The second line is derived by introducing a new dummy variable t1 = t0 — t'. This is the the optical depth measured relative to the top of the atmosphere, and the re-expressed integral is computed by integrating from the top down, rather than from the ground up. The first term on the right hand side of Eq. 4.32 represents the proportion of the upward surface radiation which survives absorption by the atmosphere and reaches space. The second term is the net emission from the atmosphere. In the optically thin limit, the integral becomes small and the exponential in the first term approaches unity; thus, the OLR approaches the emission from the ground, 1+,s. As the atmosphere is made more optically thick, the boundary term becomes exponentially small, and the integral becomes more and more dominated by the emission from the upper reaches of the atmosphere. However, to obtain the optically thick limit, we cannot use the grey gas form of Eq 4.19, since dT/dp becomes infinite at p = 0 when the dry adiabat extends all the way to the top of the atmosphere.
In the optically thick limit, t0 ^ 1, the first term becomes exponentially small and the upper limit of the integral can be replaced by to, yielding the expression
Cp where r is the Gamma function, defined by r(s) = /J0 Zs-1exp(— Z)dZ. Using integration by parts, r(s) = (s — 1)r(s — 1), while r(1) = 1, sor(n) = (n — 1)!. For Earth air, 4R/cp = f soT(1 + 4R/cp) will be close to r(2), which is unity; in fact it is approximately 1.06. For any of the gases commonly found in planetary atmospheres, r(1 + 4R/cp) will be an order unity constant. As the atmosphere is made more optically thick, the OLR goes down algebraically like t-)4R/Cp, becoming much less than the value aT^ prevailing for a transparent atmosphere. The OLR approaches zero as t0 is made large because the temperature vanishes at the top of the atmosphere, and as the atmosphere is made more optically thick, the OLR is progressively more dominated by the emission from the cold upper reaches of the atmosphere.
The calculation can be related to the conceptual greenhouse effect model introduced in the previous chapter by computing the effective radiating pressure prad. Recall that ^T^ = OLR, so aTs4()4R/cp = OLR = aTs4Tro4R/cp r(1 + —) (4.34)
ps cp whence (prad/ps) = t_1(r(1 + 4R/cp))Cp/4R. This formula implies that the radiation to space comes essentially from the top unit optical depth of the atmosphere. If an atmosphere has optical depth = 100, then it is only the layer between roughly the top of the atmosphere (t = 100) and t = 99 which dominates the OLR. For the all-troposphere model, the maximum temperature of the top unit optical depth approaches zero as the atmosphere is made more optically thick, because this entire layer corresponds to pressures approaching zero ever more closely as k is made larger.
If S is the absorbed solar radiation per unit area of the planet's surface, then the surface temperature in balance with S is obtained by setting the OLR equal to S. Solving for the surface temperature, we find that in the optically thick all-troposphere limit, the surface temperature is
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