UPDATE: Eli has started working on the graphics and text as per suggestions. When he is done he will move the original text with comments to Eli Rabett's Memory Hole and repost the modification. 2/10
Periodically Eli tries to explain the greenhouse effect to someone who doesn't want to believe in it, and like explaining atomic or nuclear structure, it really cannot be done without the person with hands over ears knowing a fair amount of stuff. The really short version is in caps at the bottom. The numbers used are based on measurements or averages of measurements. Let's list a few of the preliminaries and put quotes around concepts that anyone interested in learning more can google
Eli is going to use the "Kelvin temperature scale" which is the appropriate one for all thermodynamical stuff.
In the atmosphere it gets colder the higher you go up to about 12 km. This is called the "adiabatic lapse rate" and is a result of gravity compressing the atmosphere. **
A solid body emits thermal radiation in the infrared (IR), this is called "black body radiation" and the amount emitted as a function of frequency (or wavelength) is described by the "Planck radiation formula". The total amount emitted per unit area is proportional to the fourth power of the temperature (ecT^4) according to the "Stefan Boltzmann law". e is a constant called the emissivity, c usually written as the Greek sigma, is the "Stefan Boltzmann constant". e is close to unity for all solids or liquids.
The total amount of energy in the sunlight absorbed by the earth per unit time, has to be emitted to space for the Earth to remain at a constant temperature. The emission is all in the thermal IR. There is very little, to no overlap between the "solar spectrum" and the thermal IR emitted by the Earth. If less energy is emitted, the Earth warms, both the surface and the atmosphere until the temperature is high enough (see "Stefan Boltzmann Law") to restore the balance. If more energy is emitted, the Earth cools, both the surface and the atmosphere, again, until the temperature is low enough to restore the balance.
With the preliminaries out of the way we can look at what the emission from the Earth through the atmosphere looks like at 20 km.
Look at the figure. The x-axis is in frequency units, wavenumbers, used by spectroscopists. It is the inverse of the wavelength in cm. The y-axis is the intensity of emission in W/m^2 per unit wavenumber. The red curve is the emission, the others are "Planck function" curves for various temperatures. If you look to the right, you see that there are parts of the curve that roughly follow the Planck curve for ~290 K which is the temperature of the ground. This is radiation that is going through the atmosphere essentially undisturbed. The chaff is absorption by water. The dip at about 1000 wavenumbers is from absorption by ozone, the big dip at 675 wavenumbers is absorption by CO2. There are continuum absorptions at the high and low frequency ends, mostly due to (H2O)2 (two water vapor molecules stuck together) and other complicated things consideration of which we leave for the next class.
At 375 "ppm", the current value, emission in the center of the CO2 band is characteristic of the 220 K Planck curve, meaning that the emission to space comes from an altitude at which the temperature is ~ 210 K. Because the temperature is very low, the amount of energy radiated to space is low, about 30% of the amount of thermal radiation leaving the ground at those frequencies (210^4/290^4 = .28). If we double the amount of CO2 in the atmosphere to 750 ppm the CO2 band widens blocking MORE thermal radiationTo restore the balance the Earth system has to warm.
Let's talk about that. A lot of this discussion, as you have noticed has to do with rates at which energy is transferred. Thermodynamics requires is that no NET energy be transferred from a hotter to a colder body.
The rate at which sunlight is absorbed by the ground is ~170 W/m2. If there were no absorption of the thermal IR in the atmosphere the average temperature of the Earth would be AT MOST ~255 K*
*(Eli is making a simplification here having to do with cloud reflectivity and absorption in the atmosphere, but the bottom line is the same)
What happens when the "greenhouse gases" absorb the thermal IR? The molecules almost immediately and completely transfer that energy by collision to the nitrogen and oxygen in the atmosphere, slightly warming it, BUT, collisions also vibrationally excite the greenhouse gases, including CO2. The net result is that there is an equilibrium amount of vibrationally excited greenhouse gases in the atmosphere, and this equilibrium amount depends on the local temperature.
The vibrationally excited greenhouse gas molecules emit IR in all directions, including back to the surface. With the greenhouse effect the rate at which thermal energy leaves the surface is ~390 W/m2. The rate at which the thermal energy returned to the surface is, ~325 W/m2 (go look at the Trenberth diagrams referenced below). Therefore the NET amount of energy leaving the surface due to radiation is ~65 W/m2. Of course this neglects convection, evaporation of water and a few other things. Add everything up and you get that on NET the rate at which the surface radiates is 170 W/m2 but here is the joker
BECAUSE OF THE GREENHOUSE GASES THE SURFACE HAS TO BE WARM ENOUGH TO RADIATE 390 W/m2, AND THAT MEANS THAT IT IS AT ~290 K RATHER THAN LESS THAN 255 K
If you want a more detailed exposition http://www.atmo.arizona.edu/students/courselinks/spring04/atmo451b/pdf/RadiationBudget.pdfComments?
** the atmosphere cools because of the adiabatic lapse rate up to the tropopause at ~ 12-15 km. Above that it warms because of absorption of UV light from the sun by ozone, but the greenhouse effect is pretty much confined to below the tropopause. Eli has a post which touches on this