Learning from a simple model

Planet: S  = \lambda A + (1-\lambda) G

The factor of two for A (the radiation emitted from the atmosphere) comes in because the atmosphere radiates both up and down. From those equations you can derive the surface temperature as a function of the incoming solar and the atmospheric emissivity as:

G=\sigma T_s^4= {S\over(1 - 0.5\lambda) }

If you want to put some vaguely realistic numbers to it, then with S=240 W/m2 and \lambda=0.769, you get a ground temperature of 288 K – roughly corresponding to Earth. So far, so good.

Point 1: It’s easy to see that the G (and hence T_s) increases from S to 2S as the emissivity goes from 0 (no greenhouse effect) to 1 (maximum greenhouse effect) i.e. increasing the greenhouse effect warms the surface.

This is an extremely robust result, and indeed has been known for over a century. One little subtlety, note that the atmospheric temperature is cooler than the surface – this is fundamental to there being a greenhouse effect at all. In this example it’s cooler because of the radiative balance, while in the real world it’s cooler because of adiabatic expansion (air cools as it expands under lower pressure) modified by convection.

Radiative Forcing

Now what happens if something changes – say the solar input increases, or the emissivity changes? It’s easy enough to put in the new values and see what happens – and this will define the sensitivity of system. We can also calculate the instantaneous change in the energy balance at the top of the atmosphere as \lambda or S changes while keeping the temperatures the same. This is the famed ‘radiative forcing’ you’ve heard so much about. That change (+ve going down) is:

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