 Okay. I guess we're starting. Are we starting? We'll have presentations on Saturday afternoon. I guess we're expecting this. So we were thinking of doing it during the afternoon, say, 3 p.m. So the idea is to prepare a few slides on what you've been doing during the week and what hopefully you'll be doing in the second week and later if you want to continue this project. So again, the idea is to present what you've been doing so far and what we really want to do after. So Saturday at 3 p.m., let's say 10, maximum 15 minutes each group and then discussion about what you're doing and potential preventive interactions with groups as well. So today, Professor Jeff Valles from Exeter University talking about the structure of the mid-latitude atmosphere. Yeah. Okay, thanks. Yeah, I'm going to get onto probably mainly the tropics, actually. A lot of it would be theory they hardly sell. It's kind of the opening act. You know, rock bands have a main attraction and then they have an opening act. So I'm kind of the opening act and the main attraction starts. One next week, I guess, is a big main attraction for many people. Montsoons and Simone will talk about Montsoons tomorrow. So I'll give you a bit of background theory hopefully for understanding that. But before I do that, I wanted to show you a couple of movies which aren't related to my talk but they're actually very cool movies which I did not make. So here's one. This is actually model results. It's the ocean circulation. It's the absolute velocities of a big kitchen sink ocean model called Orca which is a European model or Nemo. And white is fast and blue is slow and you can just see. I guess there are two things. The scales of the motion are pretty small. Here's the crochet. Here's the Gulf Stream. It takes up lots and lots of eddies all over the place. Eddies with a scale of a few hundred kilometers. Is it working? Yes, but it can't get your video. Okay. So it's very pretty. I won't talk about the ocean at all today. But it seemed like a nice thing to show. The interesting thing is the next movie which superposes the atmosphere on top of it. So you can actually, we'll now see the atmosphere is a coupled model. And what we'll see is the precipitation field of the atmosphere superposed on the ocean. So here it is. There's the precip field. And there's two things to note. Just off the scale of the atmosphere is much larger. You can see these precipitation bands, which don't often go over Trieste, are much larger than the oceanic scales. And they're actually much faster than the ocean scale. The ocean is actually moving in this. It's not frozen. You have to look pretty hard to see the actual motion of the ocean. So, but if you focus on the ocean and look at a spot for a while, you will actually see the ocean move. But pretty slowly. So I mean the atmosphere is moving 100 times faster. It's scales are 10 times bigger. So it's time scales are 10 times faster. So you can see all these. Middle Attitude Westerly is belting along here. Less so in the tropics, but quite a bit of precipitation here. Here I guess is the ICCZ, which you're seeing. Not much rainfall at the equator here. Middle Attitude Westerly is bowling across. Okay. Anyway, that's a nice, just a nice thing to set scales. As I say, I won't really be talking about the ocean at all. Or probably the Middle Attitude atmosphere. I mean, after 30 years of lecturing, I still don't know how long my lectures will take. So I never prepare for a single lecture because it never works. Anyway, okay. So this is more or less what I hope to talk about. A little bit about the general circulation of the atmosphere. More about the theory of the Hadley-Sell. Quite a bit on that. At least the first lecture. Going on to tropical dynamics. A bit about radiative convective equilibrium. And then I snuck in a little bit about the... After Brian's lecture, I stuck in a little section because this is actually purportedly a summer school on multiple equilibrium. I put in a few slides on multiple equilibrium. And then some more about tropical dynamics. And probably I might or might not get onto this. And if you're interested in the theory of this, I've written a little book which you can download here. Well, there's a big book here which is 900 pages. But I've condensed it to the essence. So if you go to this directory, http tiny.cc slash valis slash essence, you can actually download the entire book for free. But it won't stop you from buying it because this is actually full of typos and mistakes. Signs are wrong. So you'll have to buy the book. Turns out most universities allow you to download the big book actually for free. Most British and American universities allow you to download the book for free from the Cambridge University Press, alas. Anyway, okay, here's the... Here's the medial overturning circulation. Perhaps the most... One of the most prominent features of the atmosphere. So H for Hadley cell, F for Ferrell cell. This is observations, if you will, a reanalysis product. The reanalysis product is... probably many of you know, it's just... you take all the observations possible and you assimilate them into a model. And the reanalysis products are used for the beginning state of a weather forecast, but ECMWF and NCAR and the Japanese GRA have gone back over 50 or so years and made reanalysis products of... of the historical record. Reanalysis products tend to be a little bit... well, they're better than pure observations because they produce fields where there are no observations. But because there are no observations or very few observations in some places, the reanalysis products tend to also introduce biases by the models, so they're not. So if you actually go to an ob... if you're talking to a real meteorologist, somebody who goes around in a plane or sends up balloons and you show them this and you say this is observations, they'll probably have a fit. They'll say, this isn't observations, anyway. But we tend to call them observations. Anyway, okay. So this is in... this Southern Hemisphere summer, Northern Hemisphere winter. So it's slightly asymmetric, a very large Hadley cell in the winter with rising motion in the Southern Hemisphere a few degrees south-south. The winter Hadley cell tends to be much longer than the summer Hadley cell. Then feral cells are apparently going the other way round in the middle attitudes and then the weak polar cell on either side. This is the zonally averaged zonal wind and the zonally averaged temperature in red. So the temperature just diminishing from the equator to the pole and diminishing with height. Westerly winds. Meteorologists have a strange habit of calling things westerly when they should be calling them eastward. It's very strange. It probably stems from the fact that people who are interested in where the winds come from because that's what determines the weather. But it's odd because you wouldn't say, I'm getting younger every day. It really makes sense, you say. Anyway, you should just be familiar with both westerly and eastward and use them interchangeably. So here are the winds. Westward winds, nearly all in the tropics, these are the surface winds. This is annual averaged, this is northern hemisphere winter, the bottom's chopped off. The winds have to average to zero. The surface winds have to average to zero because otherwise there'd be a torque on the earth and a torque on the atmosphere and then the atmosphere would spin up slightly to make the average values of the surface winds zero. So they have to be averaged to zero. Once you're away from the surface, they don't have to average to zero, of course. It's the vertical structure of the atmosphere. This is... The Americans have this habit of standardizing things and naming it after America. So this is the U.S. standard atmosphere. A strange concept. You know, like a pint of milk or a kilogram, but it's a standard atmosphere. Anyway, and these are actually observations, some of them re-analyses. But you see the structure of the atmosphere at the top of temperature diminishing up to about 10 kilometers on average, then about constant increasing slightly because of ozone up to the stratopause and then diminishing slightly. Ozone causes... is heating the atmosphere up here, but it's not the cause of the tropopause. You don't need ozone to have a tropopause. It will still be there without it. We'll hopefully come to that later. Here are some observations. Extra tropics in the green is cooler. There's a lower tropopause in the extra tropics. This one up here is the tropics. So the tropopause is about 16 kilometers up here, about 10 kilometers on average in the extra tropics, down to about eight in polar regions. And it's rather sharp, actually, in places that have tropopause. So we want to explain the structure, but I'm not going to explain it now. There have been lots of attempts over the years, and this is actually from a review article by J.J. Thompson. Thompson was the brother of Kelvin, of whom you've heard, Lord Kelvin. But they're both Tomsons. You should reference Kelvin's papers as Thompson. He became a Lord later on. So his papers are published as Thompson. People call him Kelvin. It's a terrible testament to the British class system, I'm afraid. What can I say? And Rayleigh, on the other hand, Lord Rayleigh, of whom you've also heard, was a Rayleigh from the get-go. He was a Lord from the get-go. He was the third down of Rayleigh. He was a gentleman scientist. He gave up his professorship because he didn't need the hassle and created a lab in his own house or in his own mansion, to look for experiments. Okay, enough of this. So this is... Well, this is a picture that Thompson and Ferrell put forward, essentially. And here's the Hadley cell. It's a big thing going round and round. And Ferrell, of course, is famous for the Ferrell cell. But his vision of the Ferrell cell wasn't really what we imagine it to be now. It was actually this cell sitting underneath the Hadley cell. You can just see it here going round the other way. And here are the trade winds. Middle Attitude, Westerlis. So the main thing that they had was the fact that even Ferrell had and Thompson was the fact that the Hadley cell goes all the way from the pole to the equator. And that was what they couldn't figure out. And later on it became apparent that the Hadley cell stops at about 30 degrees north. And that's what required explanation. And the explanation for that became clear. Well, much, you know, 100 years later, in the 1960s and then in the 1980s again, and that's what we'll talk about, why does the Hadley cell stop? You might think, and it's perfectly reasonable to think, that the Hadley cell, you know, heating things at the equator, it should air rises, it should just go forwards and sink and then come back. And it should sink where it's cold, which is the equator, that pole, sorry, and come back. It's not a done thing to think at all. So why doesn't it do that? But it's a sort of modernish view. This is just quipped from the book by Wallace and Hobbes. Today's view of Hadley cells, a Ferrell cell going round the other way, Tropesfate jet steam here, trade winds going west, converging at or near the equator in the ITCZ, the Intertropical Convergence Zone. And here are our middle attitude, middle attitude variability of Baroklinikedi's weather. And that weather, the cause for the weather, well, people, of course, have always known about the weather, especially British meteorologists because we get lots of it. The mathematical explanation for the weather came about really in the first half of the 20th century, first through a group in Bergen, the so-called Björkney School, who had a sort of phenomenological kind of description of it. And then into landmark papers by E.D. and Charney separately. And they each published a sole author paper in 48, 8 and 49 about the mathematical theory of weather of Baroklinikens' stability. And they were truly landmark papers. And so, okay. So, probably won't get into that much, but I do want to talk about why the Hadley-Sells stops round about 25 degrees north. Okay, and it, well, one reason it stops is that it can't get to the pole without spinning like crazy. And if it spins like crazy, well, it can't spin like crazy, for reasons we'll talk about. The Earth is spinning about once every 24 hours, something like that, especially 23 hours, 56 minutes. You think it's 24 hours, it's 23 hours, 56 minutes because the Earth is rotating around the Sun. So a day length of a sole when the Earth, the Sun is above the same place is 24 hours, but the actual rotation of the Earth, the value which gives the Coriolis parameter is slightly less. Not much difference because the year is so much longer than the day. But on Venus, they're more equal. And on Venus, the rotation period is 200 days, but the sole is actually 100 days because of the... Anyway, we won't worry about that. So the Earth is spinning, the speed of the surface here is about 400 meters per second. So you... Well, omega A at the equator is about 400 meters per second, and here's the angular momentum of a parcel of Earth sitting here, or sitting anywhere, which has a latitude, theta, but I'm not... Just to slightly confuse you, I use theta for latitude and potential temperature. Although, in fact, I'm not duplicating. This is... For those interested in Greek symbols, this is var theta, and it says something like in a squiggly theta. So let's suppose, well, as a parcel of Earth moves from here to here, in the absence of friction, if it's in the free atmosphere, it's going to conserve its angular momentum. So it's going to spin very fast as it gets up to here. I mean, not... Because there's also this cos squared factor. So it's really going to spin like crazy as it gets up to here. So its velocity, if it conserves its angular momentum, and it starts out with velocity of zero here, it's going to go like sin squared theta over cos theta. So it's actually going to become infinite at the pole. I'll be spinning extraordinarily fast, and I'll show you a graph in a second. So that would be the angular momentum conserving wind, which is most easily obtained just from angular momentum conservation. You might think, well, why does that come in the equations of motion? You know, when we write down the normal equations of motion, we don't think about angular momentum, we just think about, you know, the velocity and stuff like that. So where does it come from? Well, you can actually write down the equations of motion in a slightly... perhaps more familiar form, or perhaps less familiar. This is the zonally symmetric equations of motion. So we're averaging round a latitude line, so the pressure averages out. We're assuming there are no eddies here, so we get this equation of motion. It's a vertical velocity of small, oops, the vertical velocity of small and it's steady, then f plus zeta times v is equal to zero. Put that into spherical coordinates, we get this. And then, so one of them has to be zero. We're going to save, if there is a Hadley cell, it is not zero. So this thing in brackets must be zero and that gives you the same thing. U equals omega a sine squared theta. And as I say, well, I can... I guess I can make these slides available if people want the slides. You can make them. So that's going to give us a kind of picture of what the Hadley cell looks like. It's going to rise, presumably at the crater because it's warm. It's going to... it's going to go, polewood's going to sink somewhere. We're not, well, we say the subtropics, we don't know that just yet and come back. Now, it's going to come back near the ground. There's a lot of friction at the ground. That's where friction is. So the actual zonal flow is going to be weak. We're going to get a very strong zonal flow here. So you're going to get a large shear between here and here. And if you have a large shear, the U by dz is large. And if you've taken an elementary course in meteorology, you know the large shear gives you a large temperature gradient between north-south. So, and here's another piece of strange notation which I will use because there used to be, and I still am a half-time oceanographer, oceanographers have a habit of using B for temperature. Well, B is buoyancy, B for buoyancy, and it's G time to change in potential temperature over the reference potential temperature. So very informally, wherever you see B in these lectures, you can imagine it to be temperature. But it's buoyancy. So dU by dz is minus dB by d theta, or dT by dy. So, because dU by dz is so large, it means dT by dy is extremely large so that air is going to get colder and colder. And eventually, essentially it gets so cold it has to sink. And it turns out when you do the calculation, which we'll do soon, that it sinks at about 30 degrees north. So that's certainly one reason why the air sinks in the subtropics and doesn't make it all the way to the pole. If it made it all the way to the pole, it will be spinning extremely fast. But it turns out it does make it almost all the way to the pole on Venus, which we may talk in a minute. So, okay. So that's the essential reason now I'll do some equations. Think about equations. Here are some equations. I think you need the equations to do the calculation. At the end of the day, you should always be able to explain your equations in words, which, oh, I think you should be able to explain your equations in words. It's a good thing to try and, you know, you've done some calculations, you should be able to explain it. Even if, you know, you won't necessarily be able to do the quantitative calculation and you need the equations, you can't just explain things in words from the get-go or that's kind of pitty, you know, very nonsense if you do that. But really you need to, so I'll try and do that in this lecture. So even if you don't follow, and I always find it impossible to follow equations in lectures, almost impossible, or the details of them, you have to take them a little bit on trust. But I'll try to explain them in words. What we're going to do is suppose that the atmosphere is forced by thermal relaxation to some particular radiative equilibrium temperature. So if there were no motion, the temperature would be theta e, the potential temperature theta e. Then it would diminish like y squared as it would diminish as a girl pole would like this. So this is the radiative equilibrium temperature. But the actual temperature from thermal wind, because we know what the velocity is, so the temperature, I'm going to get b here, is slaved to the wind. The temperature doesn't get to choose what it is. We know what the wind is. So presumably it's in thermal wind balance so the temperature is slaved to that wind. So that gives us, we've determined what the wind is from thermal wind balance and angular momentum conservation. The wind is given by where it was. This guy. You're assuming that surface wind is zero? Yeah, surface wind is zero, exactly. And it's a uniform shear. So that gives us the wind. That actually gives us what the temperature is. And that temperature can't fall below the radiative equilibrium temperature. I mean, it starts out higher. This temperature is going to fall quite rapidly. When it falls, it's going to fall such that eventually it gets colder than the radiative equilibrium temperature and then it's going to sink. And if you do that calculation, you get this. And I'll show you those in this slide. And there's a slight complication. I'll come to it in a second, but so radiative equilibrium temperature is a red curve. So that's going like this, going quite rapidly. The angular momentum temperature is this blue line. So this is the temperature given by the angular momentum conservation, the actual temperature, which is slaved to the velocity field. So that goes like this. Bum, bum, bum, bum. And here, where it crosses over, that's what we presume to be the edge of the Hadley-Sal. And then after that, the actual temperature in this model just follows the radiative equilibrium temperature. So there's just motion here. And then on average over the entire Hadley-Sal, if the Hadley-Sal is sort of self-contained, the average radiative equilibrium temperature has to equal the average actual temperature because overall the atmosphere is neither being heated nor cooled. So the average of the radiative equilibrium temperature has to equal the average of the actual temperature. And that gives us the absolute value of the temperature, whether how hot it is or how cold it is. So that gives you a little bit of mathematics to do. It's just a calculation that you have. I'm not going to can't do here. Go away and do it. It's in my book always. And you get this to be the edge of the Hadley-Sal. And this particular calculation was done in the paper by Heldon Howe, who was really Arthur Howe's PhD thesis in a way. The idea of an angular momentum conservation, being a major part of the Hadley-Sal, actually comes from a paper by Ed Schneider, E.K. Schneider in about 1977. I should have referenced him here. So that gives you the edge of the Hadley-Sal. And it's parameterized by this non-dimensional number here. Delta theta H is the temperature difference between the equator and the pole of the radiative equilibrium. How many give us the rotation rate of the Earth? A is the radius of the Earth. So this is like a, it's called a thermal Rosbin number. And here the, this is what it looks like again. Same plot here. This is actually what the wind looks like. And you can see, well I've only plotted it from 0 to about 45. If we continue with the angular momentum conservation, well you can see already it's 300 meters per second. Never guess that far, of course, because it stops, stops around about here. And Ed Schneider actually thought that, well introduced partly the notion of cumulus friction, which would slow it down. That the clouds would be providing a lot of friction and slowing it down. But in fact, that's believed not to be so important these days. You can then, you can also calculate the Hadley-Sal strength from this model. And to do that in a simple way, you can suppose that the, in the thermodynamic equation of rough balances, Wd theta by dz is theta E minus theta over tau. So this is the thermal forcing on the right-hand side. Theta is the actual temperature. Theta sub E is the equilibrium temperature. So that gives you an estimate for what W is, if you know what the height is. And the height is around about the height of the tropopause, which we haven't said what that is. That's a separate calculation I hope to come to later on. Then you go through a little bit of mathematics, and you get this estimate for the vertical velocity. This is the vertical, well this, sort of this, yeah, at the equator. Well this is an overturning circulation. This is the strength. Are they? So eating the equatorial, eating down to play at all? Well, it does in a sense, well, sort of. It all comes into, I mean, here's, this is an estimate for the strength of this psi. So psi is such that d psi dy is W minus d psi dz is V. And I will show you, I'll show you once, how the Mohr's Hadle cell, this is my only slide, I think, on the Mohr's Hadle cell. But I think that Simone might talk a little bit more about this tomorrow, or you'll hear more about it. The Hadle cell, in this picture at least, and you can argue with the picture, is not driven, if you will, by moisture or convection. It's not. The Hadle cell doesn't arise because we have convection at the equator. The Hadle cell arises because there's a differential heating between the equator and higher latitudes. So if everything were the same temperature at the equator, we wouldn't have a Hadle cell. We might just have convection everywhere, but we wouldn't have a large scale overturning circulation. So the Hadle cell is driven, if you will, driven as a force word in our field. Some people, you know, what does it mean? Does it mean controlled by, or is it a mechanism? Think about who drives the car, the driver drives the car, not providing the power, but oftentimes use it in different ways. But the Hadle cell is not driven by convection or by moisture, but they make a difference. It's really driven by the differential forcing between the equator and higher latitudes. So here, for example, this red line, again, is a solution, the actual temperature solution, because that's slaved to the velocity field. Now, theta e here is the radiative relaxation temperature, the temperature it's relaxing back to in the dry model. And you can see in the dry model it's not the actual temperature, not all that much different from the radiative equilibrium temperature. Now, if we add moisture, moisture is releasing a lot of heat at the equator, less so at higher latitudes. If you want to put moisture into this framework, we can sort of imagine an equivalent radiative equilibrium temperature which takes into account the release of heat due to moisture by increasing the radiative equilibrium temperature near the equator, which is sort of a simple way of including the effects of moisture. So this is the radiative equilibrium temperature that I've just invented if we have a moist model. So the solution temperature is still exactly the same. So there's a much larger difference between the solution temperature and the radiative equilibrium temperature if we have a moist model or if the atmosphere is moist. Therefore, we expect the circulation to be actually much stronger in a moist model than in a dry model. And I think actually some people are doing projects related to the Hadley cell with and without moisture. So it would be good. One thing that you'll be able to look at is whether it is indeed strong, more strongly. And was there a question? It's been set by my imagination in this plot and just saying that when we have moisture, it will be releasing heat to the equator. Therefore, the equivalent radiative equilibrium temperature would be higher. In a sense, it's partly because the actual, and probably shouldn't take this too seriously, but there's this equal area construction so that the actual solution has to be, the area on this side has to be equal to the area on that side. So if theta e star is high on this side, it has to be lower on this side, or the red line would actually adjust to make it so. But maybe we can talk about that later if it's not entirely clear. So, yeah. The Hadley cell is driven by convection. But the Hadley cell is driven by a very general difference of convection. Yeah. That's right. Why not driven? But affected by... Yeah, affected by, certainly. Yeah, that's right. But if we had a dry model... Yeah, dry model. We also get a hardly cell. Maybe radiation difference. Yeah, that's right. Yeah. So I'm not saying it's not important. I'm just saying it's not the driving factor, if you will. I mean... The moisture is affected by... Yeah. Different... Yeah. Yeah, that's right. I mean, this... This, you know, whatever I call it here, this theta E is what I'm... Is a proxy for all of the heating. So we're calling it the radiative equilibrium temperature. But this is a proxy for all the heating which you might have. So we call it the radiative equilibrium temperature. But if you want to include the effects of moisture in it, you would just... You would modify this theta E if you wanted to try and fake it with a dry model. So... So even a dry model on the right-hand side of the thermodynamic equation, there would still be a Q... Even a dry model due to radiative... Yeah, that's right. Exactly. Exactly. Yes. So... Yeah. Yeah, so the circulation is stronger. So I'm arguing here that moisture is enhancing and not causing the hardly cell. And perhaps we'll hear more about this tomorrow, I don't know, and next week. All right. So let's just go back to this guy. Here... The shear is increasing like crazy. So... What else might stop the hardly cell? Is that this flow becomes unstable because the shear is so large. And that's... In a sense, that's the traditional view of why the hardly cell stops. And that's discussed in Lorenz's monograph. There's a famous monograph by Lorenz in 1967. It's well worth looking at. It's very hard to obtain. All the general circulation of the atmosphere. It's been out of print for years. But it's a nice sort of... It's a transition to the modern view. It talks about bioclinic instability, the hardly cell, and so on. But at that time, he doesn't know about all these modern concepts that Insik talked about yesterday, pseudo-momentum, blah, blah. Or even the cause of the westerlies was not understood at that time. But it's... But the notion there was that the temperature gradient between the equator and the pole was sufficiently large because Edie and Charney had done this, was sufficiently large that the middle attitudes would become unstable, and therefore you couldn't get a hardly cell going all the way from the equator to the pole. So... Here we are. Again, I won't go through it because you need an entire course. In my GFD course at Exeter, and the one I've used at T-shirt Princeton, we had a 12-week GFD course, and at the end of it, we were about to get on to bioclinic instability. Sarah, probably, remembers. I don't know whether you took it from me or somebody else. From me? Okay. And so... And then we get to that in the second term. So, obviously, you're not going to... I can't do the derivation, but I'm just going to say that when the shear gets to be sufficiently large, so sort of explaining it in words, it will become unstable. And so that shear... There's a formula for it. It depends, in particular, on the rotation rate and the latitude. So bioclinic instability is inhibited at low latitudes. It's inhibited for two reasons, two connected reasons. One, the Coriolis parameter itself is small because you're close to the equator. And one, the beta effect... Beta is the fdy. It's large, and on the sphere, beta is 2 omega over a cosine of latitude. So that becomes... So that's large at the... So this is the threshold for bioclinic instability. This guy is very large here. This is the actual angular momentum conserving wind. This is our solution. So when we get here, which is about 20 degrees, the angular momentum conserving wind, the shear is sufficiently large that it will become bioclinically unstable. And then the Hadley cell will terminate because you'll get eddies and all sorts of things. And that's the latitude of which that occurs. Ed needs to measure the stratification. H is the height of the tropopause. Omega is the rotation rate. A is the radius of the Earth. And you can see that as the rotation rate diminishes, or the size of the planet diminishes, the Hadley cell extents will increase. So that's another thing. And that is true also in our... So this is in our model, which was purely angular momentum conserving. Here it goes like just 1 over omega a. Here it's going like 1 over omega a to the one-half power. But these scalings, you might even... If you were doing the Hadley cell experiment, you might even be able to test which one actually works. Well, it's very hard to actually give you fair warning. It's hard to test these things in a real... Well, even in an idealized model like this, it's hard to actually test things because the results always end up being ambiguous. And it kind of looks like this, but it's got a little bit of this in it. It doesn't quite work as well as it. Oh, this is using the Phillips model of baroclinic instability, which is the Phillips two-layer model of baroclinic instability. So you have to have a... So if you have a beta, there's a critical shear for instability. That's where that's come from. That's the logic behind it. Yes, okay. It's another ambiguity. Yes, it does actually because beta is this and has a latitude in it. So that's where beta is. So I've used this value of beta in here. So I replace beta by 2 omega a cos theta over. And then I use that. And so beta and f get kind of conflated. So that's why there's a... It looks as though it's an f-scaling. It looks as though it's an f-scaling, yeah. But it does raise the other point that in other models of baroclinic instability, there's no critical shear, like in the ED model. There's no critical shear. So nonetheless, there is... So the exact calculation, or the calculation is not exact, I should say, but the notion that it will become baroclinically unstable. And it even, perhaps as a slight aside, in a sense for the professionals, the Hadley cell may sink before it becomes truly baroclinically unstable. You may have a baroclinic zone at higher latitudes, which is creating rosary waves, which are propagating equator wood. This is the full equation. These rosary waves are breaking here and causing the Hadley cell to sink even before you get into the baroclinic zone. So it's a little bit complicated to do the exact calculation. That's why we have models. You can't do everything with pencil and paper. I'll show you a few numerical simulations. This is a dry Hadley cell in a zonally symmetric model. So you have a Hadley cell. It's relatively weak, but it exists and it terminates about 30 degrees outside of the equator. This is what happens when we actually make it three-dimensional and put eddies into it, baroclinic eddies. You get a much stronger Hadley cell. It's slightly narrower, I would say. The edge of the Hadley cell is slightly narrower than it is here, slightly lower latitude. Then you get these ferrocells on either side. This is the zonal wind in these simulations. This is the zonal symmetric calculation. So the zonal symmetric calculation, the dashed line here is the angular momentum conserving wind from the calculation. The solid line is the wind from the simulation. You can see in the zonal symmetric case it almost is following the angular momentum conserving wind until you're about 30 degrees north. In the three-dimensional simulation, it's not because the middle attitudes, baroclinic and stability sense suck out angular momentum, transferring angular momentum out sort of in the way that we heard in yesterday's lecture from INSEC. Oh, here are the two pictures together. So I guess I should break, actually maybe I should break now for five minutes before talking about the seasonal cycle. Yeah, take a few questions before we go ahead. You've given us three different rationales somewhat different from a finite extent. Okay, I thought I'd only given you two. Let's talk about that. One is a thermodynamic in the thermo wing. The shear hits really large because the wings have to spin up really fast as you move toward the angle of the pole. So at some point, it gets too cold and has to sink. Yeah, that's what I said. It gets too cold and has to sink in the center. Yeah, that's one. That's the notion that this, because the wings spin up very fast, they become baroclinically unstable. So the angular momentum and the conserving circulation can't exist in the presence of that instability. That's right, because you've got this term. And then this is a slight variation on that. So this is 2.5. This is 2.5, right. Yes, that's right. But it's still due to baroclinic instability. So it's a variation on that theme. On the rotation. Exactly, lecture two. Yes. So we're on the same page. Yeah, we're on the same page. Yeah, any other... It's different, right? The scaling load. The scaling load, yeah, that's right. Indeed, yes. The scaling law is different to the extent that you believe the scaling law for this baroclinic instability. It goes like 1 over omega a to the one-half power. And here it's going like 1 over omega a. So the scaling law is different, but it's still going... And differentiating between those scaling laws, I don't think it's been truly successfully done in the literature. Yeah, I mean, people have argued, oh, you know, this, that, and the other, but it's not... A lot of... You know, it's... The real earth, unfortunately... Well, the earth doesn't behave like a real theory sometimes. So it's a little bit ambiguous as to... I mean, probably both effects are important. It's not necessarily an either law. And furthermore, in the seasonal cycle, you may have slightly different scalings for the summer versus the winter. In fact, the seasonal, which we'll come to in a minute, the Hadley cell is almost... It's strongly seasonal. So to even talk about the annually averaged Hadley cell is missing half of the story, because, you know, perhaps we should really be talking about the summer Hadley cell and the winter Hadley cell, and it really tries to go fairly quickly from one to the other. And that transition is part of what makes up the monsoon, which we'll hear about more later. Okay, yeah, you, because... Sorry. Yeah, most of it is a geometrical argument. Here, you're just going to have to go through the equations, but y comes in in a number of ways. I mean, f, if you do it, f is 2 omega sine theta, which is like y, because for small angles sine theta is theta. So, you know, f is f0 plus beta y. So it comes in there. And it comes in in the fact that we want a sphere. So it's coming in both dynamically. This is sort of a dynamical argument and a geometric argument just because we want a sphere. So you get y's for different reasons. You could actually do the problem. And if you're interested, you can look it up. Actually, my big book. Not in the little book. I did the argument in purely Cartesian geometry on the beta plane, and then you lose the spherical effect. But you still have this effect. So it comes in in a couple of different ways. Yeah, yeah, yeah. It depends what you mean by drive, of course. I mean, you can... Yeah, that's right. Yeah, there's downward motion, exactly. Yeah, that's right. Yeah, there probably is. Because, yeah, this is... This same argument comes up in oceanography, actually. People talk about the mid-yone lower turning circulation of the ocean being driven by convection off of Greenland or off of... But, yeah, if you take... I mean, here's a cloud. You've got a lot of upward motion here. You've got downward motion on the other side. You've got a hardly cell like this. We think of vertical motion in the atmosphere as being small. If you're actually in the core of a strong cumulonimbus, the vertical velocity here can be 10 meters per second. It can be really large in the middle of a cumulonimbus. And you get this compensating subsidence on either side of it, which people talk about. So this overturning circulation is not just a consequence of that. You've got all this downward motion, which is much weaker on either side. But if you take the net effect, there'll be some circulation like that. Yeah, the back, John. And when John would explain the Hadley circulation tools, he would emphasize the pattern of surface winds, right? So that if you've got barricaded headies in middle attitudes, they think they're rospy waves in the trompetes, and that leads to momentum transfer out of the trompetes. Yeah. So easterly, isn't it? Right, yes. And then the aegis-tropic flow connected with the surface wind induces an overturning circulation. Now, this process is all before Helden, whom an angular momentum can serve. Yeah, yeah. So what's your take on that? There is an element of truth to that. There is an element of truth to that. And really, I think that is, yes, I suppose that is sort of this equation, that these are providing an angular momentum source. They're sucking momentum out from the tropics. And if you can create, and that actually gives you a much stronger Hadley cell. So the Hadley cell, and there's only symmetric, the Helden-Howe Hadley cell, whoops, where are we? Let's go to the simulation. The interesting thing is, you can assume u equals 0, the ground. Yeah. Whereas Green's perspective was not to do, yeah. Right. Yeah, that's right. Yeah. I think, in my view, the Helden-Howe Hadley cell is way too weak. The Helden-Howe Hadley cell is not, the annually averaged Helden-Howe Hadley cell is far too weak. It does not conform with the real Hadley cell, even on the annual average. So I kind of regard, well, my view of the Helden-Howe model is that it is a model for the ideal Hadley cell on a planet in which there are no eddies. And its contribution is to show that the Hadley cell actually sinks at about 30 degrees north, so that I think of it as in the same spirit of these models. And it shows you that this ideal Hadley cell cannot exist. But it's not a realistic model of the actual Hadley cell, although it may have to do better in winter than in summer. But maybe we should talk about that after the break. So we'll still have a five-minute break. But now we'll come back a quarter off. But keep the questions going, because that's good. So you have your plot where you have the crossover between the barotonic instability argument and the angular momentum conserving with it.