What is the greenhouse effect?

If you want to debate the greenhouse effect, a good start is to know what it is. People argue whether there exists such a thing as the greenhouse effect,  however, here we will provide a simplified mathematical formulation that nevertheless captures the essential features. The background information is gathered from Goody and Yung "Atmospheric radiation", Oxford University Press.

The underlying principles are rather simple, the earths surface and the atmospheric layers radiate isotropically according to Stefan-Bolzmanns law but with variable absorption/emissivity. The radiation absorbed is instantly thermalized. Furthermore, the absorption a(z) varies with altitude z according to:

The ground temperature is given by the formula:

Here Fs is a measure of the incoming solar radiation. Finally the lapse rate is given by:

Where H is a height scale. We will now solve these equations numerically for three different values of a0. (H = 1, sigma = 1, Fs = 1 held constant).

The blue line is for a0 = 0, the green line is for a0 = 0.5 and the red line is for a0 = 1. As we can see, by increasing the absorption/emission parameter we heat the surface and cool the upper layers of the atmosphere. Note that for a0 = 0 we have an isothermal atmosphere. In more advanced models the pure radiative equilibrium is modified so that when the lapse rate exceeds a certain critical value convection sets in. These models are called radiative-convective models. A lapse rate neutrally stable to convection is characterized by a constant potential temperature and are often called adiabats. These differ depending on the amount of water vapour present. Of course, the equations presented in the beginning need some further explanation, but we will leave that for now and refer to the litterature. 

See also

Manabe-Strickler

Lindzen

Although we were able to give a mathematical illustration of the greenhouse effect in just a few lines, unfortunately the scientific debate hardly ever reaches this point. Many proponents of the theory don't seem to be willing to communicate it in the way I just did, or they might simply not know what they are talking about.

Usually, successful scientific debates are carried out in the spirit of Leibniz by the parties carefully explaining the axioms, assumptions, approximations and equations underpinning their theory/model. In the greenhouse debate though, neither part seems to be interested in such a discussion. However, from the skeptical point of view the danger of not knowing your target is obvious. You become more succeptible to the sophistry of your opponent, you are more likely to make mistakes and to focus on extraneous details rather than essentials.

Equipped with the equations and their solutions we may now disect some common pseudo descriptions of the greenhouse effect:

1) The greenhouse gases absorb the outgoing terrestrial IR-radiation thereby heating the atmosphere.

Comment: The greenhouse gases do not heat the atmosphere, they change the temperature profile of the atmosphere.

2) By adding greenhouse gases the optical thickness of the atmosphere increases, hence the effective radiating level is pushed to a higher altitude. Since the temperature decreases with increasing altitude, in order to get into equilibrium with the incoming solar radiation the temperature at this new altitude must be that of the blackbody temperature of the earth. The ground temperature is then recovered by following the adiabat.

Comment: This is more to the point but still deceptive. It is true that the red curve crosses the blue curve at a higher altitude than the green curve does, but a lot of information is left out, for example, it is because of the greenhouse effect that the temperature decreases with height in the first place. In essence, the description doesn't explain anything.