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P3: Radiativeforcenado (Radiative Forcing)
382 Plays5 years ago
Learn more about radiative forcing, the environment, and how global temperature changes with atmospheric absorption with this Problem Episode about you walking your (perhaps fictional?) dog around a park. This episode is distributed under a CC BY-SA license. For more information, visit CreativeCommons.org.
[Featuring: Sofía Baca, Gabriel Hesch]
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Transcript
Introduction & Patreon Tiers
00:00:01
Speaker
This is Sophia. And this is Gabriel. And you're listening to Problem Episode 3, Radiative Force NATO, an episode about radiative forcing. Because this is a Problems episode, we suggest that you listen to Episode 46, Earth Irradiated, for more information. Hi, I'm Sophia. And I'm Gabriel. And we have a few plugs before we start on the rest of the podcast.
00:00:28
Speaker
Yep, first plug I will go ahead and mention is our Patreon poster tier. That is where if you go to our Patreon, which can be found at patreon.com slash breakingmathpodcast, you can donate to this podcast if you would like to support what we do. One of the things that we have for our Patreons of a certain amount, the amount is pi to the E dollars. Yeah, $22.46. Yes.
00:00:50
Speaker
If you sign up for the $22.46 level, then you can get a poster. You will get a poster designed by Sophia that explains tensor calculus and how it's used in Einstein's general theory of relativity. It's a beautiful poster.
00:01:07
Speaker
Yeah, and the general theory of relativity is how gravity affects the space time, not just how velocity affects your perception of space time. We also have a tier where we deliver outlines is only $5. We also, if you want to just buy the poster outright though, you go to our Facebook store at facebook.com slash breaking math podcast. And it's $22 and 46 cents for a tensor poster plus 450 shipping and handling.
00:01:30
Speaker
Yep, that total is $26.96. Again, that's pi to the E dollars plus shipping and handling. Also, you can follow us on Twitter at Breaking Math Pod. We have a lot of updates that go up there frequently. We also have a website, breakingmathpodcast.com, but the website we have been notified is actually down right now.
00:01:50
Speaker
Yeah, we are working with our hosting service to see what's up. Don't go there quite right now. Wait until the next episode where we tell you to go there. There's ads there
Website Issues & Future Topics
00:01:59
Speaker
that are not our ads. We aren't supporting these people. These people are alien to us. Yep. Yep. Yeah. We've been, we've been taken over by hostiles.
00:02:09
Speaker
Actually, we've been taken over by the kind of people who you like to wipe off to the bottom of your shoe. And that's all I'm going to say about that. Okay. Okay. Okay. Yeah. So if we have an episode that is released after this episode, hopefully the website should be back up and running. So yes, moving on, moving on.
00:02:25
Speaker
And of course, this will be the last climate change episode for a while. We just wanted to kind of cover some of the... We were struck by how simple the math is for climate change. And of course, the models get very complex, but even the simple models have reasonable predictive power.
00:02:42
Speaker
Yeah, so this would be our third in a series of climate change related episodes. It is a very, very fascinating topic and a very relevant topic. There's really a lot to discuss here. Maybe one of these days we might revisit this topic. Maybe if we can have
Retro Causality & Eye Surgery
00:02:55
Speaker
say Dr. Catherine Hayhoe on our show or any of the other fabulous climate scientists to talk about what it is that they do.
00:03:01
Speaker
Well, but we have a bunch of other episodes on other topics. What are some of the other topics that we're going to do in these next few weeks? We're going to do some episodes about, we found some papers on retro causality. So we're going to do a few episodes, an episode or two on causality. We're also thinking about doing one on machine learning and inventions in prison. Yep. And we're hoping to have a guest that we've had in the past on, we're not going to say much more until we know for sure.
00:03:29
Speaker
And real quick, retrocausality, what on earth is that? Just as a little preview. It involves Bell's theorem, which we have done an episode on.
00:03:37
Speaker
Yeah, in a sideways way. Yes. But yeah, retro causality is the idea that the definition of causal relationships allows you to draw arrows that go against the progression of entropy or time. So it's almost like you have a second time stream that is just due to causality, but we're going to go into all that weirdness and I wouldn't do justice talking about it right now.
00:04:02
Speaker
Yeah, it is an extremely strange, strange idea. But the paper is, well, it's a pretty compelling paper. So it's certainly worth going into. We also might be revisiting Godel Escher Bach, that famous book. That's one of our favorites.
00:04:14
Speaker
Oh yeah. Um, yeah, there's a lot of stuff in there that we think would be fun for y'all. Yes. Um, but, uh, yeah, we're looking forward to making some, uh, good episodes. And, uh, just to let y'all know, um, I know we haven't been on in a couple of weeks or so, and we won't be for another week or two, at least another week or two after, uh, this episode, because on February 3rd, uh, Sophia is getting eye surgery and that's going to be a whole thing. Yep.
00:04:40
Speaker
So we will be on a little break for a little while, but I don't know why I referred to myself in the third person. I'm getting eye surgery. I was curious about that too, if we're all going to do a third person from now on. I was imagining that I was writing an email and I always write the emails from both of us. Oh, okay. Okay. It's all good. Yeah. So on this episode,
Understanding Radiative Forcing
00:04:57
Speaker
Sophia has designed a problems episode all about the concept of radio. I can't say the word.
00:05:03
Speaker
Radiative forcing radiative forcing so that way you all can understand it on a deeper level and And I want to give a quick shout out to there's a Harvard resource on the greenhouse effect Well, we've been using this quite a lot and this problem episode is Very much comes very much from the math that they talked about in there. It's not exclusive to them, but we wanted to give them credit Yeah, so without further ado
00:05:33
Speaker
Now this problem starts with a model of the earth and the sun and the energy that comes from the sun and heats up the earth and then from the heat of the earth leaves the earth. And we're going to use that model to talk about radiative forcing which is like
00:05:50
Speaker
let's say that we clone the earth right and we put it i don't know like a month ahead of us in the orbit around the earth or around the sun yeah or a couple days or something like that and but in that atmosphere we just decided to be jerks and dump a bunch of carbon dioxide in there
00:06:06
Speaker
The difference between the amount of solar energy that is used by the Earth system in the one with more carbon dioxide or methane or whatever versus our Earth, that is radiative forcing. It is the amount of radiative flux that is caused by an increase in atmospheric absorption and we'll break that down.
00:06:28
Speaker
Yeah. And so with this model, we're starting off, we're gonna talk about three variables. These aren't really too complicated, really. We just have to explain them and understand an example of them. The first variable is f sub s. And that's the solar radiation flux.
00:06:45
Speaker
Yes, and we keep talking about this thing flux radiative flux is the amount of light that hits like let's say we have a bunch of red and blue light and we shine it at your closet and The closet has a certain amount of light on it and you measure that with a radiometer and you take the amount of light that you measure and divide it by the area of the closet that is the amount of radiative flux on that closet and In black bodies, which is what we're gonna amount of the earth as because of the simplest model I will justify that later
00:07:14
Speaker
I actually will justify right now, it's relatively accurate and it's the best way to talk about some of these concepts and all these concepts can be made more nuanced by using more complicated GCM models. What are the units for flux in this case?
00:07:30
Speaker
Oh, watts per square meter. Okay. So, yeah, so the total, in a black body emits light of all frequencies at the same time. Okay. However, if you sum them all up together, that's a total radiative flux. And that relates to another of our constants that we're going to talk about, which is sigma, which is a Stefan-Boltzmann constant. And that just relates the temperature of a black body and flux itself.
00:07:57
Speaker
Okay. So yeah, again, that's, that's three variables. That's not really, that's not too bad. You got. Okay. Oh, that's right. A for Albedo. So I have a toddler who I'm reading alphabet books to all the time. So I'll do a science breaking math themed alphabet book for your kids. A is for Albedo. B is for breaking math. C is for constant and the speed of light. You know what? That'll
Radiative Energy Models
00:08:20
Speaker
be fine. That'll be available on our gift shop soon. Right. As soon as I'm done with the breaking math children's alphabet book,
00:08:26
Speaker
Let's really do this. Let's do this. Oh, it'll be so much fun. I'm sorry. I'm sorry. I am not distracted at all. We're recording this on Super Bowl Sunday and I like, I'm just, I'm not even, I don't even care about the Super Bowl Sunday right now. You can cut that part out, I guess. Okay. So the three variables again are F sub S, which is the solar radiation flux.
00:08:45
Speaker
Or the amount of sun reaching the Earth per area? Yes, we have the next variable, A. A is for albedo. Which is the amount of light reflected from the Earth. Yes, yes. And then we have sigma. This is a lowercase sigma that looks like an O. You know, an O with a little top hat or something.
00:09:02
Speaker
Which is the Stefan-Boltzmann constant for relating temperature of a black body and flux. I mean, where even did that come from? I mean, obviously the dude named Stefan and, you know, and Boltzmann, but, you know. I mean, all of that stuff, I think, originally comes from a light bulb factory that wanted better light bulbs in the 1800s. Oh, yeah, yeah. Actually, I think also the student distribution comes from a light bulb factory in Germany, too.
00:09:27
Speaker
So we're good on the three variables, right? I mean, are we, are we, are we solid? Yeah. And, um, now we're going to talk about on the last one, we talked about a little bit about how black bodies absorb energy. You could listen to the last episode if you want to know that, but right now we're just going to talk about three systems to set up the equations. And we're going to go a little bit into them because we didn't give them justice on the last show.
00:09:46
Speaker
Yeah, that's right. That's right. So so we've got we have three Important systems that are coming up the first system that we'll be talking about is the Sun Earth system The next system is the Sun Atmosphere ground system and on that second system that when it says ground that's the Earth's ground, right? I
00:10:05
Speaker
Yeah, the ground temperature like the surface temperature of the Earth. Okay, very good. Then after that, we have the atmosphere ground system. Not to be confused with the Sun atmosphere ground system. It's different. It's different. It's like when you solve an equation of two lines. It's like if I say that three slices of pizza and two sodas cost $7, but two slices of pizza and three sodas cost $9 and you have to figure out how many. Oh my God. Dude, you're bringing me back to middle school.
00:10:35
Speaker
school that was when I was that that's when math is mean to you and you're like not really enjoying it you know which is you know yeah yeah exactly but you know I guess to me this is kind of like talking about a wedding cake where sometimes you're talking about like you know multiple layers of the whole cake itself and then like individual layers would you say that's a good analogy or am I totally off here
00:10:56
Speaker
What was it? Oh, so so all of these models combined or like these models are kind of like talking about a wedding cake Where you've got different layers of the wedding cake and how they oh, yeah Yeah, the Sun atmosphere ground system just kind of like three layers We're assuming that the atmosphere is like infinitely thin right above the earth basically Yeah, and also the earth doesn't have to be a spirit in this model the earth could be like literally flat and the Sun could be above it flat earth, but The math all works out the same
00:11:24
Speaker
Okay. Yeah. So flat earthers. This, this, this one goes up to you. I do believe in climate science, but I do not believe in globe earth. I'm just kidding. Okay. Yeah. Yeah. Where does the sun go at night? You know, if you're, if the earth is flat and it's, uh, it's on the other part of the earth. Okay. No, no matter that. And that would mean that on this side, there is zero flux given at that time. Cause it's true. Cause
Thermal Equilibrium & Albedo
00:11:50
Speaker
there's no sun. Okay. Okay. Moving on.
00:11:57
Speaker
This episode is all about climate change and how it's related to critical reasoning. This episode is all about climate change and how it's related to critical reasoning and one of the topics we cover has to do with energy.
00:12:18
Speaker
This episode is all about climate change and how it's related to critical reasoning. To that end, our partner Brilliant.org has a course about scientific thinking, which is about how to explain the world in scientific terms. I love how the course starts by explaining the nature of simple mechanical systems and takes you all the way through more advanced topics like heat and light. To support your education in math and physics, go to www.brilliant.org slash breakingmath and sign up for free.
00:12:48
Speaker
The first 200 breaking math listeners can save 20% off the annual subscription which we have been using and now back to the episode So we're gonna talk about radiative energy a lot because the Sun radiates energy onto the earth and the earth radiates energy outward when it's hot and it's measured in watts per square meter, which is power per area and
00:13:14
Speaker
Measuring the radiative flux for each wavelength and summing it up gets the total radiative flux and the radiative flux is different at each wavelength because Okay, like let's say we get a piece of iron, right and we hit up the first color it turns is red, right? Yes, because it's emitting in an iron is almost like a black body. So Actually, we'll just talk about black bodies to to typify this or exemplify this something like that. I
00:13:41
Speaker
So it turns red first because it's generating a lot of heat, but also a few red photons which are low wavelength photons. And then it turns yellow because it starts getting yellow and green wavelengths which combine with red to make yellow in our eyes. And then as it gets more and more wavelengths, it gets whiter and whiter.
00:14:03
Speaker
So that's for a black body, but a black body is a perfect object that can absorb every wavelength and emit every wavelength. But we know from
Radiation Concepts & Climate Science
00:14:13
Speaker
experimental data that if an object can absorb a wavelength, then it can emit it just as efficiently as it absorbs it.
00:14:20
Speaker
So there's something that I'm going to talk about called the flux distribution for a black body. The flux distribution for a black body was discovered by Max Planck in 1900 by the equation. This is a bit of a wordy equation here. You don't have to take notes on this. This is something that you can just Google search. But the equation that he found was 2 times pi times Planck's constant times c squared. Which is the speed of light squared. Correct. Speed of light squared.
00:14:46
Speaker
All of that divided by L, which we just mentioned above, I don't think we actually said it. L is the wavelength to the fifth power, times E raised to Planck's length times the speed of light, all divided by Boltzmann's constant times the temperature times the wavelength. And that's all over E.
00:15:05
Speaker
Yeah, I fully get and then there's a minus one. Did y'all get that? Did everyone get that? Okay. Okay. We're good. Just Google search it just you know, anyways, yeah, so there's an equation here that exists and and the peaks of this This shows that peaks at wavelengths are inversely proportional to the temperature.
00:15:21
Speaker
Yeah. So if it's a really, really high temperature, then it peaks at a very short or very, very energetic wavelength. Correct. Yeah. All of that math, just to say that single point. If the object absorbs a certain percentage of radiation at some frequency, then it will also radiate at that ratio. So like if you absorb like 50% of red light, then what that means is that when you're heated up, you will emit 50% as much red light as a black body would emit.
00:15:48
Speaker
This can allow us to generate the emission spectrum from the absorption spectrum. Literally, multiply the absorption spectrum by the black body radiation spectrum component-wise. Like over the frequency. So you have the first graph, which is the black body radiation spectrum, which goes something like...
00:16:05
Speaker
over time and then like you might have the absorption spectrum which might be actually very like abrupt like no nothing nothing nothing and then one for a little bit then a half for a little bit then the absorption spectrum especially with the atmosphere is something that's gonna be very messy and if you but if you multiply the two then you get you know the emission spectrum what it emits when you heat it up and that's essential for climate science because the energy emitted by the atmosphere and the earth is what it's heated and
00:16:33
Speaker
Yeah, actually that's the main point of this so that now that we've talked about how how you can know the emission spectrum or the heat or the energy that objects radiate if you know that based on their Absorption spectrum and and vice versa if you know the absorption spectrum then you know the emission spectrum So now let's talk about two very specific bodies that relate to climate science and that is the earth and the Sun What do we know about the earth and the Sun?
00:16:58
Speaker
Not much. I'm just kidding. Solar radiation peaks in the visible range, which is not surprising because we're designed to look at things that reflect solar energy, which is about 400 to 700 nanometers. And its maximum in the green, which is 500 nanometers, and about half of that infrared frequency, which is less than 700 nanometers, and a very small fraction is in UV.
00:17:24
Speaker
So this is the composition of the sun's energy. It just tells us what we know is like about half is like in the infrared, which is very closely related to heat. And the reason why infrared is closely related to heat is because gases, it was we'll see in a second, absorb infrared radiation by either rotating or vibrating. Yeah, we'll talk about that in a second.
00:17:47
Speaker
Yeah, actually, that's something I found very, very interesting is I had no idea, like, you know, we talk about emissions and absorption. I didn't know what happened at an individual molecule level. I guess there is, there's all kinds of interactions, you know, including vibrations where I think of like, you know, a spring, like if a molecule were to spring, it would squish and expand, right? Oh, yeah. Not just in one direction as a spring does, but in multiple directions. But then there's rotational energy where you've got something like like spinning around, like I think of rolling down a hill, you know, and you get all kinds of heat there.
00:18:16
Speaker
Or somebody's playing tennis and they put a spin on it. Yeah, exactly. So there's all kinds of things related to heat at the individual molecular level. Fascinating stuff, I find actually. Now, we know that the Earth is not sufficiently hot to emit significant amounts of radiation in the visible range. We know that because it's not like a lump of coal that glows red in a fire.
00:18:42
Speaker
Yeah. If you looked at, I mean, sometimes the earth glows red, like at volcanoes, but that's pretty much it. Yeah. Yeah. That's not saying the earth itself as a whole. Yeah. So that's a very dramatic explanation of why it doesn't, like how we could see that it doesn't emit significant amounts in. Yeah. Now by the laws of thermodynamics, everything radiates. So the earth certainly does radiate. It's just not, I mean, as long as it has any motion whatsoever, it radiates.
00:19:08
Speaker
Um, any, any heat whatsoever, I believe. And I think emotion might be related through some kind of, I think Hawking studied that, uh, that black bodies that are traveling through space have a slightly different temperature. I'm not sure. Yeah. Interesting stuff. Yeah. Learn more about that in our black hole episode, where we literally talk about the thermodynamics of black holes, fascinating stuff as well. This is not nearly as complicated, not nearly as complicated. No, not nearly. Yeah.
00:19:32
Speaker
Although the models themselves are more nitty-gritty as we will see, I will have to say that for them, but yeah, the math itself is very basic. Now, let's talk about the thermal equilibrium of the Earth and the surrounding space.
00:19:45
Speaker
Oh yeah. So, uh, basically if the earth is at a constant temperature, yeah, thermal equilibrium is when the amount of radiative energy going into the earth from the sun and stars and everything is equal to the amount of radiative energy, which is the micro, almost microscopic amount of visible radiation as long as well as a very large amounts of infrared radiation emitted by the, um, the earth are equal.
00:20:12
Speaker
The amount of energy that is reflected is very important. It's a property called albedo, and it's reflected back into space by clouds, snow, ice, whatever. And it's equal to 0.28 for the Earth, which means that 28% of the light that reaches Earth is radiated away. In the IR range, though, it's almost 100%.
00:20:34
Speaker
Interesting. Now, this is something that we'll talk about more and more with, again, as you said, with Earth, the albedo, or rather the amount that is readied back into space is A equals 0.28. For other planets, it's completely different.
00:20:46
Speaker
Yeah. And what's interesting is that the mean temperature of Earth, because of the albedo of the Earth, and this is a very simple equation. This doesn't relate to temperature within the Earth, but it's 255 Kelvin, which is about negative 18 degrees Celsius, or I think negative 25 Fahrenheit or something like that. Okay. I don't know. I'm
Greenhouse Gases & Absorption
00:21:09
Speaker
not doing the math in my head right now. Yeah, me neither. So what about something like Venus?
00:21:13
Speaker
Now Venus, which has a famously hot atmosphere that can melt, I believe, lead? Yeah, it can melt lead. So even though it can melt lead, it has an average temperature of negative 41 degrees Celsius, or 232 Kelvin. You might wonder, how can this be? And the reason why is because of all the clouds on Venus. They reflect about 75% of the light that reaches them.
00:21:38
Speaker
Wow. Interesting. So here's the thing though. So if it's reflecting that much light, if it reflects 75%, it's closer to the sun than the earth, then yet it's still hot enough to melt lead. Yeah. And the reason why is because of the difference between the ground temperature and the atmospheric temperature. And that's represented by a separate model, as we'll see in a second.
00:22:00
Speaker
Okay. And I think it's interesting because, I mean, like we said, it's like 700 degrees Celsius or like 500 or something like that on the surface of Venus. It's hot enough to melt lead. Wow. But its average temperature is negative 41 degrees Celsius as seen by an observer in space. Wow. Interesting. So higher albedo still would mean like if it were a lower albedo, it would be even hotter.
00:22:22
Speaker
Oh yeah. Well, not necessarily on the surface though, because lower albedo might mean that the composition of the clouds are different. Clouds are actually one of the biggest varying factors in climate models. So now that we've talked a little bit about radiative effects as well as the earth and the sun, it's time to talk about something else, namely gases. Yeah, and why are gases important? Because the atmosphere is made of gas.
00:22:52
Speaker
Exactly, exactly. So, a few things about gases. First of all, the internal energy in a gas is quantized in a series of electronic, vibrational, and rotational states, at least according to our current understanding. Increase in internal energy corresponded to a change in state. Now, this is where... Right away, and there's a girl, a gamer mentioned current understanding, and I wanted to expand on that. That's a current understanding according to even the standard models.
00:23:16
Speaker
Yes. Yeah. So that's pretty solid. Yeah. When I say that, it's not like it's going to change next year. Well, it could. It could change next year. But according to the standard model, that is basically as rigorous as it currently gets. Again, you learn something new every day. This is something that I was unaware of.
00:23:33
Speaker
Each of these three quantized mode of a gas corresponds to a different amount of energy. For example, in order to increase the electronic energy of the gas, am I saying that right?
00:23:47
Speaker
Oh yeah, the electronic internal energy in a gas. You would do that through applying UV or ultraviolet radiation to it. And this is actually interesting because one of the early experiments in quantum physics was charging gold foil using UV light and showing that it can't be done even with a ton of red light because of the... We go into this more in detail, I believe. Do we go into detail on that in the Black Hole episodes?
00:24:13
Speaker
I don't, uh, we should. If we didn't, we should have. Listen to the whole thing to just check. Just, yeah, listen to everything, everything we've done and come back to us. How's that sound? But finish the episode, no. Yeah, yeah, yeah, exactly. Yeah. Um, yeah. Then, uh, for like, uh, vibrational changes in energy, you need near infrared energy, which is like very, very low red to like, uh, 20, like 700 nanometers to like 20 micrometers.
00:24:38
Speaker
Correct. Yep. And that's just, again, the, should we call it, lack of a better term, the springiness of the individual molecule? Yeah, the buzziness, springiness, how much they vibrate. Sure. And then lastly, you have the rotational energy. If you want to change the rotational energy of molecule, then you'll have to apply something greater than 20 micrometers. So that's far infrared. Yeah, far infrared. I believe that in some, I'd really tell microwaves work actually. Okay.
00:25:02
Speaker
Okay, I mean you like blast even radio waves like lower frequency radio waves and then change the rotation of molecules Yeah, so interesting and then it says here that visible changes There's nearly no changes in visible light of them is that that's oh what's interesting is that the visible spectrum Doesn't cause many changes in gas states. It doesn't change UV. I mean doesn't change electronic Internal energy nor does it change vibrational or rotational? significantly
00:25:31
Speaker
Couple of considerations here. So if the charge is distributed symmetrically, then it cannot acquire vibrational states since the vibrational states affect the dipole movement, which affects the structure of the molecule. Let me say that again. So you have different atoms and you've got different gases. And the valence electrons are arranged differently in each of them. There are some of them like the noble gases where you've got like eight
00:25:58
Speaker
valence electrons, right? And there's an issue here. If they're distributed evenly, then you're really not going to be able to add a whole lot of vibrational energy to it. Is that right?
00:26:07
Speaker
Yeah, well, let's take nitrogen for an example. Nitrogen is two nitrogen molecules connected together. It's charged and distributed perfectly evenly, like perfectly evenly, like exactly evenly. And because of that, basically it can't bend or anything like that. And since it can't bend, there's nothing to vibrate really.
00:26:29
Speaker
Imagine having a molecule in front of you and it's just made of like little balls connected by very, very tight springs. The more you can like, if you did nitrogen, if you tried to toying nitrogen, there would only be one vibration and it would be throughout the whole thing. But if you did something like carbon dioxide, you could bend it back and forth and because you could bend it back and forth, it wobbles.
00:26:55
Speaker
Okay. And so just the kind of wobbliness is correspondent to vibrational energy. So basically, again, you're going to have different shapes based on the arrangement of electrons. And some of these shapes have more vibrational energy than others, than others. Yeah. Or more receptive to vibrational energy. Correct. Yes. And that, yeah, that includes carbon dioxide.
00:27:16
Speaker
Yeah, and molecules that can acquire a charge asymmetry by stretching or flexing. So actually, let me do carbon dioxide as like a counterexample. So carbon dioxide is two carbons on either side, two oxygens on the other side, one carbon in the middle. So it has like the negative charges on the outside and the positive in the middle. So it's perfectly symmetrical. So in that state, it can't acquire any vibrational energy. However, if it gets knocked out of that state, like for example, if the
00:27:45
Speaker
carbon gets closer to one of the oxygens just perchance or it bends in the middle, kind of like a joint, then it can acquire more vibrational energy in that state. And so those are greenhouse gases. So hydrocarbons, O3, N2O, H2O, carbon dioxide. Yeah, and although water is a greenhouse gas, it's a very, very short-lived one.
00:28:12
Speaker
as we'll see that affects how potent of a greenhouse gas something is. Yeah, as you said earlier, molecules that cannot acquire charge asymmetry by flexing or stretching, things like N2, O2, or H2 are not, therefore, greenhouse gases.
00:28:28
Speaker
And the efficiency of absorption in our atmosphere is actually about 100% in the UV due to the electronic transitions of oxygen and O3 in the stratosphere. And it's about 100
Climate Models & Policy Implications
00:28:39
Speaker
% in infrared because of greenhouse gases. But between 8 and 13 micrometers, near the peak of terrestrial emission, there's like a weak window between 8 and 13 micrometers, but like a peak for O3 at 9 point. So basically, it's a complicated atmosphere is what we're saying.
00:28:57
Speaker
and you need models that reflect this. But we can go and do some simple models too. Yes. So with all that said, this is a part that if you've made it this far on the podcast, let me just say congratulations. You've been listening well or just playing it in the background. For this next part, we're going to talk about a simple model.
00:29:18
Speaker
So the simple model, uh, we start by pretending that the atmosphere is like a shell above the surface of the earth that can absorb energy. I mean, it already is above the earth, but what we're talking about is like, like separate from the earth. Yeah. Like, like, uh, imagine like a ring. Yeah. Imagine like you got rid of the white part of the shell of an orange and all you're left with is that orange on the outside and the fruit on the inside. That's kind of the model we have. Sure. Okay.
00:29:45
Speaker
Cool. So yeah, in this simple model, it implies something or it implores something that we call Kirchhoff's law. As an electrical engineer, we use Kirchhoff's law, but it's used in many, many things about the flow of energy. Kirchhoff's law basically states that the electricity that goes, the current that goes into a node is going to be equal to the current that leaves that node. They have to be the same. It's sort of a conservation law, if you will.
00:30:13
Speaker
Yeah, and it applies here because we're saying that the shell that the atmosphere is pumps energy outwards and inwards, outwards towards space and inwards towards the Earth. Correct, correct. And they have to be, yeah, what goes out has to be what comes in and vice versa. And we also have a sun that is generating, that is giving a constant amount of solar radiation.
00:30:34
Speaker
So in this simple model, if we were to calculate the incoming solar radiation, we would simply have to add the energy that's radiated, the energy that's radiated from the atmosphere into space, that's the outer layer, plus the energy that's radiated from the Earth into the atmosphere. Again, in the simple model, the atmosphere is above. So it's basically, it's Torkov's Law. What comes in from the Earth is equal to what is absorbed outward.
00:31:01
Speaker
And we know the amount that is radiated from Earth into the atmosphere and from atmosphere into space, because as we mentioned before, the amount that's absorbed is the fraction absorbed is the fraction radiated. And we use Kirchhoff's law also to show that the temperature of the, the amount of energy absorbed by the atmosphere from the Earth's surface temperature is equal to the total amount of energy being irradiated away from the atmosphere, both into space and into Earth.
00:31:27
Speaker
And we'll have more details on that in the paper to come. But basically we get this formula that the temperature of the Earth is equal to the mean solar flux times one minus the albedo, all divided by four times sigma, which is a constant that relates the fourth power of temperature to a black body's energy. So we're assuming that the atmosphere is a black body. This is called the gray atmosphere model.
00:31:57
Speaker
So it's over four sigma times one minus F, which is the amount of the fraction absorbed by the clouds divided by two, all to the one quarter. So as you can see, if we change the fraction that is absorbed, and we might have an app where you can play around with this actually, it'll change the temperature pretty significantly, which explains why Venus, even though it's colder than the Earth, is hotter at the surface.
00:32:22
Speaker
That'd be a very cool app, actually. Yeah, and so we showed that the global surface temperature is about 288 Kelvin. Which is what, Fahrenheit? I think it's like 40 or 50 degrees. Okay, yeah, we can do a quick... Yeah, it's like 58 degrees. Cool, very good. Right on.
00:32:38
Speaker
Let's talk about, obviously, there were some simplifications made in that model. We have improved models as well. Yeah. For example, when we were calculating the previous thing, the amount of energy absorbed by the atmosphere of the Earth's surface temperature and stuff like that, we could have looked at how it affects every single different spectrum differently and then sum it up together. That's something that we did not do. We just assumed the Earth was black and then the atmosphere was black.
00:33:05
Speaker
Yeah, yeah, basically we're saying is that the model can be approved by not considering the earth and atmosphere as black bodies and still improve further with something called GCMs or general circulation models. Basically those provide 3D equations for energy, mass and momentum. Man, that gets real math-y. For those of you guys who like spaghetti of math equations, this is right up your alley. You'll love this.
00:33:30
Speaker
Oh, yeah. But what's interesting, though, is that this model that we're talking about here, you can see in the link that we linked to earlier, that it actually predicts a slightly more global warming than we've had in the last century that actually happened. But it's still pretty fairly accurate, even this small, almost insignificant model.
00:33:50
Speaker
Yeah, yeah, yeah. So the simple models are still very, very good. And they only get more and more specific. It's like we're just getting more refined models now, like with the GCMs, the general circulation models. You know, I have it on good authority that some of our listeners might be like us, fairly, fairly nerdy, and they like the nitty-gritty details. Should we tell them some more of the points that are in the more advanced models?
00:34:13
Speaker
Yeah, well basically it's one of the more nitty-gritty points that it has in that thing is optical depth, which is how much flux has absorbed and scattered over time. You could take the different layers of the Earth's atmosphere, you know, the stratosphere, ionosphere, troposphere, and I think thermosphere. You could take all of those into consideration. Like, there's so much stuff you could do. Like, the world is your oyster when it comes to this stuff. Yeah, yeah, yeah, yeah.
00:34:38
Speaker
But what's interesting too is that although there are disagreements in the predictive surface warmings resulted from a given increase in greenhouse gases, all GCMs tend to show a linear relationship between the initial radiative forcing and the initial ultimate perturbation at the surface temperatures. Let's break that down. What is radiative forcing? That is just the radiative perturbation associated with an increase in a greenhouse gas.
00:35:01
Speaker
So it's saying if we add one pound of carbon dioxide to the atmosphere, how much radiative energy increase are we going to see at the surface? And we relate things to carbon dioxide because it's kind of a baseline, and also I think it's most prevalent greenhouse gas. But I mean methane, over a period of 20 years, is 62 times as potent as carbon dioxide as a greenhouse gas. Over 500 years though, because it has a sort of lifeline, it's only eight times as potent.
00:35:30
Speaker
But if we want to look at a really potent greenhouse gas, check out sulfur hexafluoride. Oh, wow. What do you have for that one? Let's see. A lifetime of 3.2 millennia. In over 20 years, it's 16,500 times as bad as carbon dioxide. 24,900 in 100 years. In over 500 years, 36,500 times as potent as carbon dioxide. Wow, that is insane.
00:35:53
Speaker
So basically what we see then is over a hundred year time horizon reducing use of sodium hexafluoride emissions by one kilogram is as effective from a greenhouse perspective as reducing carbon dioxide emissions by 24,900 kilograms. That's important for designing regulatory goals.
00:36:11
Speaker
And another important, I mean, also we have to, we can't just, if there's like only point, if there's only one milligram of sodium hexafluoride released into the atmosphere per year versus a billion pounds of carbon dioxide, we still have to pay more attention to carbon dioxide. So that's another, that's not the actual stats as you might've guessed, but it's something to take into consideration. And that's how it can help us design better factories. And once we have regulatory, regulatory goal points when it comes to greenhouse gases in the amount, in the,
00:36:40
Speaker
We can even design factories in terms of radiative forcing, taxum in terms of how much radiative forcing they generate. There's so much stuff we can do with this data. And I think that's a big takeaway from this. Oh, and these are the models which tell us that we are generating enough carbon dioxide to create a significant effect.
00:37:03
Speaker
As we've seen, global warming is happening and we are mostly to blame. Actually, I believe that
Conclusion & Climate Discussion
00:37:08
Speaker
if I'm not mistaken, humanity is entirely responsible for the current warming as we should actually be in a cooling trend. Oh, I did not know that. We have learned about how radiative forcing relates to climate science, how gases interact with light, and what we might be able to do about it.
00:37:24
Speaker
I'm Sophia. And I'm Gabriel. And this has been Breaking Math. Just a reminder, we are on patreon.com slash breakingmath and facebook.com slash breakingmath podcast, where you can see new releases, things like that. Facebook, breakingmathpodcast.com. What else?
00:37:41
Speaker
So this has been a very exciting series for me. We have interacted with some real, real top level researchers. I'm not sure I'm at liberty to say who we've been chatting with. We'll say that just in case it doesn't pan out. Absolutely. Yeah. This topic is a huge topic the world over with the Australian wildfires that have been exacerbated by climate change to California fires.
00:38:04
Speaker
California fires, flooding in low-lying areas, all kinds of things. Even Syria can be attributed in a large part to the rapidity of climate change. Correct. If it changes the availability of natural resources, then it'll have an impact on things like
00:38:23
Speaker
Economics because I mean if you want to see how if you want to watch something I would check out by CGP Grey on YouTube It's the rules for rulers and it talks about her resource distribution can lead to either democracy or tyranny correct correct absolutely
00:38:41
Speaker
So yeah, and on our Twitter recently in the last six months, I have followed probably two dozen climate scientists, including people like Dr. Catherine Hayhoe, lead author of the US climate assessment from 2018, also Dr. Michael Mann, as well as a slew of others as well. So send us your climate change related questions and questions for any of these doctors. And I mean, you know, we'll see if we can find answers to them.
00:39:11
Speaker
I'm Sophia. And I'm Gabriel. And you're listening to Blakey Ma- I'm faking math.