This is Part 2 -- Part 1, on Earth's radiating equilibrium temperature, is HERE.
On the previous page we saw that, in the absence of an atmosphere, surface equilibrium temperatures would be quite cold. This situation is depicted in the diagram below. Here, there is no atmosphere to mediate energy exchange between the surface and space.
Note that the diagram depicts energy in terms of "100" incoming -- it's easier to keep track of where it is all going this way -- but we know this is really (1367 / 4) = 342 W/m2, which yields a surface temperature of -18oC, using the Stefan-Boltzman Law (as before).
Now the next figure shows what we can do with an atmosphere that lets insolation pass freely onto the surface, but which greatly interferes with the outgoing energy from the surface (in fact it absorbs it all!).
If we continue this series to equilibrium, we get:
But why should the atmosphere do this? Well, as we know, the incoming energy is short-wave, and the outgoing is long wave -- this is due to the fact that the Sun is a much hotter emitter than Earth, and so the wavelength of the incoming energy is much shorter, as we know from Wien's Law. So, it turns out the atmosphere is a much better absorber of LW than it is of SW, especially the water vapor, carbon dioxide, and methane -- these are the most potent, natural "greenhouse gasses".
So now we can see that we can have a much higher temperature -- in this case it would be equivalent to ((1376 / 4) * 2) W/m2, or = 58oC using Stefan-Boltzman!
Although this is high, it is not quite as silly as it seems -- the diagram below shows estimates of the real energy flow in the system if we assume that radiation is the only means we have of transferring energy. Note that the atmosphere does absorb some of the incoming energy, e.g. by the ozone cycle in the stratosphere.
Note here the radiating surface temperature is equivalent to ((1367/4) * 1.45) W/m2, or 33oC using Stefan-Boltzman!
Now we seem to have the opposite problem we had before -- too hot, rather than too cold!
The problem is, we have been assuming that electromagnetic radiation is the only means we have of moving energy around in the system. But this is just not so.
Think of your own body, and how you keep cool in hot days . . . you can radiate energy away (I = σ . T4), or you can use conduction (heating air that is in direct contact with your skin -- on a cold day you get wind chill from this), or you can use evaporation of perspiration (really, latent heat drawn from your skin). The latter two mechanisms mean we don't need to have such a high surface temperature to establish equilibrium -- and that's exactly what happens to Earth's surface: evapotranspiration consumes energy, as does conduction-convection (totaling 29 of our "units") -- the remainder (116 units) is equivalent to: (1367/4) * 1.16 = 396 W/m2, which yields a radiating temperature of 15oC using Stefan Boltzman.
The figure below shows the situation in terms of actual energy units -- but you'll see the diagram above is the same thing, in percent form.
You can obtain some further information on the energy budget from NASA.
There are a few other things we might take note of if we wish to make things even more realistic:
1. The incoming solar radiation ("insolation", i.e. in-sol-ation) can be divided into two components, if we wish: direct and indirect (scattered) -- the latter is responsible for the color of the sky and the fact that the Sun does not appear white (as it does from the Moon, where there is no atmosphere -- also black sky on the Moon).
2. Our balance is for an "average" location -- but there will be considerable variation on account of surface variations, particularly deserts, where an absence of water necessitates higher surface temperatures (the energy has to go somewhere, if not into evaporation, then into other heating).
3. In reality there is considerable variation in cloud types, and the following diagram shows how each plays different roles:
Further details on this are available from NASA.
3. There is considerable geographic variation on insolation and albedo (resulting in lower temperature, as we move north or south from the equator):
ABSORBED SOLAR ENERGY
LONG WAVE EMISSIONS TO SPACE
4. The long wave emissions are much more evenly distributed than the incoming radiation. This is another consequence of the atmosphere -- heat (particularly latent heat) can be moved around from place to place in a way radiation cannot -- so that evens things out, and makes the Earth a far more equable place than it would be otherwise (i.e. the Poles and not quite as cold, and the Tropics are not quite as hot). The figure below shows this fact (i.e. if radiant energy were the only output, then "in" and "out" would balance at all locations on the diagram).
Further information on a number of issues can be had HERE and HERE.