Losing myself in the greenhouse effect

If you like to go down a Python rabbit hole with me, to explore some of the basic dynamics of the greenhouse effect, Please continue. If not, get out while you still can!

Ok, so this page of Kump et al.'s "The Earth System" (third edition, 2010) briefly presents a simple model of the greenhouse effect: the "greenhouse effect of a one-layer atmosphere".

It basically shows (with interesting, but ultimately, unnecessarily complex equations) that if…

• … the shortwave radiation of the sun passes through the atmosphere and heats up the surface of the earth
• … the surface emits this energy as longwave radiation, and emits only in one direction (upwards)
• … the atmosphere absorbs all longwave radiation emitted by the surface
• … the atmosphere, simplified as one layer, re-radiates this energy in two directions (upwards and downwards),
• … the temperature (in°K) of any radiating blackbody depends on the flux at which it radiates (following the Stefan-Boltzmann law: F = σ.T⁴)

… then it follows that …

• … the surface will always be emitting twice as much radiation as the atmosphere (because the atmosphere will be sending half of what it receives up, and all that it receives is what the surface emits)
• … a T that is 2^(1/4) higher, will result in a F that is twice is high (again, see the Stefan-Boltzmann law)
• … so the temperature at the surface must be 2^(1/4) higher than in the atmosphere

Kump et al. then write :

This example is not meant to be realistic. The real atmosphere is not perfectly absorbing at all infrared wavelengths, so some of the outgoing IR radiation from the surface leaks through to space. This effects tends to make the ΔTg (MV: the greenhouse effect) smaller. Conversely, a more accurate calculation would subdivide the atmosphere into a number of layers. Including more layers tends to make ΔTg bigger …

This let me to start a little side project: modeling the greenhouse effect with multiple partly transparent layers. Please enjoy this notebook.