aTt = 4(1 - ap)S0 + *T4e = 2aTt, (2-8) where we have used Eq. 2-7. Therefore

So the presence of an absorbing atmosphere, as depicted here, increases the surface tem-peratureby a factor 21/4 = 1.19. This arises as a direct consequence of absorption of terrestrial radiation by the atmosphere, which in turn re-radiates IR back down to the surface, thus increasing the net downward radiative flux at the surface. Note that A 4 is of the same order—in fact in this simple model, equal to—the solar radiation that strikes the ground. This is true of more complex models, and indeed observations show that the downwelled radiation from the atmosphere can exceed that due to the direct solar flux.

Applying this factor to our calculated value Te = 255 K, we predict Ts = 21/4 x

255 = 303 K. This is closer to the actual mean surface temperature of 288 K but is now an overestimate! The model we have discussed is clearly an oversimplification:

• For one thing, not all the solar flux incident on the top of the atmosphere reaches the surface; typically, 20%-25% is absorbed within the atmosphere (including by clouds).

• For another, we saw in Section 2.2 that IR absorption by the atmosphere is incomplete. The greenhouse effect is actually less strong than in the model assumed previously, and so Ts will be less than the value implied by Eq. 2-9. We shall analyze this by modifying Fig. 2.7 to permit partial transmission of IR through the atmosphere—a leaky greenhouse model.

2.3.2. A leaky greenhouse

Consider Fig. 2.8. We suppose the atmosphere has absorptivity e, that is, a fraction e of the IR upwelling from the surface is absorbed within the atmosphere (so the case of Fig. 2.7 corresponds to e = 1). Now again if we insist that in equilibrium the net flux at the top of the atmosphere vanishes, we get

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