edit2: ok I just read the wiki article. Everything they tell you about how the greenhouse effect works is wrong. It’s not that the atmosphere somehow blocks the outgoing radiation, as that would violate the second law by allowing the earth to heat up relative to it’s surroundings.
That can’t be right. The atmosphere does block most of the outgoing radiation — its transmissivity for the Earth’s longwave radiation is only about 20% — and if it were transparent to radiation it couldn’t exert a greenhouse effect at all. Also, a thought experiment: if we had an electric oven plugged into a solar panel orbiting the Sun, the oven could heat itself relative to the surrounding space just by using light from the Sun, and that wouldn’t violate the second law.
Maybe the second law is the wrong way to look at it. The second law says that the sun can’t cause you to heat up hotter than the sun on average. (you can do tricks with eat pumps to make parts of you hotter than the sun, though)
It also says you can’t do tricks with surface properties to change your temperature. (in the absence of heat pumps)
The atmosphere does block most of the outgoing radiation — its transmissivity for the Earth’s longwave radiation is only about 20% — and if it were transparent to radiation it couldn’t exert a greenhouse effect at all.
Ok I’m still a bit confused about this. I suspect that this effect alone is not enough to cause a greenhouse effect. Let’s think it through:
Assume the 0.2 missing from transmissivity is all absorptivity (t, a, and r add up to one). And that we model it as simply an optical obstruction in thermal equilibrium.
The sun’s radiation comes, some of it goes to the atmosphere, some to the earth. If the atmosphere magically ate heat, the earth would get less radiation. However, it does not magically eat heat; it heats up until it is emitting as much as it absorbs. The longwave from earth also gets eaten and re-emitted. About half of the emitted goes to earth, the rest out to space.
So our greenhouse layer prevents some power P1 from getting to earth. Earth emits, and P2 is also eaten. The emitted P3 = P1+P2. Earth gets P3/2. The sun is hotter than earth so the power at any given wavelength will be higher, so P1 > P2, therefor P3/2 > P2, which means on net, heat is flowing from the greenhouse layer to earth. However the earth is receiving P1 less from the sun, and P1 > P3/2. So the earth cools down relative to the similar earth without “greenhouse” effect. This makes sense to me because the earth is effectively hiding behind a barrier.
Therefor, if a greenhouse effect exists, it cannot be explained by mere atmospheric absorption. Unless I made some mistake there...
The assumption that is not true in that model is the atmosphere being in independent thermal equilibrium.
If we instead make the atmosphere be in thermal eq with earth, there is no effect; the earth acts as a single body, and absorption by atmosphere is the same as absorption by ground.
If we instead model the atmosphere realistically as a compressible fluid, things become more interesting. I’m not going to do the math here, but the model goes like this: the atmosphere at ground is eq with the ground. If a piece of air gets heated up at ground, it expands and floats up. As it goes up, there is less pressure from the air above it, so it expands, which does work, which cools it down. It cools down at the adiabatic lapse rate, which is the temperature gradient in a well mixed compressible fluid in a gravitational field. At the tropopause, our piece of air has reached about −40. The tropopause is where the atmosphere stops being opaque to greenhouse rays. Therefor, the tropopause is approximately the emission/absorption surface for greenhouse radiation, and the earth gets the other stuff.
So what does this mean for greenhouse? If we make the atmosphere absorb more, it shifts the average radiation surface upwards to colder air. If the surface of our body is too cold, it must heat up to maintain thermal eq. So it heats up. The lapse rate enforces a certain temperate rate between earth and atmosphere, so you can see if you move the equilibrium point up, the earth has to heat up. As for the second law, the atmosphere is acting as a heat pump.
Therefore global warming.
Even this model is a bit broken. if you heat up some air at the top of the atmosphere, it stays up there and stops mixing. I think this is what the tropopause is. I have no idea how to model this.
Maybe the second law is the wrong way to look at it.
I think so. In practice, changing the surface properties of a body in orbit can affect its temperature. If we coated the Moon with soot it would get hotter, and if we coated it in silver it would get colder.
So our greenhouse layer prevents some power P1 from getting to earth. Earth emits, and P2 is also eaten. The emitted P3 = P1+P2. Earth gets P3/2. The sun is hotter than earth so the power at any given wavelength will be higher, so P1 > P2, therefor P3/2 > P2, which means on net, heat is flowing from the greenhouse layer to earth. However the earth is receiving P1 less from the sun, and P1 > P3/2. So the earth cools down relative to the similar earth without “greenhouse” effect.
Two key complications break this toy model:
P1 > P2 doesn’t follow from the Sun having higher spectral power. The Sun being hotter just means it emits more power per unit area at its own surface, but our planet intercepts only a tiny fraction of that power.
The atmosphere likes to eat Earth’s emissions much more than it likes to eat the Sun’s. This allows P1 to be less than P2, and in fact it is. P2 > P1 implies P3/2 > P1, which turns the cooling into a warming.
This makes sense to me because the earth is effectively hiding behind a barrier.
The barrier metaphor’s a bit dodgy because it suggests a mental picture of a wall that blocks incoming and outgoing radiation equally — or at least it does to me! (This incorrect assumption confused me when I was a kid and trying to figure out how the greenhouse effect worked.)
The assumption that is not true in that model is the atmosphere being in independent thermal equilibrium.
It’s a false assumption, but it’s not the assumption breaking your (first) model. It’s possible to successfully model the greenhouse effect by pretending the atmosphere’s a single isothermal layer with its own temperature.
The second model you sketch in your last 4 paragraphs sounds basically right, although the emission/absorption surface is some way below the tropopause. That surface is about 5km high, where the temperature’s about −19°C, but the tropopause is 9-17km high. (Also, there’s mixing way beyond the top of the troposphere because of turbulence.)
That can’t be right. The atmosphere does block most of the outgoing radiation — its transmissivity for the Earth’s longwave radiation is only about 20% — and if it were transparent to radiation it couldn’t exert a greenhouse effect at all. Also, a thought experiment: if we had an electric oven plugged into a solar panel orbiting the Sun, the oven could heat itself relative to the surrounding space just by using light from the Sun, and that wouldn’t violate the second law.
Maybe the second law is the wrong way to look at it. The second law says that the sun can’t cause you to heat up hotter than the sun on average. (you can do tricks with eat pumps to make parts of you hotter than the sun, though)
It also says you can’t do tricks with surface properties to change your temperature. (in the absence of heat pumps)
Ok I’m still a bit confused about this. I suspect that this effect alone is not enough to cause a greenhouse effect. Let’s think it through:
Assume the 0.2 missing from transmissivity is all absorptivity (t, a, and r add up to one). And that we model it as simply an optical obstruction in thermal equilibrium.
The sun’s radiation comes, some of it goes to the atmosphere, some to the earth. If the atmosphere magically ate heat, the earth would get less radiation. However, it does not magically eat heat; it heats up until it is emitting as much as it absorbs. The longwave from earth also gets eaten and re-emitted. About half of the emitted goes to earth, the rest out to space.
So our greenhouse layer prevents some power P1 from getting to earth. Earth emits, and P2 is also eaten. The emitted P3 = P1+P2. Earth gets P3/2. The sun is hotter than earth so the power at any given wavelength will be higher, so P1 > P2, therefor P3/2 > P2, which means on net, heat is flowing from the greenhouse layer to earth. However the earth is receiving P1 less from the sun, and P1 > P3/2. So the earth cools down relative to the similar earth without “greenhouse” effect. This makes sense to me because the earth is effectively hiding behind a barrier.
Therefor, if a greenhouse effect exists, it cannot be explained by mere atmospheric absorption. Unless I made some mistake there...
The assumption that is not true in that model is the atmosphere being in independent thermal equilibrium.
If we instead make the atmosphere be in thermal eq with earth, there is no effect; the earth acts as a single body, and absorption by atmosphere is the same as absorption by ground.
If we instead model the atmosphere realistically as a compressible fluid, things become more interesting. I’m not going to do the math here, but the model goes like this: the atmosphere at ground is eq with the ground. If a piece of air gets heated up at ground, it expands and floats up. As it goes up, there is less pressure from the air above it, so it expands, which does work, which cools it down. It cools down at the adiabatic lapse rate, which is the temperature gradient in a well mixed compressible fluid in a gravitational field. At the tropopause, our piece of air has reached about −40. The tropopause is where the atmosphere stops being opaque to greenhouse rays. Therefor, the tropopause is approximately the emission/absorption surface for greenhouse radiation, and the earth gets the other stuff.
So what does this mean for greenhouse? If we make the atmosphere absorb more, it shifts the average radiation surface upwards to colder air. If the surface of our body is too cold, it must heat up to maintain thermal eq. So it heats up. The lapse rate enforces a certain temperate rate between earth and atmosphere, so you can see if you move the equilibrium point up, the earth has to heat up. As for the second law, the atmosphere is acting as a heat pump.
Therefore global warming.
Even this model is a bit broken. if you heat up some air at the top of the atmosphere, it stays up there and stops mixing. I think this is what the tropopause is. I have no idea how to model this.
I think so. In practice, changing the surface properties of a body in orbit can affect its temperature. If we coated the Moon with soot it would get hotter, and if we coated it in silver it would get colder.
Two key complications break this toy model:
P1 > P2 doesn’t follow from the Sun having higher spectral power. The Sun being hotter just means it emits more power per unit area at its own surface, but our planet intercepts only a tiny fraction of that power.
The atmosphere likes to eat Earth’s emissions much more than it likes to eat the Sun’s. This allows P1 to be less than P2, and in fact it is. P2 > P1 implies P3/2 > P1, which turns the cooling into a warming.
The barrier metaphor’s a bit dodgy because it suggests a mental picture of a wall that blocks incoming and outgoing radiation equally — or at least it does to me! (This incorrect assumption confused me when I was a kid and trying to figure out how the greenhouse effect worked.)
It’s a false assumption, but it’s not the assumption breaking your (first) model. It’s possible to successfully model the greenhouse effect by pretending the atmosphere’s a single isothermal layer with its own temperature.
The second model you sketch in your last 4 paragraphs sounds basically right, although the emission/absorption surface is some way below the tropopause. That surface is about 5km high, where the temperature’s about −19°C, but the tropopause is 9-17km high. (Also, there’s mixing way beyond the top of the troposphere because of turbulence.)