Mathematical Geoscience: Radiative Balance Models - Oxford Mathematics 4th Year Lecture

What this episode is about

This course on mathematical geoscience focuses on utilizing differential equations to model and understand natural phenomena, specifically in three key areas: climate (including the carbon cycle), river dynamics (predicting floods), and ice behavior in the climate system. The initial lecture addresses Earth's average surface temperature and the radiation balance that influences it, contrasting it with other celestial bodies like Mars.

Key takeaways

• Earth's average temp is a cozy 15°C, but it could've been a chilly -60°C like Mars.
• Radiation defines our climate: the sun beams in shortwave brilliance, while Earth emits longwave whispers.
• Earth's albedo—a reflective personality trait—averages 0.3, but surface hues can change its mood dramatically.
• Stefan-Boltzmann: a law as vital as gravity, dictating how warm bodies radiate their warmth (or lack thereof).
• Mathematical models may seem dry, but they’re the pulse of climate understanding, from carbon cycles to ice cap dynamics.

What is discussed, and when

1. 0:00 – 5:00: temperature, climate models, carbon cycle
2. 5:00 – 10:00: temperature, albido, greenhouse effect, Stefan Boltzman
3. 10:00 – 15:00: temperature, CO2, water vapor, density
4. 15:00 – 20:00: temperature, greenhouse effect, climate models, density
5. 20:00 – 25:00: temperature, greenhouse effect
6. 25:00 – 30:00: temperature, albido
7. 30:00 – 35:00: temperature, density, tropopause, troposphere
8. 35:00 – 40:00: Pressure, temperature, density, hydrostatic equilibrium
9. 40:00 – 45:00: density, temperature, troposphere, ideal gas law
10. 45:00 – 50:00: troposphere, Pressure, temperature, stratosphere
11. 50:00 – 52:31: temperature, water vapor, greenhouse effect, gamma

Notable quotes from "Mathematical Geoscience: Radiative Balance Models - Oxford Mathematics 4th Year Lecture"

• Parts. on the first section is going to be about climate and we're going to start that today. So we're going to build some very simple climate models and in particular models…

  about climate models · at 0:53

• This is a word that we're going to talk about quite a bit. And this for Earth at the moment is around about 0.3.

  about albido · at 6:10

• We're going to have to modify this to account for the greenhouse effect because this tells us how much gets emitted from a surface and some of what gets emitted from the Earth's surface gets absorbed by the atmosphere and then.

  about greenhouse effect · at 8:22

• T is going to be the temperature of the surface. That's what I'm aiming to try and find. this sigma is a constant called the Stefan Boltzman constant. so this is about 5.67 *…

  about Stefan Boltzman · at 9:22

• In fact, more important in some ways is the concentration of water vapor which is a important greenhouse gas and it depends on various other things like methane and other gases that occur in.

  about water vapor · at 11:19

• Is less than t0 then dt by dt is positive. So what will happen is that the temperature will increase and you will move this way and you'll move this way until you get to that…

  about temperature · at 24:08

• Latitude. which I'm not worrying about here. at least this value does. the region below this tropopause is where most of the weather happens. That's where we form clouds.

  about troposphere · at 34:27

• This is what's sometimes called the lapse rate. It's the rate of change of actually it's I think there's a minus sign there. It's the rate of change of temperature with respect to height. So.

  about gamma · at 50:27

Who and what is mentioned in this episode?

Concepts & topics

• temperature — Concept · Key argument · 34 mentions · first at 1:28
  Main focus of the lecture, explaining the average temperature of Earth's surface and atmosphere.
• greenhouse effect — Concept · Core topic · 5 mentions · first at 8:24
  Central to understanding Earth's temperature regulation and climate models discussed in the lecture.
• climate models — Concept · Framework · 2 mentions · first at 0:56
  Tools used to describe and predict Earth's climate and temperature variations.
• troposphere — Concept · Key argument · 5 mentions · first at 34:42
  Lower layer of Earth's atmosphere where most weather phenomena occur, discussed in detail.
• Stefan Boltzman — Concept · Framework · 2 mentions · first at 7:31
  Law used to describe how a surface at a certain temperature emits radiation, key to understanding Earth's temperature.
• albido — Concept · Key argument · 5 mentions · first at 5:49
  Fraction of sunlight reflected by Earth's surface, affecting the planet's temperature.
• water vapor — Concept · Key argument · 3 mentions · first at 11:23
  Important greenhouse gas in Earth's atmosphere, affecting temperature and climate.
• gamma — Concept · Key argument · 2 mentions · first at 9:48
  Greenhouse factor modifying the Stefan Boltzman law to account for the greenhouse effect.
• carbon cycle — Concept · Core topic · 1 mention · first at 0:53
  Part of the course's first section, discussing models of Earth's carbon cycle.
  Some sense out of it. And the course is going to be divided into three main parts. on the first section is going to be about climate and we're going to start that today. So…
• CO2 — Concept · Key argument · 1 mention · first at 11:15
  Greenhouse gas affecting Earth's temperature through the greenhouse effect.
  In fact, more important in some ways is the concentration of water vapor which is a important greenhouse gas and it depends on various other things like methane and other gases that occur in.
• density — Concept · Key argument · 11 mentions · first at 14:25
  Property of the atmosphere discussed in relation to its vertical structure and temperature.
  I'm just going to make a note. So the density of the air is also not a constant, but I'm going to treat it that way. It's about 1 kilogram per cubic meter. and CP is another physical.
• Pressure — Concept · Key argument · 7 mentions · first at 36:08
  Atmospheric pressure discussed in the context of hydrostatic equilibrium and ideal gas law.
  Later on. It's equivalent to saying that we have a vertical force balance where the pressure at any point within the atmosphere u is proportional to the weight of all the air that's on top of it.
• tropopause — Concept · Key argument · 3 mentions · first at 33:38
  Boundary between troposphere and stratosphere where temperature reaches a minimum.
  Occurs at around about 10 km height somewhere called the tropopause.
• stratosphere — Concept · Key argument · 2 mentions · first at 34:58
  Layer of the atmosphere above the troposphere, discussed in the context of temperature variations.
  Where most of the contents of the air contents of the atmosphere is. But there is some bit above. This bit above here is called the stratosphere.
• ideal gas law — Concept · Framework · 2 mentions · first at 35:54
  Law used to describe the relationship between pressure, density, and temperature in the atmosphere.
  This describes a behavior of a gas which the atmosphere obviously is and this says that there's a relationship between the pressure and the density and the temperature of a gas.
• hydrostatic equilibrium — Concept · Framework · 2 mentions · first at 38:19
  Concept used to describe the balance between pressure and gravitational force in the atmosphere.
  Okay, the next thing that we need to know is that the atmosphere is roughly in hydrostatic equilibrium.
• Solar Radiation Flux (Q) — Concept · Represents the solar radiation received by the Earth · 1 mention
  Essential for understanding the energy balance in climate models
  To be some radiation coming from the sun. I'm going to treat that as a constant. And I'm going to call this capital Q.

Full transcript with timestamps

0:12 So this is a course um so it's called mathematical geoscience uh and the aim of the course is really to use mathematical models um relatively simple mathematical models to describe and understand aspects of the natural world. Um, we're going to do that mostly through differential equations. There's some more seats around here if you want to come around around the back. Um, we're going to mainly use differential equations. So, ODEs and PDEs. Um, and we're going to have quite a lot of physics and chemistry that we need to sort of translate into the language of differential equations and then make some sense out of it. And the course is going to be divided into three uh main parts. on the first section is going to be about climate and we're going to start that today. So we're going to build some very simple climate models and in particular models of the carbon cycle and then the second section will

1:03 be about rivers and we're going to talk about how we can predict floods and then the last section of the course will be about ice um which is a very important part of the the climate system and which has some interesting dynamics. So we're going to talk about that at the end of the term. Um so I'm going to start with the the the first lecture. We're going to talk about very simple question uh which is what is the average temperature of the earth's surface or the average temperature of its atmosphere and why. Okay.

1:37 Now obviously that that varies depending on where you are on the earth and it varies over time but we're just going to talk about what's what's the average. Okay. And the average temperature at the moment is around about 15° centigrade. Why is it 15 and not say -60 like it is on Mars? Okay. Now that turns out to be mainly a question about radiation. Radiation is the transfer of energy by electromagnetic waves. And we receive radiation from the sun, lots of it. Uh but we the earth also emit radiation back out to space. And we do that at a rate that depends on our temperature and that leads to a kind of an equilibrium or at least the possibility of an equilibrium where those two things can balance each other and that turns out to be the main thing that determines the average temperature. So the first section of the course is going to be about radiative balance models.

2:42 And I'm going to start um as I often do with a picture. So here is the Earth. The Earth we're going to treat as a sphere with a radius R. It's going to turn out not to matter what the radius of that is, but um it's about 6,400 km, the radius of Earth. And I'm interested in the atmosphere which I'm going to treat as a very thin sort of spherical shell around the planet. And I'm going to treat this as having a thickness which I'll call D uh which is going to be about 10 kilometers. I'll talk about that a bit more later on.

3:29 And okay then somewhere over here is the sun. So this is the sun and that is emitting radiation um towards us. The sun emits what we call shortwave radiation. Um so that's um these are electromagnetic waves that have a relative relatively short wavelength. Uh in this case it means that they're in the visible part of the spectrum. We can see the light from the sun. It's it's light. Um but also a lot of what comes from the sun is ultraviolet which we obviously can't see but is transmitting a lot of energy. And I um that contrasts with the radiation which the earth is emitting.

4:25 So the earth is also emitting radiation. We can't see that. That's because it's not in the visible part of the spectrum. It's mostly in the infrared part of the spectrum. We call this longwave radiation.

4:47 And that's because the Earth is at a lower temperature than the sun. And lower temperature objects tend to emit longer wavelengths of radiation. Okay. So, what I'm going to try and do is to write down a a energy balance model for the the heat that's contained in the atmosphere. Okay. And so, importantly, there's going to be some radiation coming from the sun. I'm going to treat that as a constant. And I'm going to call this capital Q. It is about 1,00 let me get the number right. About 1,370 watts per square meter.

5:28 Okay, what is a unit of power per square meter means this is the amount that's coming and hitting us per square meter that we look at. Now a certain fraction of that just gets reflected back off the earth and doesn't get absorbed by the earth. So some of some of this is kind of useless to us because it just bounces back off. And that's a fraction which is called the albido. So there's a certain amount that just gets reflected back off again which I'm going to write as a * q where a is something called the planetary albido.

6:10 This is a word that we're going to talk about quite a bit. And this for Earth at the moment is around about 0.3. And it depends very much on what the surface looks like. It depends on how dark it is. So if if you imagine one of those pictures that you will have seen of the Earth from space, you'll see that the oceans look very dark and the deserts and clouds look quite bright and ice caps look quite bright. The so where where it's brighter, it's reflecting more sunlight. That means the albido is large and where it's darker in the ocean and forests, it's absorbing more and the albido is um lower. And so this is like a global average.

6:57 So for earth is about 0.3. Okay. Um now then the earth is going to emit radiation and that depends on the temperature. I'm going to call the radiation coming from the earth um QE and I'm going to have to write a bit about this which I'll do over here so we can see the picture still. So, I'll say Earth emits radiation. And we're going to introduce a piece of physics here, which is something called the Stefan Boltzman law, which tells us how a surface at a certain temperature emits radiation. So, I'm going to say according to the Stefan Boltzman I think Boltzman might have two ends.

7:56 Um law for what we call a black body. Now, the Earth is not black, but the black body is the word that gets used to describe a surface which is not internally heated. Um, which doesn't have some reactions going on inside it. Okay. And I'll tell you what this Stephan Boltzman is in in a second. We're going to have to modify this to account for the greenhouse effect because this uh tells us how much gets emitted from a surface and some of what gets emitted from the Earth's surface gets absorbed by the atmosphere and then reitted by the atmosphere on its way out. So only a fraction of what gets released from the Earth's surface gets out to space. And that's the phenomenon that you will hopefully have heard of called the greenhouse effect. So I'm going to introduce that in quite a simple way here um which we'll see. So I'll say modified to account for the greenhouse effect.

9:06 Um, so I'm going to write this QE is going to be gamma * sigma * T ^ 4 where T is going to be the temperature of the surface. That's what I'm aiming to try and find. Um, this sigma is a constant called the Stefan Boltzman constant. Um, so this is about 5.67 67 * 10 - 8 watt per square meter per kelvin to ^ 4. So that's just an empirical parameter that's come from the Stefan Boltzman law. Uh and the gamma is a greenhouse factor. So this is how we're modifying it. Gamma is going to be less than one.

10:08 um is the greenhouse factor. Um so that's the fraction of radiation that's emitted from the surface that actually makes it out to space. Okay, that depends on the composition of the atmosphere because what the atmosphere is composed of determines how much of this longwave radiation gets absorbed. And in particular, you'll be aware that this depends on the concentration of greenhouse gases which are particularly good at absorbing certain wavelengths of radiation.

11:15 So obviously it depends on concentration of CO2. In fact, more important in some ways um is the concentration of water vapor uh which is a important greenhouse gas and it depends on various other things like methane and um other gases that occur in smaller concentrations and we're going to talk a bit more about um some of that in later lectures. Okay. Right. So, what I'm now going to do is using all of that, I'm going to write down an energy balance equation for the heat that's contained in the atmosphere. Okay? So, we're going to write down energy balance for the atmosphere.

12:09 Okay? Okay. And so what I'm going to write down is that the rate of change of the heat content of the atmosphere is going to be equal to the amount of energy that is gained coming from the sun minus the amount that's emitted through radiation out to space. This was like a conservation law, conservation of energy. So I'm going to write down the rate of change and then I want to write down what's the heat content of this atmosphere. Right? I'm going to write that as the volume of the atmosphere times the heat content per unit volume which is proportional to the temperature.

12:46 Okay. So first of all I need to say what's the volume? I'm going to approximate this because the atmosphere is so thin compared to the radius of the earth. I'm just going to take the volume to be the surface area of the earth times the thickness. You have a question. Yeah.

13:02 But the temperature I think I interpret it like on the surface. This is a great question because I'm going to talk about that later on in the lecture. So the question was uh the temperature should vary with with height in the atmosphere and that's certainly true. So the temperature generally gets colder as you go up through the atmosphere. So we're doing we're being extremely crude here. We're just going to say that the whole thing is roughly at the same temperature. And then we're going to talk later on about obviously it's colder at the poles than it is at the equator and that's because there's more sunlight arriving at the equator than there is at the poles and it varies with with the height as well. So we're going to come back to that but for the moment I just want to know what's the average everywhere.

13:48 It's also not right to say that the atmosphere is of a definite thickness. It's actually quite hard to define what the thickness of the atmosphere is because the air just gets thinner and thinner as you go out towards space. Um, so that's also kind of an approximation. Yeah, thanks. So bear with me and then we'll come back and make it more detailed. We do the simple things first. So I'm going to write down the volume of this spherical shell and I'm going to say that that's going to be 4 pi r 2 * d. Right? Hope that makes sense as a an approximation of the volume. And then I'm going to multiply by um the density of the air times the specific heat capacity of the air times its temperature.

14:36 I'm just going to make a note. So the density of the air is also not a constant, but I'm going to treat it that way. It's about 1 kilogram per cubic meter. Um and CP is another physical constant which is the heat capacity of the air which is around about 10 3 uh jewels per kilogram per kelvin. We're going to need those in a second when we try and work out some numbers. Um okay so this is the rate of change of this energy. Okay, this is going to be equal to the amount of energy we absorb from the sun. Now, this was an energy flux. It was watts per square meter. So, I need to multiply it by the area that it's being received over. If you think about the geometry here, if we're standing on the Earth looking at the sun, the area that this thing is hitting is um is a circle of radius r.

15:38 Okay. So, in fact, maybe it's better to imagine standing on the sun and looking at the Earth, right? And you'll see the Earth as a circle of radius R. So, here we're going to have pi R 2 * Q* 1 - A because a factor a fraction a of it got reflected back off again so it doesn't get absorbed. Okay, so that's what's coming from the sun. And then I'm going to subtract off uh the radiation we emit out to space. This is emitted from the whole surface area of the earth that has surface um area 4 pi r 2. So I'm going to have 4 pi r 2 times this rate. This so this was again this was in watts per square meter. Um so gamma sigma t to 4 The temperature, by the way, this is going to be important in a second. In this um Boltzman law, the um the temperature has to be in Kelvin, right?

16:46 It's proportional to the temperature to the power four for reasons that are not entirely obvious, but um that temperature has to be the absolute temperature. Okay, so this is now a differential equation for temperature T that we're going to solve. We're going to treat everything else as a constant. We can simplify it a bit. If you notice the there's factors of R squ everywhere. So I can just get rid of those and pi as well. So I'm going to rewrite this. I'm going to write it as capital C * DT by DT uh is going to be equal to I'm going to divide by four as well a quarter q 1 - a minus gamma sigma t 4 and capital C is just some constants that I've grouped together That's going to be d row cp.

17:53 We can think of this as like a an integrated uh heat capacity of the atmosphere which in fact allows us to kind of get away with the fact that row is not really constant and d is not really defined. If we think of d times row actually as being the depth integral of the density of the atmosphere that that is a kind of well- definfined quantity. Okay. And that has a value that is around about 10^ 7 jewles per square meter per kelvin. Okay. So this is our first very very simple climate model and it only includes radiation. Um but it's enough to tell us at least roughly what the temperature of the planet is. Like I'm going to treat everything in here as a constant apart from capital T.

18:54 So it's quite a straightforward ordinary differential equation. And the first thing that you should think about differential equations like this is whether they have any steady states. This one has a steady state. steady states of the solution that's independent of time. So that term would be zero. So that's when we have a balance between the radiation from the sun and the radiation emitted by the earth. And that tells us what the temperature should be. So we have a steady state. I'm going to call that t. Okay. Okay. So just by rearranging that that's q * 1 - a / 4 gamma sigma all to the power of 14.

19:46 Now if we didn't have the greenhouse effect this gamma would just be one. We wouldn't need it. So I'll just make a note about what happens if in that case if gamma were one. That's what would happen if the earth didn't have an atmosphere. Um then if we put in the numbers that I've given you here uh we'd find that this T0 is about 255 Kelvin. It's about minus min -20 Celsius. So if that was the case then most of the planet would be would be frozen. Um if we go the other way and say well we kind of know what this temperature should be because we can measure it and I told you earlier on it's around about 15° centigrade. So it's about 288° um Kelvin. So actually maybe I should just make a note around 273 roughly Kelvin is the same as 0° C.

21:10 So the observed average temperature which is around about 288 keeps going up obviously um that suggests and we're working slightly backwards here that suggests that this greenhouse factor must be around about if you put in the numbers you get about 0.61 61.

21:39 If you remember, that was a fraction of what's emitted from the Earth's surface that gets out to space. And obviously as you increase the concentration of greenhouse gases in the atmosphere, what you're doing is decreasing that gamma, which is slightly confusing that that that when you make the greenhouse effect stronger, you decrease this greenhouse factor. That's just the way that it's been defined. And that as you decrease gamma, then that obviously makes this temperature go up. Okay? And that's a very simple um explanation of of global warming.

22:23 Um the second thing that you should ask if you have an OD like this having found that it has a steady state is whether that steady state is stable or unstable. This one, it turns out, is stable, which is a good thing because otherwise we wouldn't it wouldn't live in that steady state if it was unstable. So um is the steady state stable? And the answer is yes. And I want to show you two simple ways that we can see that. that you will hopefully have seen before, but this is just um techniques that we're going to use quite a lot in this course. I want to make sure that you're familiar with these. The first um is a graphical method.

23:21 Uh and so this we're going to just plot according to that equation. We're going to plot dt by dt as a function of t. Well, which is what this equation tells me. The tdt is this function of t which is a constant minus constant time t to the 4. So it's just a cortic which is going to look something like that. The steady state was defined as where ddt is zero. So that's where this line crosses the axis. So that was t0 and what you can see from this graph is that if you are at some temperature that is less than t0 then dt by dt is positive. So what will happen is that the temperature will increase and you will move this way and you'll move this way until you get to that point. So if I were here I would move the temperature would increase and I move towards that point. If I was over here at a larger temperature dt by dt would be negative. So my temperature would decrease and I come back this way.

24:36 So wherever I start, I'm going to always end up at that point. That shows that uh that state is stable. And generically for this type of differential equation, if you've got the um the graph crossing the axis going downwards, then it's going to be a stable steady state. Okay. The second method is slightly more formal um and that is a linear stability analysis. And so you hopefully will have seen this idea before. So the idea is that we're going to perturb ourselves from the steady state by a little bit and then we're going to look at the dynamics and work out does that perturbation decay or grow over time. And that tells us in this case it's going to tell us that we decay back towards the steady state.

25:31 So we're going to do that by writing the temperature as the steady state temperature plus a pertubation. And I'm going to call that perturbation t primed where t primed is small um by which I mean it's much smaller than the steady state value t0 and then we just stick that into the equation. The equation then then so t0 is independent of time. t primed is the thing that depends on time. So we have dt primed dt is equal to 14 of q 1 - a those were just constants minus gamma sigma t0 + t prime to the^ 4 and because t prime is small we're going to linearize this term this nonlinear term we're going to approximate it using the fact that t primed is small that's basically like doing a tailor expansion or in this case a binomial expansion. So that's going to be quarter q 1 minus a minus sigma gamma t 0 to the 4 minus 4 sigma gamma write them the same way around gamma sigma t 0 cubed * t primed

27:03 plus higher powers of t primed I'll write that as h o t higher order terms And but if t prime is small those terms are going to be small. So we can ignore them and we just get this bit here. These two terms exactly cancel out uh by the definition of t0 being the steady state. So that term and that term uh cancel out. So this is this is an approximation now approximately minus4 gamma sigma t0 cubed * t primed and we just ignore the higher order terms. So we have linearized the equation at that point. The reason for doing that is that if we have a linear equation like this, then we know that that just has exponential solutions, right? dt by dt is minus some constant time t primed.

28:01 That just has a solution which is proportional to the exponential of minus that constant time t. So t prime is proportional to e to the minus I'm going to write this as t over t0. where t0 is that constant um which is uh c over 4 gamma sigma t0 cubed but this is a time scale when I've written the exponential like this is the time scale over which the perturbation decays So this tells us two things actually. First is that it tells us that the the this perturbation decays in time because there's a minus sign there. So that means that this is a stable steady state. Um but it also actually tells us a bit more because it tells us how fast can we expect it to decay which for these sorts of problems is quite a useful thing to know because it tells us how quickly would we expect the system to respond to a perturbation.

29:11 If we work out using the numbers what this thing is, this turns out to be about 30 days, which means that the atmosphere, the temperature of the atmosphere responds relatively quickly, depending on your perspective. Um, to a change, say the sun was to double its strength, it would get hotter quite quickly. Uh, for example, that's obviously not likely to happen. Um, but for example, when there has been huge volcanic eruptions in Earth's history that's put loads of ash into the atmosphere, that has the effect of changing the albido um because that blocks some of the sunlight. So, it means that we don't receive as much sunlight and that leads to a cooling over that sort of time scale.

30:04 Uh, so that's something quite useful that comes out of doing that. Okay, so that that's obviously extremely simple model. Later on, we're going to come back. In fact, over the next couple of lectures, we're going to we're going to kind of make this more complicated. Um, in particular, we're going to think about how other things apart from the temperature here could vary in time. So that the the radiation coming from the sun, the Q, it doesn't change that much, but it does change a bit. Partly because the Earth's orbit is not completely circular.

30:36 the albido um can change a lot that we talked about the fact that that depends on the what the surface of the earth looks like. So that can change especially depending on how much ice cover there is. Um and that's partly why the earth's climates changed a lot in the past. Um and this gamma obviously that can change with the composition of the atmosphere and that's um something that we're well we're doing a kind of large scale experiment on that at the moment obviously but we're going to talk a bit about how we can actually predict gamma um from considerations of what's going on in the atmosphere. Uh and then this will then tell us what's the temperature changes that we can expect from that.

31:20 This model can also be applied to other planets. And if you look at the problem sheet, I think the problem sheet asks you to think about what happens with um other planets where the main difference is that you have a different um Q because if you're a different distance from the sun, you have a different amount of radiation that reaches you. Okay. So I want to move um on let's do this over here um to come back to the question that was raised about the fact the temperature really varies with with height in the atmosphere.

31:52 So I'm now going to talk about the vertical structure of the atmosphere. So in reality the temperature and the density which I both treated both of those as as um just single values here they really depend on depth and let me just sketch how roughly how they do depend on depth and then we'll talk about why that is. So I'm going to plot the density row as a function of height zed. This is roughly roughly exponentially decaying. So looks something like that. And its value at the surface is the value that I used earlier on. Um it's about 1 kilogram per cubic meter.

32:46 So you you well you'll be aware maybe that if you climb Mount Everest, the air gets pretty thin at the top which is why you generally have to use oxygen. Um and that's a reflection of the fact that the density is much less. Um and then I can plot the temperature also as a function of depth. This is in some ways kind of more interesting actually. So the temperature does get colder as you go up. um it actually reaches a minimum at some at some height and then turns around and increases again and then reaches a local maximum and then decreases towards outer space.

33:30 This minimum occurs at around about 10 km height somewhere called the tropopause. So this is around about 10 km and that's around the height that a plane usually flies at least a big a big plane. I don't know if you know the temperature outside a plane. I used to in the olden days they had told you on the screen on the airplane what the outside temperature was which is usually about -50° centigrade. I I'll call it 220 Kelvin here. This is at the ground. This is where it's about 288 uh Kelvin at the moment. Obviously, this depends also on latitude. Um which I'm not worrying about here. Um at least this this value does. Um the region below this tropopause is where most of the weather happens. That's where we form clouds.

34:40 That's where we have storms and so on. Where if you fly, you're usually above the clouds. Um, this is called the troposphere and that's the region that I was really thinking of as the atmosphere in the first model that we wrote down. This is where most of the contents of the air contents of the atmosphere is. But there is some bit above. This bit above here is called the stratosphere. And you do sometimes get some clouds up there. kind of high high clouds and then this local maximum called the strata pores and then above that there's a messphere and then there's there's there's various other um spheres as you get higher where the atmosphere is really really thin. So most of the interest of of of um so weather forecasters and so on is is with this troposphere down here.

35:32 Okay. So why do they look like this? And this um can be understood with some quite simple pieces of physics. So let me try and explain um some of these. Okay. So the question is why? Okay. So the first thing that we need um is the ideal gas law. This describes a behavior of a gas which the atmosphere obviously is and this says that there's a relationship between the pressure and the density and the temperature of a gas. And it says that the pressure is uh row rt divided by m a at least that's how I'm going to write this. This is the density times a constant time the temperature divided by another constant.

36:33 Um so R is called the universal gas constant. That's about 8.3 juwles per kilogram per mole. And m a is the molecular mass of the atmosphere which this is averaged over the uh all of the composition of the atmosphere. Uh and this is around about 29 * 10us 3 kg per mole. A mole if you remember your chemistry is a measure of an amount of a substance. Um well this is really the molar mass I guess molar mass of the atmosphere. Okay so this um this symbol here is a P. That's the pressure. This symbol here is a row. That's the density. In this course we have lots of pressures and densities and they can they can look very very similar. So, I try to put a sort of stalk on the top of my P and I try to make my density very rounded to make it a row. Um, but just just be aware that we're going to talk about P's and rows quite a lot. Uh, and you need to try and find some way of distinguishing them.

37:59 T is again temperature has to be measured in Kelvin for this to work. Okay, so we're going to use that. I'll call that equation one. Okay, the the next thing that we need to know is um that the atmosphere is roughly in hydrostatic equilibrium. That's a concept that comes from fluid mechanics which we'll talk about a bit later on. It's equivalent to saying that we have a vertical force balance where the pressure at any point within the atmosphere u is proportional to the weight of all the air that's on top of it. And this is not exactly true, but it's a very good approximation for how the pressure varies with height in the atmosphere.

39:09 seem to have chosen a pen that doesn't rub off. Give me a good workout. We come back up here. Okay. So the next the next thing that we need to know here is that we have hydrostatic equilibrium. Okay, this is an approximate vertical force balance. And we're going to express this with the differential equation. dp dz is equal to minus row g. So this is the rate of change of pressure with height is equal to minus the density times gravitational acceleration g which I haven't written down anywhere. So g is about 9.8 m/s squared.

40:31 Okay, I'm going to call that equation two. So um we can understand just with these two pieces of physics we can understand uh why the density looks like this at least roughly because the temperature although it varies it doesn't vary by that much on an absolute scale. Um so if we treat this temperature as a constant and then this ideal gas law just says that the pressure is proportional to the density. If I stick that in here, the pressure is proportional to a density. Then I can basically replace that with a drow dz. And then this equation says droz is proportional to minus row, which gives me an exponential solution, an exponential decay with zed.

41:29 So if we treat um temperature as roughly constant now obviously this is not that great an assumption um because it's not that constant um but it's not too bad as so roughly constant Um then if we stick equation one into equation two we'll get d row dz uh is equal to minus m a g over r t0 times row. I've just rearranged to get this. This stuff here is all then I'm treating as a constant.

42:30 Okay. And that gives me that row is some constant. Let's call it row s for row at the surface times e to the minus and I'm going to write this as z over z. It's a bit like what I did before where Z 0 is just this constant. So Z 0 is going to be R T0 over M A G. Uh and I'm writing it like that because this has got something that's got units of length and it's the height scale over which the density decays. So if we put in the numbers, this is around about 8 kilometers.

43:31 Okay, which which ballpark is similar to the 10 km height of the tropopause and that's why I used that value of 10 km as the depth of the atmosphere in that very simple first model because that's the the length scale of which the density decays. This this um is some sometimes called the scale height of the atmosphere. Uh it's a term that gets used a lot in meteorology. Okay. Uh then the last thing that I want to do today is try and explain I'm not going to explain all of this but I want to explain this temperature decrease here uh within the troposphere. And for that we need an energy equation.

44:19 So that the the last piece of physics we need is the energy equation. We already wrote down an energy equation for the whole atmosphere. That was this thing over here. But now I want to do it sort of depth dependent. And when you do that, it turns out radiation is not so important. Actually advection of the air. So movement of the air and um compression and decompression of the air are the things that are the most important. So I'm going to write down a more complicated looking energy equation. Um this is going to be I'll write it like this. O CP * capital D by DT of T minus capital DP DT is going to be equal to the divergence of K grad T minus the divergence of some Q.

45:32 So this is a form of a heat equation that you might have seen before. This term here represents conduction in the air. That term turns out to be really negligible. So in fact I'm just going to ignore it. We're going to include that term in some of the other models that we look at. This term here you might not have seen before. This term is uh absorption and emission of radiation. So Q here is the radiative energy flux. Now this was really important on the scale of the whole atmosphere. Um but it turns out not to be so important like locally. So I am at least within the within the troposphere. It is more important in the stratosphere. Uh so this is radiation and and this is not such a great approximation but I'm going to ignore that for the moment as well.

46:25 These terms here this capital dt if you haven't seen this before this is something called a material derivative. So maybe I just make the note here. Capital d by dt. This is again a concept from fluid mechanics. This is the rate of change uh of with respect to time following a fluid particle. So you can write this as partial d by dt plus u dot grad t. U is the velocity of the fluid. Gradient of t means the gradient in space. So this represents the rate of change following a fluid.

47:07 And the same the same thing here. Uh and in in the troposphere this does turn out to dominate this equation. Well partly because there's nothing else left. So it kind of has to. Um and that's because in the troposphere you have a lot of convection. This is why we get interest in weather. um convection represents vertical motion of the air in the atmosphere and that causes um this equation to be dominated by the vertical advection terms that appear in here. So there's the vertical velocity times the vertical gradient in the temperature and the same thing in the pressure and that turns out to dominate these terms.

47:46 um so I'm going to say in the toposphere um convection the convection is driven by the fact that the air has been is hotter uh we're not going to get into this but the the air is hotter lower down and hotter air generally is lighter and therefore wants to rise compared to colder air and that sets in motion some convection um

48:22 vertical adection to dominate. Okay. And so if we average over time, the time dependency is quite important. If we if we ignore time dependence just to understand what's the sort of base state um then we're going to have here row cp time dt dz minus dp dz is approximately zero. Right? So the vertical directionction causes the the zed derivatives in these two terms uh to dominate and to balance each other. So I didn't say this term here represents heat the heat due to the compression or expansion of the gas. So as you change its pressure that changes its heat content. That again if if you've seen fluid mechanics for a like a liquid you you generally don't have that term because it's very small but in a gas that's important particularly on this scale.

49:31 Um, okay. So, this is the approximation of the energy equation that I'm going to call equation three. Uh, and if I combine that with equation two, which has come off the top of the board, equation two told me what dpdz is, minus row g. So I can put that in here and then that's going to tell me what dtz is.

50:07 If I rearrange it, it tells me dt dz is uh minus g over cp two constants. This I'm actually going to give a name. I'm going to call it gamma capital gamma d uh which is the name it's often given. This is what's sometimes called the lapse rate. It's the rate of change of actually it's I think there's a minus sign there. It's the rate of change of temperature with respect to height. So it's how quickly uh does it get colder as you go up. And if we put in the numbers this um turns out to give us around about 10 Kelvin per kilometer. If I use those units, say if you go to the top of Ben Nevice, which is order a kilometer uh higher than sea level, then the temperature is about 10° colder on average.

51:04 Um okay, then this this actually I've ignored something here which does turn out to be a bit important. This is this gamma D. The D represents dry. So what we've done here was to ignore water vapor. Uh and it turns out that the condensation um of water vapor to form clouds uh in the atmosphere is quite an important term in this energy equation that I didn't write down. So although I crossed out various terms that I didn't think were important, I actually ignored writing down a term which is important. Uh and that's because I don't want to talk about that yet until next week and when we will talk a bit about how clouds form. But they do they do affect this.

51:49 So this is this is how the temperature varies when there's very little water vapor in the atmosphere. When there's water vapor that turns out to change this slightly and you get actually slightly lower rate of decrease of the temperature uh with height. Okay. So that's it for today. On Wednesday we're going to talk about modeling the greenhouse effect in more detail. So we're going to try and work out what this gamma actually is. See you then.

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What is "Mathematical Geoscience: Radiative Balance Models - Oxford Mathematics 4th Year Lecture" about?

This course on mathematical geoscience focuses on utilizing differential equations to model and understand natural phenomena, specifically in three key areas: climate (including the carbon cycle), river dynamics (predicting floods), and ice behavior in the climate system. The initial lecture addresses Earth's average surface temperature and the radiation balance that influences it, contrasting…

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17 people, works and ideas are mentioned, including temperature, greenhouse effect, climate models, troposphere, Stefan Boltzman, albido. Each one on this page links to its own page with every episode that mentions it.

How long is "Mathematical Geoscience: Radiative Balance Models - Oxford Mathematics 4th Year Lecture"?

The episode runs 53 min; the transcript is about 6,938 words in 58 timestamped sections.

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