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I'm trying to understand this "toy model" of the greenhouse effect.

The model predicts the surface temperature of the Earth given the sunlight intensity and the emissivity of the atmosphere, $\lambda$. To do this, they use a two-energy model of light. There's shortwave radiation from the sun, and there's thermal radiation from terrestrial sources. They way they define $S$, the radiation intensity from the sun, allows for the atmosphere to reflect some shortwave light back into space, but the model doesn't allow for the atmosphere to absorb any shortwave light, meaning that all of $S$ makes it to the ground. Then the ground is treated as a blackbody. It logically follows that it emits radiation of intensity $G$ upwards, of which $\lambda G$ is absorbed by the atmosphere. Atmosphere emits $1/2 \lambda G$ back down to the surface because it is also a blackbody.

What I don't understand is the complexity they seem to invoke to write equations describing this. They use equations for surface, atmosphere, and planet.

$$S + \lambda A = G$$ $$\lambda G = 2 \lambda A$$ $$S = \lambda A + (1 - \lambda) G$$

These are 3 equations, but aren't there only 2 unknowns?

My question: how do you draw the necessary system boundaries to get closure for this system? Also, is there a requirement on the temperature of the atmosphere? I think the source argued that it need to be cold for it to have insulation properties, but I don't think that's reflected in the math, since we require (radiation in) = (radiation out). That would seem to imply the temperature of the atmosphere is arbitrary.

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Radiation from the Sun is in this field usually called shortwave radiation or solar radiation, not thermal, which is used only for terrestrial radiation. – gerrit Jan 15 '13 at 17:15
@gerrit Okay, well I edited so that I'm not flipping back and fourth between terms, but whether we call it thermal or not, my understanding is that the radiation from the sun is thermal, it's just at like 3000 K. – Alan Rominger Jan 15 '13 at 17:29
Your understanding is correct. However, atmospheric scientists don't call it that, so to do so can cause confusion. – gerrit Jan 15 '13 at 17:43
open question: why is the second equation right? Shouldn't it be $\lambda G = 2 A$? Doesn't the current form violate conservation of energy at the atmosphere? What happens as $\lambda$ goes to zero? $A$ should go to zero as well, but it doesn't here. It seems like this equation should be wrong. – Alan Rominger Jan 15 '13 at 19:13
$A$ is half the energy the atmosphere would emit if it was a perfect blackbody, right? If $\lambda$ goes to zero you decouple the atmosphere from everything else and you're left with only $S=G$. Then it doesn't matter what $A$ is. – jkej Jan 15 '13 at 19:36
up vote 3 down vote accepted

The third equation is just a sum of the two first:

$$ S+\lambda A=G \Rightarrow S=-\lambda A+G\\ \lambda G=2\lambda A \Rightarrow 0=2\lambda A-\lambda G $$

Add them up and get:

$$ S=-\lambda A+2\lambda A+G-\lambda G=\lambda A+(1-\lambda)G $$

which is the third equation. So, still only two independent equations and two unknowns.

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