Aug 02 2007
Simple Model of Greenhouse Gases
|-----------------------------------------| | | |===================================== Ta | | | | | |===================================== Te | |-----------------------------------------|
Consider the system above. Excuse the ASCII ‘art’, I didn’t want to make an image. With my artistic skills, an image probably wouldn’t be better than the above anyway…
The system consists of a bottom slab that is at a temperature Te, and a top slab which is at a temperature of Ta. The absorptivity of the top slab over the spectrum of the emitted radiation of the bottom slab is α. Assume the spectrum averaged α is equal to the spectrum averaged ε. Also assume that the top slab does not reflect radiation from the bottom slab; the reflectivity is 0. The transmissivity is thus 1 minus the absorptivity (1-α, or 1-ε). The top slab is transparent to an incoming source of radiation (S). By conservation of energy, the incoming energy must be balanced by the energy emitted by the bottom slab and transmitted through the top slab plus the radiation emitted by the top slab.
S = σTe4(1-ε) + εσTa4
Now apply it to just the top slab:
2εσTa4 = ασTe4 = εσTe4
If we solve these equations for Te4:
Te4 = S / σ(1-ε/2) or
Te4 = 2S / σ(1+τ)
Let’s look at the extremes where τ=0 and τ=1.
For τ=0, Te=303K
For τ=1, Te=255K
In the case where there are no greenhouse gases (τ=1), the temperature of the bottom slab is equal to the radiative effective temperature. If there are greenhouse gases (τ>1), the temperature is always greater than the radiative effective temperature.
[This post based extensively on the work done in Bohren and Clothiaux, Fundamentals of Atmospheric Radiation, pp 31-33. Rewritten under fair use.]
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3 Responses to “Simple Model of Greenhouse Gases”
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Assuming sigma is 5.67e-8 W/m^2/K^4 implies that S is 239 W/m^2. What does this value represent?
239W/m2 is the total solar radiant energy per area averaged uniformily over the globe, and taking into account the Earth’s albedo.
The solar constant is about 1369W/m2, multiply by πR2 to get the total energy intercepted by the Earth, divide by the surface area, 4πR2, to get the average energy per area. Then multiply by 1-A, where A is the Earth’s average albedo (0.3), and you get about 239W/m2.
Thanks,
Makes sense.