 As well as a balance at the top of the atmosphere to get the outgoing radiation equal to the incoming, there has to be a surface energy balance. And once you satisfy the surface energy balance, then that determines where this intercept is, then given the lapse rate, it determines where the chopper pauses. So one difference between the middle attitudes and the tropics is that the lapse rate is different. Then it gives you this prediction, so the tropopause height must go up with global warming, so more greenhouse effects gives you more warming. And of course the planet with the biggest greenhouse effect in the solar system, depending on how you define an atmosphere, is Venus, which is kind of similar to Earth in some ways, but different than others. Radiatively, the surface pressure is 92 bars, 100 times that of Earth. It's all carbon dioxide. So when we talk about global warming, we're talking about going from 300 parts per million to 600 parts per million, Venus has got about a million parts per million. So, you know, already it's 1,000, 3,000 times more concentration, 3,000 than 100 times more atmosphere, so 300,000 times more carbon dioxide. So the greenhouse effect, now there's so much greenhouse, the pressure, there's so much greenhouse that if Venus were a little bit colder, it would actually condense and rain carbon dioxide because it would, the pressure would exceed the saturation vapor pressure of carbon dioxide. So almost on the verge of actually getting carbon dioxide rain on Venus, the reason we don't get carbon dioxide rain on Venus is actually because it's so warm, because there's so much carbon dioxide, so the surface temperature is about 700 degrees Kelvin, so that's above the saturation vapor pressure. But if Venus, if carbon dioxide were not such a greenhouse gas and the temperature would cooler, some of the carbon dioxide would condense out and you would get rain. Anyway, that's a snippet, so surface pressure is 92 bars, enormous greenhouse effects, so we expect the tropopause to be higher by our previous arguments, and ta-da, there it is. These are actually profiles from the pioneer mission. It's kind of interesting, they had trouble with these of course because they get crushed on the way down, but some of them were successful. So these are profiles in various places in Venus. Here's about the tropopause, 60 kilometers high. And Venus, as far as I know, has no, it has clouds, it doesn't have towering cumulonimbus as far as I know. It has lots of clouds actually above the tropopause up here. So it has convection. No cumulonimbus, no ozone, still has a tropopause. Nearly all the planets in the solar system have one form of a tropopause or another. It's a very robust feature of the planetary atmosphere. The other interesting thing about Venus is that these are all in different, quite different latitudes, different locations, but the temperature is almost the same in all of them. Venus is enormously isothermal until you get to very high latitudes. And part of that, the reason for that is because the hardly cell is so big as we talked about earlier. The hardly cell goes to about 60 degrees north and south and in the period of that hardly cell, in the extent of that hardly cell, the temperature is varying very little indeed. And that's, the lack of temperature variation has been confirmed by later measurements than these. OK, the last, I'm probably not going to get on to the middle latitudes, guys. So I'm probably not actually going to talk about the bolus flux after all. Oh well. See, I told you my timing wasn't very good. Why? Oh, because it's not in radiative equilibrium. If it were in radiative equilibrium, it would be uniform, but obviously it's not in radiative equilibrium. I don't know what determines this particular thing here. You actually have clouds up here which are emitting in for VED, so it's certainly not. So the trouble pose isn't really sharp. It's kind of a merger up here. But yeah, a lot of clouds up here. Sulfuric acid clouds. Not nice, rain clouds. Venus is not in radiative equilibrium? No, the entire planet, of course, is in radiative equilibrium. OK, just that one? Yeah, just the, but up here it's probably not in radiative equilibrium, but of course the entire planet is in radiative equilibrium. And in fact, the incoming, the net incoming solar radiation on Venus is very similar to that of Earth. Venus is closer to the Sun, so it gets more gross incoming solar radiation, but it has quite a high albedo because it's covered in clouds. So it reflects more. So the net incoming solar is quite similar to that of Earth. And nonetheless, pretty toaster down here. You wouldn't want to, you get crushed and roasted. I don't know. I guess the crushing would kill you first. I don't know. Anyway, OK, let's, let's not worry about that. I don't know whether this will be the last. I just wanted to have a little, since we've been talking about radiation and since Ryan talked about isalbedo, I thought I'd talk a little bit about water vapor radiative feedback in a very simple way. Isalbedo feedback is sort of relatively simple to understand. Some initial change. So it gives you cooling, increased snow and ice, high reflectivity. Less solar radiation, more cooling. Round and round you go. Water vapor radiative feedback. Again, not of a similar ilk. Suppose you have a warming perturbation. You get increased atmospheric water vapor. More greenhouse effect, more warming. We tend to argue them these ways. There's no reason why these arrows can't go the other way round. I mean, isalbedo feedback can be a warming feedback. And the greenhouse effect can be a cooling feedback. If you reduce the temperature, you get less water vapor and it gets colder and so on and so forth. So it's nice to have these feedbacks. I mean, the cloud feedbacks are a bit of a diversion. One of the problems with climate science is that there is no nice loop. We kind of, you know, what would the loop be? You know, if it warms, do you get more clouds or less clouds? And part of the problem there is it depends upon the time. And part of the problem there is it depends upon the type of cloud in detailed ways. So low level clouds tend to have a high albedo, but not much of a greenhouse effect. So they would cool the earth. High clouds might do the opposite. Not a particularly big albedo effect. They do have a greenhouse effect, so they might warm the earth. Thick clouds, who knows what they do? Who knows how they change with global warming? So that's why there's no lecture on clouds. None for me anyway. Let's try and think about just multiple equilibrium due to water vapor. This is part of the simplest possible EBM. You can see Epsilon sigma T to the fourth is S1 minus Alpha. T is temperature, say, Earth has one temperature. Sigma is Stefan's constant. S is incoming solar radiation. Alpha is the albedo. Epsilon is something there, because the earth is not a black body. So if we imagine T is the surface temperature, then certainly we need Epsilon to be less than one in order to get the surface temperature more or less equal to the observed surface temperature. If we just take Epsilon equals to one, putting known values for the right-hand side, we get a well-known value for the surface temperature of about 255 Kelvin. Some 20 or so degrees colder than it should be. So we want to fix that. We fix it with an Epsilon. So I saw a bit of feedback just makes Alpha, the albedo, a function of temperature. I want to make this emissivity a function of temperature. And I'm going to do that because there's more water as the atmosphere gets warmer, you get more water vapor, more greenhouse effect. That's more or less the amount of water vapor in the atmosphere with the zero-off order approximation to how much water vapor there is at any given place is a temperature. If the temperature is higher, a given volume can hold more water vapor. And that's the classiest claperon relation. So where it's hot, you get more water, where it's cold, you get less. So for example, we don't always process that. So we think of England as being rainy, which kind of is. But in fact, if you compare London to New York, London has more rain days. The average rainfall in London per year is something like 50 centimeters per year. New York, the average rainfall is about 100 centimeters per year. It's twice as much in New York. And the reason for that is it rains heavily, more heavily when it does rain in New York than it does in London. Because it's hotter by and large. So temperature is the zero-off order approximants, how much water vapor there is. The solution of the classiest claperon equation tells us that the water vapor pressure increases approximately exponentially with temperature. Not exactly exponentially, but more or less. So let's go back to our, let's also not worry about convection. We could put that in, but let's not worry about it. Now let's go back to our radiative equilibrium model, which tells us that the surface temperature is equal to the emitting temperature times 1 plus tau nought. The tau nought is the optical depth of the surface. So a bigger optical depth gives us a bigger surface temperature. This emitting temperature here is just solar constant times 1 minus the albedo. So we know that. We'll take albedo to be fixed, 1 plus tau nought. So our epsilon, in our previous slide, we had epsilon sigma t to the fourth. Now our sigma t to the fourth is this times 1 plus tau, so our epsilon from the previous slide is just 1 plus tau nought. The inverse of 1 plus tau nought. So for the simplest model, let's make our water vapor feedback. Let's make our tau nought a function of temperature A plus B times the saturation vapor pressure, which is this exponential function of temperature e to the gamma t. So then we can put this in here. So then we have sigma t to the fourth equals s, 1 minus A, 1 plus tau, and tau is A plus B is this exponential function of temperature. So now we have a more complicated equation to solve with the possibility of multiple equilibrium. And because we can imagine we can get a cold dry climate would be one solution with a low chopper pause, of course, or a warm wet climate. Now whether we actually get that on Earth, you certainly couldn't use a simple model like this to predict whether you get that on Earth, but we get the idea that there might be more than one solution. So we can just, well, one thing we can solve this equation. It's a bit tricky to solve that analytically because we've got an exponential function here, which is sigma t to the fourth here, but we can solve it graphically or numerically. So I'll do it two ways. When I'll just plot the surface temperature, I'll just plot the left and the right-hand side and see where they intersect. And that's what I've done on this left-hand plot here. This is actually just the surface temperature. The dash line is the surface temperature increasing, and this is the right-hand side. So we actually get two solutions there and there for a given solar constant, a warm, wet solution and a cold, dry solution. And then if I actually plot it for a whole range of emitting temperatures, and this is the ground temperature, this is the emitting temperature which we know, we get two solutions. And it's kind of interesting that we get two solutions and that in this solution is sort of what we expect that if we turn the solar constant up, the surface warms, this solution here, if we go from here to here, turn the emitting temperature up, turn the solar constant up, the ground temperature goes down. And we saw that kind of behaviour in some of Brian's Curves yesterday with our Albedo feedback, and it's very similar. And if you go to here, to kind of see it, if you go to this curve, I now have some coloured chalk. So here's a surface temperature. Here's one curve. Now suppose I increase the emitting temperature, so this is my emitting temperature, which is really like the solar constant, one minus. If I increase the solar constant, that curve is going to go like this. So the surface temperature of the lower solution will go up. The surface temperature of my other solution will go down. So we get this additional branch. And it turns out that this branch is actually unstable. If you're actually sitting on here, if you found the solution, and somebody comes and kicks you, you'll actually fall down to here. So it's kind of interesting. You can see what's happening. Suppose you're sitting here, we're on earth, and earth moves towards the sun. A major mistake in a geoengineering project. And we move towards the sun. We get warm and warm and then... What happens here? So in this model, you then cease to find a solution. You cannot find a solution, temperature. There isn't a solution. It becomes infinite in this model. Well, that will be an extreme example of a runaway greenhouse. You're just getting warmer and warmer. Now, of course, there's something wrong with this model. This is one of the limitations of grey radiation. We're assuming that the entire emission here, the sigma t to the fourth, which is a function of tau, which is a function of water vapor, all of the emission is affected by the presence of water vapor. In fact, things aren't quite as bad as they seem, because there are windows, appropriately named windows, where radiation can escape from the earth's surface more or less unimpeded by the presence of absorbers in the atmosphere. So there are infrared windows, and we can add an infrared window to our model by, I suppose, just by... I don't think I've actually written down the equations. No. By, in a sense, making the emissions have two bands to it, one of which is affected by the absorption of water vapor and one of which the radiation can escape anyway, no matter what, which is a little bit more realistic, and then I put that in to the model. So look at this curve, and then any guesses what will happen when I add another band, any... any penalty for being wrong. What's going to happen is we're going to get another band up here. So these curves kind of look really quite similar in a qualitative way to the curves that you get with isalbedo feedback. So this is one branch, perhaps a branch. We're on... This is an unstable branch, and this is a very hot branch. Now, probably Earth doesn't have a behaviour like this. You'd have to do a more complicated model to do this. You can calculate this with a little Python code so I can make this Python code available to anybody. It's like 100 lines of Python to do this. And it iterates to find the solutions. But, you know, you're going along here, Earth is getting closer to the sun, and you get to this point, and then you would shoot up to here to find the solution. So this is a slightly less extreme runaway greenhouse. But it's actually seriously believed that Venus went through this process by looking at the isotopic composition of its atmosphere. It looked like it might have had a liquid ocean a long, long time ago. But that... The liquid ocean boiled, the atmosphere became saturated, it got warmer and warmer. In fact, it got so much warmer that the water vapour was lost to space, disassociated, and was lost to space, and now we no longer have any water vapour on Venus. But we still have a very strong greenhouse effect. So... So that's... That's multiple equilibrium. You can also, although I probably should have shown this, but I haven't. You can actually add then isalbedo feedback to this model by making alpha a function of temperature. So you can have a joint isalbedo feedback and a moist radiated feedback. And what happens then is that one or two, either this branch or this branch, will themselves bifocate. And this branch itself should probably more likely, because there's no isa up here, would bifocate and you get a very cold, you get two branches because of the isalbedo down there. So it's kind of fun to play with these models. And it was one way I taught myself Python was to write up this little model. It's quite fun. Then in my last ten minutes, I will, or whatever, I'll try and finish the 12-ish. I will actually talk about scales of motion. Back to dynamics. Scales of motion in the tropics and middle attitudes. And just introduce you to one important concept called the weak temperature gradient approximation. The weak temperature gradient approximation says temperature gradients are weak in the tropics. I think that term is coined by Chris Brethren. It turns out the idea actually is due to Charney. In meteorology Charney, all loads go back to Charney in a sense. It's also a nice notion of scaling. B is temperature again. These are the equations of motion for an atmosphere. Actually it's a Boozinesque atmosphere, but it doesn't matter. U is horizontal velocity Coriolis. The advected derivative pressure. This is the hydrostatic equation. It's the thermodynamic equation. So n squared is the stratification. W is the vertical velocity. And there's the mass conservation equation. This is the Boozinesque equation. It's actually almost the same equations occur in pressure coordinates. So if you're actually a dyed in the world meteorologist and you like pressure coordinates, then you can just change z for p and you more or less get the same equations. You still get this incompressible equation in pressure coordinates. But if you're not a dyed in the world meteorologist, then the notion of actually using pressure as an independent variable seems totally weird because pressure is a field. You wouldn't think of using velocity as a coordinates. But anyway, so I kind of like to think of this is my oceanographic heritage coming back. I like to think of the Boozinesque equations. So then you just scale the equations so what that means is you say that horizontal scales are about a scale L, z is some scale height, H, et cetera. And then what happens then is you get a bunch of non-dimensional numbers which pop out and the equations are written in non-dimensional form. These familiar non-dimensional numbers pop up. The Volsby number, the Berger number, the Richardson number, famous names. Richardson was the, well, Volsby, we all know who Volsby was, a famous Swedish meteorologist founded the departments in Chicago and MIT largely. Richardson was LF Richardson, Lewis V Richardson was a British meteorologist at the turn of the century, last century. He was one of the, he envisioned numerical weather prediction. That's what he's most famous for. And he actually tried to do numerical weather prediction by himself, by hand. So he differenced the equations and then tried to solve them by hand to give a forecast. And it was, he got a pretty bad forecast. A very bad forecast because, well, not only his resolution was wrong, he didn't know about gravity waves, so he got enormous gravity waves dominating his solution. And it was completely wrong, but it was a pioneering effort. And he actually envisioned, and you can read about this in a biography of him, that the future of numerical weather prediction will be there to be like a concert hall, like it felt like this. I'll be sitting here, and I'll be saying, you calculate the derivative over in this grid point and you do this and you do the thermodynamic equation. And you'd all be calculating by hand, and the results would be passed to somebody who would collect them together. And then you'd go forward a time step and then you'd all repeat it, you see? It would be, I mean, not only did he envision numerical weather forecasting, he actually envisioned parallel computing. Now you have a thousand people in the hall and they're all doing calculations, and there's a little controller here, and there's a little, you know, a high network band passing between the two. So, anyway. And then, of course, the first successful numerical weather prediction was passable new weather, was Charny again. Fiwrtoffton von Neumann in the early 50s using a very simplified set of equations in Princeton. So, Richardson was a pacifist. He drove ambulances, and he resigned from the UK Met Office because he became part of a Ministry of Defence, so he didn't want an interesting fellow. OK, anyway. But think about back to the weak temperature approximation. Let's take the momentum equation and the hydrostatic equation and scale them. So that tells us that the pressure, this equation, scales as F times U times L because this vertical derivative is a, this derivative here is a horizontal derivative. So it's d5 by dx, so you get an L here. And then the buoyancy is 5 z, so it's FUL over H. That's where you get a middle attitude, simple scaling. The tropics, there's no F. Key difference, F is 0 to leading order, sort of. So, this scales, this term here scales like U squared over L. So, the pressure scales like U squared. So, now U squared, F, if we compare how big these two fields are, F times U, F naught, which is the middle attitude value of U, is going to be bigger. I'm sorry, it's going to be smaller. It's going to be bigger than U squared. U squared is less than F naught UL. U squared is less because F is big. Okay? So variations of pressure and temperature are smaller in middle attitudes than they are in the tropics. Right? U squared is less than FUL. Small rosby number of assumption. So that is the weak temperature gradient approximation. Variations of pressure and temperature are smaller. There's actually a bit of a hidden assumption here. The hidden assumption is that the actual winds are similar, or at least not any bigger, in the tropics than they are in middle attitudes. If they were bigger, you could kind of invert the argument or something like that. You might say, okay, well let's suppose the pressure gradients are the same in middle attitudes and the tropics, then you would have to have very big winds in the tropics. But in fact we argue or it is conventionally argued that the winds are about the same, so the pressure gradients are less, and therefore the temperature gradients are less in the tropics. That's the essence of the weak temperature gradient approximation. That comes from Charney. I forget exactly when. Does it work? Here is a snapshot of pressure, temperature and wind. This is pressure, essentially, geopotential temperature. The wind on a particular day, I chose this day because it happened to be my birthday. I thought it was a suspicious day. So I went to the Rinald, and there was a nice blocking high over the UK then. But you can see, look at the wind, look at the geopotential. Almost no gradient of pressure in the tropics. Compared to the winds, I mean just look at, you can just blur your eyes, stand back and look at the, the same number of contours in each plot. So we're not deceiving you, the same number of contours in each plot. So there's a lot of variability of the winds, but very little variability of the geopotential, very little variability of the temperature. So that's the weak temperature gradient approximation. Is this surface temperature? No, it's not surface temperature. It's probably 500 hectopascals. But I cannot remember. Similar to a magnitude, because of the courier's parameter, and the phi should be a small root of the tropics. Yes, phi should be. It's a simple consequence of geometry. So why are you... Wait a minute. No, no, you have to invoke the dynamics. You have to invoke f. If the winds are similar, phi has to be smaller, because there's no courier's parameter in here. Yeah, that's the argument. It's a simple argument. I'm not invoking it. It's not a complicated argument. Geostrophic wind, yes. The wind is geostrophic in middle attitudes, but it's not geostrophic in low latitudes. So, but if the wind is approximately the same in the tropics, in the middle attitudes, the pressure gradient has to be smaller. So that's a relatively simple argument, which seems to hold. And this was extended, actually, by Sobel, Nielsen and Polvani. Because what you want to get, and this is going to be my last slide, so I'll finish at 12. What meteorologists want to get are simplified equations that they can actually integrate and step forward. And here are just the shallow water equations, sort of in full form. They're primitive equations, if you will. They've got three time derivatives. One for the height field, one for the vorticity, and one for the divergence. If you make this weak temperature gradient approximation, what you actually end up with is this time derivative and this time derivative go away, and you end up with a single time derivative for the vorticity. So you've actually ended up filtering gravity waves, and it's a formally at least simpler set of equations for the equations of motion in the tropics. It's sort of analogous to the quasi-geostropic equations in middle attitudes, and that you've made some filtering assumptions, got rid of some time derivatives. But it's actually become less... It's not as useful in some ways as the quasi-geostropic approximation, because the difficulty of the tropics from a scientist's point of view is that it's very hard to make simple equations relevant, and relevant equations tend not to be simple. So there's a bigger gap there. I mean, you want to have both. But the closest that the tropical meteorologists come to in making a simplified set of equations is to use something like these equations, along with this quasi-equilibrium assumptions I was talking about before, though I didn't use that name, in which we impose a particular vertical structure for the atmosphere, because we assume that the convection relax it back to the most adiabatic lapse rate, so we get rid of all the vertical structure in one fell swoop, and then we make these equations, the temperature gradient equations. There still tend to be a complicated set of equations. So you wouldn't actually use... Well, it depends where you're coming from, whether you think they've been successful or not. And that's the first half of my lectures. So I'll stop there. Thank you.