Radiative Processes in the Climate System - Climate Dynamics - page 81

0.1 S += σ T

4

2 σ T

4

(4.18)

ABS S A

and

0.9 S += σ T

4

σ T

4

. (4.19)

ABS A S

Solving Eqs. 4.18 and 4.19 simultaneously, we get T S  298 K and T A  254 K.
Again, T A is the same as in Case II, in which no solar radiation is absorbed by
the atmosphere. The radiative balance at the top of the atmosphere must be
preserved—the OLR ( F 4
UP

σ= in the slab atmosphere model) must always
balance S ABS , so T A cannot change unless S ABS , changes. But since the atmo-
sphere is now absorbing some solar radiation, it must absorb less terrestrial
radiation from the surface to keep its temperature the same as in Case II. Since
the atmosphere is opaque to the terrestrial radiation, the only way for the at-
mosphere to absorb less longwave radiation is for the surface temperature to
be lower than in Case II.

A

GREENHOUSE CASE V

A major shortcoming of the preceding examples is that the atmosphere is as-
sumed to be isothermal, since the single slab has a uniform temperature. We
know, however, that temperature falls off at a rate of approximately 6 K/km
in the troposphere ( Fig. 2.8 ). Changes in atmospheric temperature with height
can be taken into account in the simple slab model formulation by using more
than one atmospheric slab. Consider, for example, the two-slab case shown in
Figure 4.12, in which each atmospheric layer is assumed to be transparent to
solar radiation and opaque to longwave radiation. In this case, the analytical
solution requires solving a system of three equations and three unknowns:

S ABS = 236 W/m 2 F UP = σ T 4

T 2 = temperature of upper atmospheric slab

F DOWN = σ T 4

F UP = σ T 4

T 1 = temperature of lower atmospheric slab

F DOWN = σ T 4

S ABS = 236 W/m 2

F UP = σ T 4

Figure 4.12 Greenhouse slab
model, Case V.

T S = surface temperature