in the lower upward troposphere, lower stratosphere, depending on your latitude, the coherence of wave activity is rather weak with the stratosphere. It's not coherent with what's going on above. It's more coherent with what's going on below. If you pick 100 hectopascals, there's little coherence below with wave activity at 100 hectopascals with what's going on in the troposphere and there's a huge amount of coherence with what's going on in the stratosphere. Okay, so what is that telling you? So if you recall in that Pulvanean wallpaper, they said they correlated 100 hectopascals with the wind in the stratosphere. What this is telling you is that that's essentially correlating the event to itself. They're taking 100 hectopascals and saying the wave activity that's coming up is strongly correlated with the wobbling of the vortex. What we're saying here is that that level is internal to the event itself. So whatever nonlinear dynamic is going on in the stratosphere, that's generating one of these events, if you pick 100 hectopascals, it is the event. So you have, if you wanted to say something about what's coming in from the lower stratosphere, you need to pick a level that's lower, something like 300 hectopascals, at which point you don't get any coherence with what's going on above because it's dependent on the stratospheric nonlinear interaction. So just to summarize, sudden warnings are not typically associated with anomalous troposphere flexes. The stratospheric basic state strongly matters. Sudden warnings have vertical wave flex signatures of some sort of resident or vortex interaction dynamic. The 300 to 100 hectopascal region seems to be kind of a crucial layer for determining whether one of these events happens or not, not so much of what's going on in the troposphere. And the deterministic predictability limit is somewhere in the seven to 10 day range. Now it's a little bit longer than in troposphere, right? In the troposphere, I don't, you know, 10 days is probably asking a whole lot out of a forecast system model. But because of radiative damping timescales, things like that, the stratosphere, that gets stretched out a little bit to about 10 days. So in terms of seasonal predictability, that's interesting because if you have 10 days of predictability on predicting one of these events, and then once an event happens and you've got 60 days of anomalous weather in the troposphere, it gives you something into the future that you can use. So with that, I could take any questions.