The Physics of Climate Change: Three Blackboard Lectures on Simple Models in Climate Science - Day 1

[00:00:00] Speaker 1: So yeah, I guess the lectures were born out of a perception that there's maybe some frustration at not being able to understand climate simply. It's simply a first principle. The only things we can use are our climate models that are hard to understand. And this decision to do this was really born out of conversations with Rob Sokolow and Stefan, especially Mike Celia, who said, you know, it feels like unless we turn to models, we can understand a new thing. You know, what can we understand on the blackboard? And I said, well, I think there are some things we can understand. You know, climate models have simple models that they use to understand the simulation, but they haven't really been pulled together and condensed in a way that's accessible to people outside of the field. So that's what we're trying to do here. So in addition to these lectures, I wrote up some notes. And so those are available on the archive, which is an internet preprint repository. So here's the URL. And if you just Google archive in my last name, you'll find it. And these notes include more details and footnotes and references. And so sometimes I'll refer to these. All right. So let's start with just a zero order climate model. So if we're going to try and understand the Earth's climate, and in particular, the surface temperature of the Earth. So let's start with just a zero order climate model. So if we're going to try and understand the Earth's climate, and in particular, the surface temperature of the Earth, what are the minimum ingredients we would need? All right, so as most of you probably have seen at some point before, one of the ingredients we need is the energy balance of the planet. There's a certain amount of energy coming in, and it has to be bounced by energy going out. So we've got sunlight coming in, and this is globally averaged and including reflection by ice and clouds and things like that. So it's a globally average net incoming solar radiation. So this is coming sunlight. This has a magnitude of about 240 watts per meter squared. So this has to equal the outgoing, what we call the thermal radiation that Earth emits to space. Everything that's warm that has any temperature emits thermal radiation. A simple formula for that is something that you may have seen in undergraduate physics. It's a constant times the temperature to the fourth. And this is the emission temperature, and you can think of this as sort of an effective temperature at which the Earth is radiating. And so this is sometimes known as the outgoing long wave. Long wave just means that it's basically an infrared that our eyes can't perceive it. Outgoing long wave radiation, abbreviated OLR. And so this constant is known, and so you can solve for this emission temperature. This is a number of people may have seen before. It gives you about 255 kelvins. That's about minus 18 Celsius. So clearly that's not the surface temperature of the Earth, right? It's too cold. This is significantly less than the observed surface temperature, which is about 288 kelvin, or 15 Celsius. This is the observed value. Observed. But this makes sense, because this outgoing long wave radiation isn't emitted by the surface of the Earth. It's emitted by the greenhouse gases in the atmosphere, right? A greenhouse gas, by definition, is one that just absorbs and emits this thermal infrared radiation. And indeed, this is a typical temperature for the atmosphere. So that makes sense. But we're interested in surface temperature. That's kind of the central variable, the most important variable in climate science. So this gives us a sort of characteristic atmospheric temperature, but how do we relate this to the surface temperature? So that's the first question that we want to answer. In a first-order quantitative way. We're not going to get the exact right answer, but we want a simple model that will get us in the ballpark. All right. With that, we need to understand a little bit more about how the atmosphere works. So here's the picture. So the pieces that we'll need here are an ocean. Can people see if I write down here? And above the ocean, there's this layer of atmosphere about a kilometer deep that's in close contact with the surface and feels the effects of the surface on short time scales. We call that the boundary layer. And I'll abbreviate that with BL. And actually I think I want to try this as a solid line. So the boundary layer is just the lowest part of the atmosphere, about a kilometer deep. Then we have the rest of the atmosphere. And then up here is space. [00:05:46] Speaker ?: All right. [00:05:47] Speaker 1: So, of course, we've got the sun. And by the way, I should say here that our simplified model for the Earth is that first we're going to consider the tropics. The tropics takes up about half the area of the Earth's surface. And it's where most of the radiation from the sun is coming in. It's where most of the infrared radiation is going out. So we should take it as a proxy for the holer to first order. The other thing we do is since the tropics is mostly ocean, we just idealize the surface as being all ocean. So this sort of tropical atmosphere above a tropical ocean is the standard of the whole thing. All right. So the sunlight's coming in. And it's all being absorbed in the ocean. So now in a steady state, the ocean has to get rid of this energy. Right? Now you might think that it'll just warm up and it'll emit thermal radiation and that that's how it'll get rid of this incoming sunlight. And indeed it does this. Right? You can measure a lot of upwelling thermal radiation near the surface. But we have these greenhouse gases. Right? And by definition, they absorb and emit thermal infrared radiation. And so, and they're very effective at doing this lower down in the atmosphere where their concentration, especially water vapor, is very high. So what we have are water vapor molecules, which I'm going to draw in a sort of Mickey Mouse-like cartoon. We've also got carbon dioxide. This chemical configuration is linear. So we've got these greenhouse gas molecules absorbing the upwelling and emitting downwelling. And these guys basically cancel out the first order. So this is visible. This is sunlight. But this is in the infrared. IR. Right? So the planet, the ocean surface can't cool by radiation because of the greenhouse gases. It has to find another way to cool off to get rid of all this energy that's absorbing from the sun. What does it do? It does the same thing we do. It sweats. It evaporates. Right? Evaporative cooling is the key. And so, the ocean is going to evaporate water molecules. I've got the water molecule here again. But now in a different capacity, not in a radiated capacity, but in more of a pure thermodynamic capacity. So the ocean is cooling by evaporating. It evaporates into the boundary layer. It also warms up the boundary layer a little bit because the ocean surface is maybe one degree warmer than the air in the boundary layer. So you've got this ocean moistening and warming this air. But then this warm moist air is going to start to convict. It's going to start to rise. [00:08:21] Speaker ?: Right? [00:08:23] Speaker 1: But as this air rises, it's going up to lower pressures. When you lower the pressure on the air parcel, it cools. Right? As it expands. You guys know that from letting the air out of your bike tire. Right? The air that comes out is cold because it's expanding. But, the moisture that cools is going to condense. It's going to make clouds of precipitation. So these water molecules are coming together. And I'm going to try and draw a raindrop here. So that's a raindrop. And these raindrops are more than incidental. Why? This is actually a surprising thing that if you're not, if you don't study the atmosphere as an atmospheric scientist, you might never appreciate this. But rain plays a key role in the energy cycle of the atmosphere. Why? Because the heat that the ocean lost when the water evaporated, the heat that the ocean had to put in to getting the water to evaporate, the same heat that you put in when you boil, when you might boil water off of a pan, that latent heat that's in the water, you get back when the water condenses. So every time a raindrop condenses and falls to earth, it's heating the atmosphere. Right? And that heat actually came from the sun. So each raindrop is basically concentrated sunshine. Right? And water vapor is the conduit by which that sunshine gets concentrated and then lofted to the upper atmosphere. And once it's up here, so this raindrop formation is heating the atmosphere, so I'm just going to draw squiggly lines here just to represent that it's heating the atmosphere. And once you're up this high, now there's not that many greenhouse gases above you getting in the way. So once this heat is transported aloft, now you can radiate it back to space in the infrared. And that's how the energy cycle of that atmosphere works. So you see there's really sort of two distinct processes which I've intentionally split in two sides of this diagram. So on this side we have radiative processes on the left and on the right hand we have convective processes. And so we describe this state as one of radiative convective equilibrium. And we'll abbreviate it as RCE, radiative convective equilibrium. So this is our paradigm for energy flowing earth sounds. [00:10:59] Speaker ?: Alright. [00:11:00] Speaker 1: So now knowing that, we want to go back and answer this question. So what does understanding radiative convective equilibrium do for us? Well now, let's consider one of these convecting parcels. And now we'll use some sort of intro-level thermodynamics to try and get this relationship. So what's this thermodynamics? Let's take the first law of thermodynamics. So the heat into or out of the parcel of air. It looks like that. So hopefully that's familiar. So Q is the heat in or out. U is the parcel's internal energy. P is the pressure. B is the volume. And by the way, I'm assuming that the mass of the parcel M, which is at the density times the volume, is conserved. Is conserved. Is conserved. Is conserved. [00:12:12] Speaker ?: All right. [00:12:13] Speaker 1: So we've got the first law. And then we have an ideal gas. We have a couple more equations we can write down. So we can say that the total energy is the mass times the specific heat at constant volume. And we also have the ideal gas law. And I'll write the ideal gas law this way. Is the pressure equal to the density times the gas constant. So this is a gas constant for air. Really for dry air. And in later lectures, we'll need a gas constant for water vapor times the temperature. Alright. [00:12:48] Speaker ?: So if we. [00:12:49] Speaker 1: And so the details of this are in the notes. But if we. Can I ask a question? Question please. Is the gravitational energy irrelevant? Um. So. Somebody's kind of ripped the water off. That's right. So the way to say that is. [00:12:59] Speaker 2: So for. [00:13:00] Speaker ?: This is the internal energy. It's just the internal energy. [00:13:01] Speaker 2: Yeah. Sure. Sure. [00:13:04] Speaker ?: Well what happened to the gravitation? [00:13:05] Speaker 2: So gravitation is going to come in. Through P. And hydrostatic balance. [00:13:08] Speaker 1: I'll write that down in a second. Alright. So. So. [00:13:12] Speaker ?: So you plug these guys in and do a little bit of rearranging. [00:13:12] Speaker 1: You get this. The CP. Is just. The gas constant plus CV. [00:13:16] Speaker ?: Uh. [00:13:16] Speaker 1: And these are. The. [00:13:18] Speaker ?: And you get this. [00:13:18] Speaker 1: The CP. Is just. The gas constant. Plus CV. Uh. And these are. And these are. You get this. The CP. Is just. The gas constant. [00:13:29] Speaker ?: Plus CV. [00:13:30] Speaker 1: Uh. And these are. You get this. The CP. [00:13:34] Speaker ?: Is just. The gas constant. [00:13:36] Speaker 1: Plus CV. Uh. And these are. So. [00:13:40] Speaker ?: You get this. [00:13:41] Speaker 1: You get this. The CP. Uh. And these guys. Have units of. Joules. Per kilogram. [00:13:48] Speaker ?: Okay. [00:13:49] Speaker 1: Alright. So. Now. We add. Two more. Ingredients. Two more. [00:13:56] Speaker ?: Ingredients. Two more. Ingredients. We assume. That. When this. Parcel is. Rising. [00:14:04] Speaker 1: That. There's. [00:14:06] Speaker ?: No. Heat. [00:14:08] Speaker 1: So. That. That. That. There's. No. Heat. Exchange. With. The. Environment. So. It's. It's. [00:14:22] Speaker ?: Adiabatic. So. [00:14:24] Speaker 1: That means. That. DQ. Equals. Zero. [00:14:29] Speaker ?: We also. [00:14:30] Speaker 1: Assume. That. Pressures. Are. Hydrostatic. Which. Means. The. Change. And. Pressure. As it rises. Just. Equal. The. Rises. Certain. Distance. [00:14:48] Speaker ?: DZ. [00:14:48] Speaker 1: Then. [00:14:49] Speaker ?: The. [00:14:49] Speaker 1: Change. In. [00:14:51] Speaker ?: Pressure. Is. Just. Equal. So. [00:14:56] Speaker 1: Now. [00:14:57] Speaker ?: If. [00:14:57] Speaker 1: I. [00:14:58] Speaker ?: Take. [00:14:58] Speaker 1: This. Equation. DQ. Equals. Zero. [00:15:03] Speaker ?: I. Can. [00:15:05] Speaker 1: I. Substitute. This. In. [00:15:09] Speaker ?: Here. [00:15:10] Speaker 1: What. You'll. Find. Is. [00:15:14] Speaker ?: A. [00:15:15] Speaker 1: Quantity. Called. The. Dry. Lapse. Rate. And. It's. Defined. To be. Minus. DT. DZ. So. How. Fast. Is. The. Partial. Is. The. C. P. Is. G. Over. C. [00:15:42] Speaker ?: Sub. P. So. You. [00:15:45] Speaker 1: Can. Plug. [00:15:47] Speaker ?: In. [00:15:48] Speaker 1: Numbers. So. G. Roughly. Ten. Meters. [00:15:54] Speaker ?: Per. Second. [00:15:56] Speaker 1: Square. C. P. Is. About. C. [00:16:02] Speaker ?: P. [00:16:03] Speaker 1: Is. About. Thousand. [00:16:06] Speaker ?: Per. Kilogram. [00:16:08] Speaker 1: This is telling you. If your parcel had no moisture in it. How fast would it cool as it rises? That's what it's telling you. And you get it just from. Soft. Mole level. Thermalia. Right? Um. And if you go outside. Um. You know. Pretty much. Most places over the earth. Uh. Certainly during the day. And you measured. How fast the temperature is decreasing with height. You find a number. More or less like this. Especially close to the surface. [00:16:34] Speaker ?: Right? [00:16:37] Speaker 1: Alright. But. There's nothing in here about moisture. And I said that. As the parcel rises. It cools. But then it condenses water. And that water releases latent heat. So you have to take that into account. So I won't go through the details here. [00:16:49] Speaker ?: You can do a various. An analogous calculation. Which you have to include. An additional term for water vapor. Makes things a little more complicated. [00:16:54] Speaker 1: The calculation is done in the notes. What you find. Is you get a moist lapse rate. So instead of gamma d. It's gamma m. Instead of being constant with height. It turns out to vary. A bit with height. Not by orders of magnitude though. Near the surface. It might be something like. Four kelvin per kilometer. And then as you get higher up. [00:17:07] Speaker ?: Well. [00:17:08] Speaker 1: Most of the water vapor is gone. And it behaves just like a dry parcel. And it behaves just like a dry parcel. And it behaves just like a dry parcel. Right? And it behaves just like a dry parcel. And it behaves just like a dry parcel. And it behaves just like a dry parcel. So this is an analogous calculation. [00:17:17] Speaker ?: Which you have to include an additional term for water vapor. [00:17:18] Speaker 1: It's a little more complicated. The calculation is done in the notes. And then as you get higher up. [00:17:21] Speaker ?: Well. [00:17:22] Speaker 1: Most of the water vapor is gone. And it behaves just like a dry parcel. So high up in the atmosphere. It's close to this. 10 kelvin per kilometer value. So it varies a bit. Not a ton. So what we'll do. In terms of trying to build. A simple quantitative model of RCE. We'll just take an average. So we'll let. Gamma m bar. Be 7 kelvin per kilometer. And that'll just be an average. Typical value. That we'll use. All right. So this is telling us. How much? We'll just take an average. [00:17:43] Speaker ?: Not a ton. [00:17:44] Speaker 1: Not a ton. So what we'll do. [00:17:46] Speaker ?: In terms of trying to build. A simple quantitative model of RCE. [00:17:48] Speaker 1: We'll just take an average. So we'll let. Gamma m bar. Be 7 kelvin per kilometer. And that'll just be an average. Typical value. That'll just be an average. Typical value. [00:17:59] Speaker ?: That we'll use. All right. [00:18:01] Speaker 1: So this is telling us. How a moist parcel. How its temperature changes. As it rises throughout the atmosphere. But it turns out. That the temperature. Of this convecting parcel. Is the temperature. Of the tropical atmosphere. Right. If. The radiative cooling. Coming from here. Cools the atmosphere. Much lower. Than temperatures. Consistent with this. Then convection. Will just. Start to pop up. And it'll kind of. Adjust the atmospheric temperature. Profile. Back to roughly. That value. So even though. In relation. For a convecting parcel. It turns out. To really. Describe. The whole atmosphere. So that means. But this is. DT. DZ. Right. So this is telling me. That. My atmospheric temperatures. As a function of Z. This lower case. A. Is just. For the atmospheric temperatures. This should just be. My surface temperature. Minus. This guy. Times Z. Because this is just. DT. DZ. Right. [00:19:00] Speaker ?: It's just telling me. The rate. [00:19:02] Speaker 3: The temperature. Is increasing. Excuse me. Yes. Hi. Back here. Quick question. Before you. Proceed. Hydrostatic balance. Is somehow. A little fishy. Because. This is not actually. A static atmosphere. By definition. It is dynamic. And. Convecting. Is there a way. To. Establish. Information. To use. [00:19:26] Speaker 1: In this. Calculation. So. You can. Take into. Account. [00:19:32] Speaker ?: That. You can't. [00:19:34] Speaker 1: Take into. Non-hydrostatic. Effects. Into. Account. Here. But. They. Only. Give. Corrections. So. The atmosphere. As we'll see. Is maybe. About. Fifteen. Kilometers. Deep. [00:19:53] Speaker ?: And. [00:19:53] Speaker 1: If you. So clearly there's some approximation going on here. But as long as we stay in the bottom 10, 15 columns of the atmosphere, this is a very good approximation to the sort of full expression that takes into account monohydatic. [00:20:05] Speaker 3: Is it easy to derive that bound, or is that more of an unobserved reality? [00:20:10] Speaker 1: No, it's not hard to derive. I mean, I won't do it here. But it's not in the notes. I think there's a reference in the notes that does it. And it's simple. It can be done on a blackboard. Good. Any other questions? And this is a really important picture. So I hope this energy flow of the atmosphere comes in through radiation, gets converted into the latent heat of water vapor, and water brings that up, transports that heat to the upper atmosphere, where it is then free to radiate away. That's the key. Very good. Okay, so now we know how atmospheric temperatures are related to surface temperatures. Right? And we somehow have to connect this with the emission temperature. But the emission temperature is, it's an average temperature of the atmosphere. So if I think of, so if TEM is kind of an average atmospheric temperature, Well then, the sort of simplest thing to do, is to assume that this occurs halfway through the atmosphere. So I know, so I know, so this occurs at an emission pressure, which is just one half of an atmosphere. Right? So I assume that this is an average atmospheric temperature. I assume it occurs at an average atmospheric pressure. Halfway up in pressure coordinates. So there's a quick calculation you can do, I'll skip it here. It just involves hydrostatic balance. It's simple, it's in the notes. And what you find is that the height, at about half an atmosphere, and this, yeah, the height at half an atmosphere, is about five kilometers. Okay? Well, now I'm good to go. So now, I just rearrange this, and say that the surface temperature. So what I'm going to do is I'm going to evaluate this equation at this emission height. Right? Where I should have the emission temperature. So T S equals T emission, plus gamma M bar, the average lapse rate, times Z emission. Right? Evaluating this at the emission height, and rearranging a little bit. Now let's plug in some numbers. What do we get? By the way, I never did this calculation before preparing for this lecture. This is not the model that's in the textbooks. I have not seen a single, I mean, I didn't invent this. There are other people who kind of know this. But in terms of this being in a textbook, I haven't seen it. And so in preparing these lectures, I was like, well, what's the simplest model we could do? And then I wrote this down, and you'll see, so what do we get? So this is 255 Kelvin, plus 7 Kelvin per kilometers, plus 5 kilometers. 7 times 5 is 35, plus 255 is 290. So this is essentially, you know, two, maybe three equations that gets you in the ballpark, and it does it for the right reason. Right? There's other books that write down even simpler models that get the wrong answer, and then they introduce some fudge factors, some emissivity for the atmosphere, some tunable parameter. They say, oh, if I set this parameter equal to 3 quarters, then I get the right answer. Right? It's not for the right reason. All right. So this is sort of part one of this lecture, and we're right on time. This is good. So this is our zero-quarter picture of the atmosphere. It gives us surface temperature. And so for the second part of the lecture, I want to talk about, well, really what the lecture was entitled, global warming has this change, and what are the timescales on which it changes? It's warming fast and slow. So any questions on this before we move on to that? Yes, Jamie? [00:24:18] Speaker 4: So you made a bunch of approximations, like if TM is average. Yep, yep. So are those good? Yes. Yeah. But that's something that you need to do a lot more work to check. [00:24:34] Speaker 1: Exactly, that's right. So basically what I'm doing here is, and this is, you know, so in the notes, you know, you give references, and you say, you know, why is this reasonable, and there's lots of caveats, and I posted these notes online, and then the first feedback I got was, there are too many footnotes in this. And I thought, are there too many footnotes? But I thought, how else do you do it, right? If you're going to prevent something simplified, you know, people are going to have questions, and you have to have some answers or references to the many, many questions you could ask, right? Because I made approximations all over the place here. But what I'm trying to do is give you, you know, from someone who studies this, you know, their best sense, right? Everything that's on the board here, you know, is sort of consistent with, I guess, things that I have seen and have read, both in my reading other papers and in my own research. So I don't think anything here is a lie. [00:25:23] Speaker 5: Another question. Yes, please. So in the sense that you just mentioned, the blackboarding or something? [00:25:30] Speaker 1: Yeah, you assume a black body, or you assume a gray body, right? That it absorbs at all frequencies, but maybe it's not totally opaque. Maybe it can, you know, let's a certain amount of light through. And that is true for the atmosphere, but to first start, I think the best approximation is to think of it as a black body, right? Because there's maybe something like one-sixth of the radiation that's emitted at the surface. Now it's even less than that. It's about 10% of the radiation emitted at the surface actually makes it all the way to the top. [00:25:53] Speaker 5: But the atmosphere, though, it's not, it depends on that in-house gas. That's absolutely true. [00:26:00] Speaker 1: So I'm ignoring all that here, and then tomorrow's lecture we'll take those ideas here. [00:26:06] Speaker 2: Yeah, Mike. How should I think about the fact that you have ocean everywhere? Should I be thinking about land in some different way, or how does that come into the story? [00:26:14] Speaker 1: Good. So the main reason I did that is so, because I wanted you to think about the surface cooling mainly by evaporation. And if your moisture limited, then it gets more complicated. So it turns out that the ocean temperatures, because there's so much more ocean than land, especially in the tropics, that what's happening in the ocean dominates the climate. And so convection warms the atmosphere here, and then this air either blows over the land or directly communicates this heating to the land surfaces. Now, land temperatures themselves can vary a lot, right? If you're someone that's very dry, the surface temperatures are higher, because you are indeed moisture limited. But if you want to think about what sets the temperature of the Earth's oceans, and also the upper atmosphere everywhere, then I think this is a good start. [00:27:09] Speaker 6: So, I mean, one thing that is often neglected in these simple models is absorption shortwave by the atmosphere. Yes. And that's a term that globally is as large as convective heating. [00:27:24] Speaker 1: Yes, correct. [00:27:25] Speaker 6: Does it matter? [00:27:27] Speaker 1: Not for this. It'll matter. It's not in it, but does it? No, no, but it doesn't matter for this. Okay. So it does matter. In all these lectures, I'm going to ignore the fact that water vapor actually absorbs a significant amount of sunlight. It's not sunlight you can see, right? It's near-infrared sunlight. It's below the red part of the spectrum, but it's sort of not part of thermal radiation, at least not at temperatures, terrestrial temperatures. That's what makes light bulbs so efficient, right? Is that, like, just regular incandescent bulbs, is that you see some visible light, but most of the light that's being emitted from a light bulb is in the near-infrared. So you can't see it, but there's a lot of energy being pumped out there. So water vapor absorbs a significant amount of that near-infrared light. The reason it doesn't matter here is because it doesn't change this fact, right? That this cancellation, even if these numbers were larger or smaller, they still cancel out. And so you might change how big this arrow is, but the atmosphere and the surface are so tightly coupled, right? Let's see. So if there were no shortwave absorption in the atmosphere, you'd have more sunlight hitting here. [00:28:41] Speaker 6: That's right. It's... So I guess the shortwave that's not absorbed by the ocean is absorbed by the atmosphere, so the water doesn't have to bring it up. Exactly. So it's absorbed by the water in situ. [00:28:53] Speaker 1: That's right. All right. [00:28:55] Speaker 6: Then, second question. What about the meridional sort of inhomogeneity of this planet, right? So most of the emission to space, the net emission to space, happens at high latitudes. Most of the convection happens at low latitudes. That's right. And so you have the transport there that is... I mean, you know, we... I'm ignoring it. Yeah, I know, but the Earth is not a point in a very clear way, right? Uh-huh. Right. I don't know if... [00:29:24] Speaker 1: I mean, I guess the one thing you could say is if you're... So you can do a calculation where you don't allow the atmosphere to transport heat from the tropics to the poles, and you get a somewhat different distribution of temperatures. And you might ask, if you average that, does it give you something very different? I actually haven't seen that calculation done. Right? You get warmer tropics. You get colder poles. Is the average significantly different? I actually don't know. So that's actually an excellent question. So maybe I'm lying in that regard, right? That I'm ignoring inhomogeneity of the Earth's surface. My guess is that it can't be too far off. Maybe plus or minus 5 Kelvin. But I haven't done that calculation. A good caveat. Now the questions are coming. [00:30:10] Speaker 5: I'll take a couple more, and then we'll go on, and we can... Yeah. If we replace the ocean with land, and use a dry lapse rate of 10 Kelvin per kilometer, with those 7 UUs, and we can predict the temperature, surface temperature, like 15 Kelvin warmer than the ocean surface. Uh-huh. Can you say that the land surface can be 15 Kelvin warmer than the ocean surface? Oh, you mean like if we look at land surfaces now? Yeah. Can you say that we can use this model to predict the surface temperature with ocean? [00:30:50] Speaker 1: So not quite like that. It's a little more complicated, because it turns out that this part of the atmosphere is the same over ocean and land, the first approximation, but this part is different. Round your layer. But it turns out, so you can't just say that, oh, if I just average over land, it's going to look 15 Kelvins warmer, because it's not like it's a dry adiabath all the way, because the temperature structure of the atmosphere over land is influenced by the temperature structure of the oceans. But it turns out that if you do a simple experiment where you say you warm the ocean temperatures by 4 Kelvin, the land warms a little bit more than that, and you can't understand it via basically the difference between a dry and a moist adiabath. So that difference is important, but you can't apply it in quite as simple a fashion as you're trying. Jane? [00:31:37] Speaker 2: I guess I'll still ask this, but I'm wondering why sensible heat fluxes aren't part of this picture? Why is it because they're so much? [00:31:45] Speaker 1: They're just small. [00:31:46] Speaker 2: They're small. So because the... [00:31:50] Speaker 1: So what Jane's asking about, she's asking about the fact that the ocean, as I said, the ocean does directly just warm up some parcels in addition to moistening them, and she's asking, where is that in the energy budget? And it turns out to be almost an order of magnitude, or a factor of five, smaller than the V. [00:32:10] Speaker 4: All right, one last question. Can you say something? Is it using this model something relevant about what happens if you just add a little one to your view? [00:32:19] Speaker 1: No, this model's very bad for understanding radioporce. So we'll need other models for that. I'm going to keep going, and we'll save it to the end. I just want to get to the end, and I don't want to keep people super late. Sorry, David. You already had, like, two questions. Don't get greedy. Okay, so... Let's go here. So what I want to do now is I want to build... So you guys may have heard about, you know, committed warming versus the warming that we're seeing now, or the transient climate response versus the equilibrium climate sensitivity. So what I want to do is build a simple model that explains what those things are and allows us to put some numbers to put some numbers to it. So that's what we're going to do. So this is this warming fast and slow. This is a two-box model for the ocean. All right, so what do we got here? So now I'm basically going to take this, just the ocean, expand it into two pieces. So this top part, I'll call the mixed layer. And then the part below, we'll purchase the deep ocean. And this mixed layer is the part, you can think of it as the part that absorbs sunlight and radiates radiation. And the deep ocean just doesn't see the atmosphere at all, or the sun at all. It's sort of analogous to the atmospheric boundary layer. It's the part that's sensitive to what's on the other side. All right, so now I want to ask, so how does this RCE system respond to forcing? And by forcing, I mean a change, a disturbance to this energy balance at the top of the atmosphere. And what CO2 is doing is it's reducing this. If you really insisted on understanding that in this model, you could think of it as changing this height, raising it up a little bit. And if you raise it up, you'd reduce the emission temperature, you'd reduce this, you'd be out of balance, you'd be getting more sunlight than outgoing long-wave radiation. And that's what we call a forcing. So I'll draw that as a downward arrow. So let's just think, a thing that we often do in climate science is consider the forcing from a doubling of CO2. So if I just took my atmosphere in steady state, and now all of a sudden I flipped a switch, I doubled the amount of CO2. All of a sudden there's a certain amount of, there's an energy imbalance with more energy coming in than going out, right? If you double CO2, this is to the tune of about three and a half watts per meter square. So it's the forcing from CO2. So how does the system respond? So we're considering perturbations here to a steady state climate. So I'm going to get some temperature perturbation. I didn't know the perturbation was a prime, right? So it's perturbation to some equilibrium temperature. So that's in the mixed layer. I have some perturbation at depth, right? It's because there's an energy imbalance, and it's the mixed layer that feels it, right? By definition. The mixed layer is the part that feels the sunlight coming in that is emitting a parent and evaporating as well. So what happens? So this guy warms up. And if it warms up, then you're increasing TS, and if you're increasing TS, then in a basically one-to-one fashion, according to this model, you're increasing TA. And so you're going to be emitting warm. So what I can do is basically think about linearizing this equation for the outgoing long-wave radiation. And so I get a perturbation to the base guy. I'm increasing the outgoing long-wave radiation by increasing the surface temperature here. And we're writing it as some constant beta times the perturbation temperature. So for those of you who aren't familiar with Taylor expansions, I'm just making the first-order Taylor expansion for perturbation to the OR. And so you can think of beta as being the derivative of outgoing long-wave radiation with respect to surface temperature. So if I warm up, I emit a little bit more radiation to space. But the other thing that happens is that if these guys warm up unevenly, then this guy's going to warm up first, and then it's going to start to warm up the deep ocean. So I'm also going to get a flux of energy that way. And again, I linearize it, and I just write it as proportional to the temperature, the perturbation temperature difference between them. So if they both warm up by one Kelvin, then I get no flux. But if this one warms and this one doesn't, then I get a flux down. Estimate. So first, let's get a feel for these numbers. So let's estimate beta and gamma. So as I mentioned, we could just, we could use the simple climate model on the board to make an estimate of beta, right? That's the beautiful thing about simple models. So I'm calling this, let's call it RCE. So using our simple RCE model, right? Well, this is the derivative of OLR with respect to surface temperature. But I'll use the chain rule from calculus and write that as the derivative of OLR with respect to the emission temperature times the derivative of the emission temperature with respect to the surface temperature, right? But this is four sigma T emission cubed. And this is just one, right, by this equation, or by this equation now. Right? So that gives me this, and then you can evaluate that, and this gives you something like three and a half watts per meter squared per Kelvin. So for every Kelvin in a surface warming, this is the additional amount of radiation at the top of the atmosphere that you see. All right. But what happens if you use, say, global climate model? So what is beta in a GCM? It's a lot smaller. It's about one watt per meter squared per Kelvin. And it turns out that gamma is pretty close to that, too. So that's the numbers we get from models. So to this simple RCE model, it's getting us the right order of magnitude, but it's almost a factor of four off, right? So we're clearly missing some physics. So that's what tomorrow's lecture is about, in part, is there's a discrepancy here. How do we understand it? So we'll learn more about that next time, and we'll improve our estimate by incorporating some additional physics. The point here is just to show you that this model can at least get you the right order of magnitude. All right. Order one watt per meter squared per Kelvin. All right. Great. So now we want to write in some equations. Let me move over here. So I want to write down equations for these two layers for the way their temperatures are going to evolve in time in response to this forcing. But to do that, I need to know their heat capacities. The heat capacities are actually going to be the central thing. So to write that down, I need a few numbers. I need the density of water. So that's 1,000 kilograms per meter cubed. I need the specific heat of water. And that's about 4,000 joules per kilogram Kelvin. And then they need heights for these guys. And this is the key thing. So the fact that there are fast and slow components to global warming, as we'll see, comes from the fact that this guy, height of the mixed layer, about 100 meters, with this guy, about 4,000 meters. So over an order of magnitude difference. So that's the big difference between these two components. All right. So with those numbers in place, I can now write down our model. So first I'm going to write down an equation for the mixed layer. So I have the density times the heat capacity times the height. So this kind of gives me the total heat capacity of this whole layer. And then I have the time derivative of its temperature. And this equals all the forcings on it. This equals these arrows. So I know I've got my CO2 forcing coming in. But then in terms of outgoing arrows, I have this beta times the perturbation temperature. And I also have this gamma times the difference of the perturbation temperatures. Between these two ones. And the units of this are watts per meter squared. So hopefully that equation is easy enough to interpret. You just have a change in temperature from positive forcing and then the response, which is negative. Opposite the sine. So then we do the same thing for the deep ocean. So I have the same density, same heat capacity. But now I have the depth of the deep ocean. I have the deep ocean's temperature now. And now I only have one arrow, which is the incoming flux from the mixed layer. So that's positive. So now, what can we do with this? So imagine that, let's just look at the mixed layer. Imagine that the deep ocean wasn't there, that its temperature wasn't changing. So imagine that this is zero. And actually imagine that, what I want to do is I want to get a time scale for how quickly this guy responds. So imagine the system, but without any forcing. Forget about the forcing for the time being. Forget about the deep ocean. Then I only have terms. I just have dt prime is proportional to some constants times minus t prime. And so you can use that to get a time scale. And all it is, is these numbers divided by these sort of response coefficients. So characteristic time scale for the mixed layer. So I'll call this tau ml. So it's just, it's basically thermal inertia terms. Rho times c times the height. Divided by these response coefficients. Natal plus gamma. And if you're not totally sure where this expression comes from. Set this to zero, set this to zero. You get a simple differential equation, which you can solve. And it has a time scale. The point about this whole model is that, you know, these came from GCMs admittedly. We don't, you know, we have a first principles estimate for this that's too big. And we don't have any simple models for estimating this. So there is some GCM input to this. But the point is that just knowing these two numbers, I can now estimate the time scale of which the mixed layer responds. So you plug these numbers in, and you get something like six years. [00:45:13] Speaker 5: Yes? So if you're going to ignore a deep ocean, maybe you set the temperature to be the same if you don't have anything considered. [00:45:19] Speaker 1: So I guess what I'm saying is, if you, I'm just trying to figure out a characteristic time scale. So if I return the whole system, the response is a sort of sum of exponentials, each of which has a characteristic time scale. Sure. So that's the complete solution. I'm trying to just isolate those two pieces. And so it turns out that the way to get one of the time scales is just by ignoring the deep ocean and just asking if I just perturb the mixed layer, if there's, you know, without any deep ocean, what's the time scale on which it would respond? But if I think it's zero, it's like you're radiating to, you know, zero helms. But you can just assume that there's no deep ocean, right? That the entire ocean surface is only 100 meters less, that you won't have the gala time at all. That's what I'm saying. [00:46:09] Speaker 3: I think I understand his concern. Yeah. It's the primes or the deviations from the baseline temperature pre-forcing. So he's saying set the perturbation deep ocean zero, assume it's not responding, rather than not there necessarily. Is that fair? Yes. Yeah. [00:46:28] Speaker 1: Okay. So we'll do the same, play the same game for this equation. We get a deep ocean time scale. Same density of water, same heat capacity, but now I have the depth of the deep ocean. And you'll notice that when I look at this equation, I don't have a beta term, right? Because the beta term is just how the mixed layer talks to the atmosphere. Deep ocean's not involved with that. So I just get these guys divided by gamma. And this is something like 500 years. So I have two. So this is global warming past and so on. You have this fast time scale over which the mixed layer equilibrates. And then the slow timescale over which the deep ocean equilibrates. What you can do then is ask about intermediate times. So consider some intermediate timescale, which is really what we have, right? Most of the carbon emissions that most of the C2 we put in the atmosphere has been, you know, I guess we usually say pre-industrial starts at 1850, but most of the emissions, the majority of them are since 1950. So it's really been 70 years, maybe. So that's right in this, you know, it's right in between these two timescales. So we consider intermediate timescales such as the one we're experiencing right now. So it's some tau that's less than tau deep or greater than tau mixed layer. What do we find? Well, the mixed layer is going to be in equilibrium because it's had sufficient time to equilibrate to basically kind of the average forcing over this period. And it's been not enough time has passed for the deep ocean to really respond. So these are fair things to say on this intermediate timescale. The mixed layer has equilibrated. The deep ocean hasn't felt it yet. But now I can go back to my equations, right? So if I take these two qualities and I look at this guy, I set this to zero and I set that to zero. And then I solve. And that's an equation that only has these three terms and I can solve for the mixed layer temperature perturbation. That's what I get that the mixed layer temperature perturbation is the fourth thing divided by gamma plus beta. Now I can plug in numbers. So three and a half watts per meter squared per kelvin. I said that gamma and beta were each about one. Sorry. Watts per meter squared. I said that gamma and beta were each about one watt per meter squared per kelvin. So this is two watts per meter squared per kelvin. So this is going to be about 1.8 kelvin. Is this how much global warming we've observed today? Probably about a degree. This is almost two. But we haven't doubled. And in fact, there's another calculation you can do. Very simple. Using the fact that the forcing of... Due to CO2 is logarithmic in the concentration. But pre-industrial was 280. Right now we're about 400. The factor between them is roughly the square root of 2. And so that tells you that you should have about half the warming that you get at a double. That puts you right in the ballpark. Right? So this is all consistent. Now, this wasn't first principles, right? I got these numbers from GCAs. But the point is it all hangs together. All right. A couple more things to say. So this is on the intermediate timescale. This is describing where we're at now. So let's say we stop changing the CO2 concentration in the atmosphere. We fixed it. Say at 400 ppm. And then just waited, you know, the 500 years. Right? What would we find? Well, in this case, so this is tau greater than tau d. So let's go back to our equation. So now, both these guys are zero. We also know that these guys will be the same. So the temperature difference will be zero. They come into equilibrium with each other. Right? So this equation is just zero equals zero. It's not telling us anything. But this equation, right? This guy is zero. This difference is zero. So the only two terms left are this. That tells you on long timescales, T prime ML equals F2X over just B. So instead of 3 1⁄2 divided by 2, this is 3 1⁄2 divided by 1. This is something like 3 1⁄2 Kelvin. So these two numbers have names in the literature, in case that's helpful. Maybe you guys have heard of these. So this is essentially the transient climate response, also known as the TCR. So this is the equilibrium climate sensitivity, also known as the ECS. And so sometimes you hear the phrase committed warming. [00:53:25] Speaker ?: And so sometimes you hear the phrase committed warming. [00:53:25] Speaker 1: This is what people are talking about. How much more warming will you get when you let the deep ocean equilibrate? And the reason the deep ocean equilibration matters for the surface temperature is that on the intermediate time scale, right? This guy warms up, but the mixed layer has two places to which to export energy, right? It can send extra energy out to space, but it can also sequester some energy in the deep ocean. And so it can kind of come to a sort of temporary equilibrium at a smaller temperature, because it's got two places to dump its excess heat. But eventually this deep ocean has had enough of that. It gets full, it doesn't want any more. This difference goes to zero, and this reservoir is no longer available. And so now this same forcing, there's only one place for it to go this way, and the temperature must be correspondingly higher. And that's exactly what I've said in words. It's exactly the difference between this expression, right, where the forcing has two places to go, gamma and beta, and dividing by a bigger number, and so it gets a smaller temperature perturbation. Versus here on long timescales, there's only one place for the perturbation to go. Divide by a smaller number, you get by a bigger temperature perturbation. Yes? [00:54:38] Speaker 7: I'm having a really hard time understanding why the heat in the surface ocean would want to go down to the deep ocean. You know, it's stably stratified, it's very happy to be there. Anything that would be just going... [00:54:51] Speaker 1: So there's just an oceanic circulation, right? [00:54:54] Speaker 7: Which is much slower. [00:54:56] Speaker ?: Total. [00:54:57] Speaker 1: That's right. [00:54:58] Speaker 7: Which is much slower than the Pregnancy. I just can't imagine that the deep ocean is going to come to its equilibrium in Pregnancy. Let's see. So what is the mechanism of heat transport from the surface down? [00:55:11] Speaker 1: That is a great question. And so I told you I didn't have a simple model for gamma. So I emailed someone at GFDL, you know, and I said, do we have any simple models for gamma? Beta, you know, I can estimate. And they said no. So it's a number you get from models, and the models have the whole deep ocean circulation in there. So you can think of all that physics, the entire ocean dynamics is being wrapped up in gamma. And I wish there was a simple thing I could draw on the board to give you that one watt per meter squared per Kelvin. But to my knowledge, and I'm not an ocean expert, and all the ocean experts are actually at the ocean scientists conference this week, so none of them are in the audience. I'm just strapped. And, you know, I only have one person. Maybe I should have emailed more. You know, but to my knowledge. [00:55:54] Speaker 7: It has the physics in it. I mean, it's all in there, right? It's all in there. [00:55:57] Speaker 1: This came from a GCM. So it's in there. [00:56:00] Speaker 8: That's a good question. Ben. Yeah, well, now getting out of the physics and anthropology and time scale here, it's long enough that, I mean, going to the concept of commitment, you might expect enough carbon to be returned to vegetation. That's good. [00:56:16] Speaker 1: So Ben is pointing out when I say that this is the committed warming, it's not really a fair statement, because this is assuming that the CO2 concentration stays fixed in the atmosphere. But we know that the ocean takes up carbon, the land takes up carbon. So there is a carbon cycle, which I have not modeled for you at all. So committed warming is in part talking about this, but then you also have to take into account the fact that CO2 concentrations, you know, in sort of idealized model world, I'm doubling the CO2 concentration and just holding it fixed. It's not how the Earth system works. So thank you for that. Anna and then Jane. [00:56:50] Speaker 4: So the temperature you computed is the emission temperature. I mean, this is part of the front side of the left-hand side of the, like the radiative... What's this? Yeah. [00:56:59] Speaker 1: This is the surface temperature. [00:57:00] Speaker 4: The surface temperature. I mean, you don't need to go through the cycle of, like, the convection and... [00:57:06] Speaker 1: You do, right? But I erased it, but, you know, what we had was that... So they're related into this one-to-one way. So that was part of the reason why I went through that model in the first place, is to say, look, the atmospheric temperatures and surface temperatures are extremely tightly coupled. So a difference in one is a difference in the other. That's a good question. And it doesn't hold everywhere across the Earth, right? In the Arctic regions, convection is not coupling the surface to the atmosphere. And so you cannot say that they move in lockstep. And in fact, they don't. And polar amplification is all tied up. [00:57:41] Speaker 7: So you're assuming that the lags rate is not changing because of the forcing? [00:57:45] Speaker 1: I do, and that's not quite true. But it's a small perturbation test. Yep. Jamie? [00:57:51] Speaker 2: I haven't seen something basic, [00:57:53] Speaker 7: so I don't understand why TV prime is zero when you're at that intermediate time scale. I would think if the mix layer is equilibrated, has it the heat version started to warm up a little bit? [00:58:04] Speaker 1: No, you've dumped a lot of heat in there, but its heat capacity is so large that its temperature is not appreciably different. [00:58:10] Speaker 2: Okay, so you have dumped heat in, but effectively it's different. [00:58:13] Speaker 1: Exactly, because the heat capacity is large. Right? So over this whole 70 years, right? This guy's warmed up. Heads is being fluxed down. But the square of the wire is so big that on these timescales, it's effectively infinite. So there's no temperature response, even though there's been heat sequestration. [00:58:30] Speaker 7: But if you're close to 500 years... Then you'll see it. Okay, so we're saying it's like at 50... [00:58:36] Speaker 1: Yeah, exactly. [00:58:37] Speaker 9: That's right. Yep. Yep. Wrong. Are there two kinds of questions that people ask? One is hold the concentration constant, which is what you've been doing. Yes. The other is stop the meeting. Yes. Which is somehow the atmosphere scrub. Let's suppose you're going in that part. You just couldn't get anything too old or something. But now you're not doing anything. Okay. The system does cool off. The atmosphere will... Yeah, but the concentration, sorry. [00:59:05] Speaker 1: Concentration will drop. If there's some mechanism for taking it out by either an ocean or the land or... [00:59:13] Speaker 9: Right. What am I saying? So I stop the emissions of people because I don't have that. Right. Well, I guess I'm thinking primarily about carbon dioxide. So it does get scrubbed out. Yes. And that's carbon cycles. Exactly. It gets scrubbed out. Then I get a temperature decline. And that's a different calculation, [00:59:31] Speaker 1: but you can't do it with the carbon cycle. Exactly. It's a different calculation. You have to have some model for the carbon cycle. But you could do that, and that would be interesting. Yes. Agreed. And you and I have talked about how you get these interesting results when you include the carbon cycle that the committed warming is a function just of the cumulative emissions, and that's not obvious. [00:59:50] Speaker 9: If I jump out of that result, it would be really important. [00:59:53] Speaker 1: It would be great, and I have seen some attempts at a simple model, but I haven't really delved into it. But that would be a great next step. It's to include the carbon cycle. And I would address Ben's question as well. [01:00:03] Speaker ?: Yes. [01:00:04] Speaker 1: Yeah. [01:00:04] Speaker 6: So, and I mean, maybe this is getting off point, but it's going back to this gamma and the time scale. [01:00:10] Speaker 9: Yeah. [01:00:11] Speaker 6: So it's definitely, so it's not the sinking branch of the circulation that is bringing warmth down, because you have to increase, warming the deep is increasing potential energy. So what you need to do is do work on the ocean. So what is driving this is whatever does work on the ocean, so the winds. So I think there would be a way to estimate gamma by looking at the winds. I think that's, or, I mean, so it's not the convective part, but it's rather, it's the work done that allows you to increase potential energy of the ocean. [01:00:49] Speaker 1: Right. Somehow it's a function of this energy difference between the temperature difference between the two. [01:00:57] Speaker 6: Yeah. So you grow the temperature difference and if you have the same work going on it, then... Then those things, yeah. Yeah. [01:01:03] Speaker 1: It'd be great to sit and write something down. Yeah. [01:01:05] Speaker 6: I don't think it's intractable. Right. [01:01:08] Speaker 1: Okay. Yeah, I mean, it definitely may be the case that no one has really wanted to do that, right? Build a simple model for gamma. So it may not be that it's impossible or very difficult, just that no one has really tried. [01:01:24] Speaker 7: As opposed to like some seasonally resolved or... Well, I guess I'm just thinking about... I mean, probably. I'm just thinking about how much the fact that like land has... Ah, good. [01:01:50] Speaker 5: How much does it alleviate? [01:01:52] Speaker 1: Yeah, yeah, good. Well, I mean, so since ocean is two-thirds of the land surface, right, the errors... So you're right there, errors, and actually, a version from GFDL, who I emailed, he sent me back, you know, he was using numbers, but he used numbers where you average the depth of the global ocean which you took into account where you said, oh, land has zero depth. [01:02:10] Speaker ?: So, you get a different number, right? But it's not an order of magnitude different. It's like two-thirds. [01:02:18] Speaker 1: It doesn't change any of the conclusions, but it might change some of the... It's about seven minutes past, maybe we'll officially end so people who want to leave and can leave. And then I'm happy to stick around and answer questions and just up on the board and get through it. So, thanks a lot. Thank you.