Radiative Processes in the Climate System - Climate Dynamics - page 79

S ABS = 236 W/m 2 F UP = σ T 4

T A = atmospheric temperature

F DOWN = σ T 4

S ABS = 236 W/m 2

F UP = σ T 4

Figure 4.9 Greenhouse slab model,
Case II.

T S = surface temperature

Use the radiative equilibrium condition to generate a set of two equations
and two unknowns, namely, T S and T A , by setting the heat input equal to the
heat output for the surface slab and for the atmospheric slab:
For the atmosphere,

σ T

4

= 2. σ T

4

(4.12)

S A

For the surface,

S += σ T

4

σ T

4

. (4.13)

ABS A S

Solving Eqs. 4.12 and 4.13 simultaneously gives T S  302 K, and T A  254 K.
Note that the OLR from the system is coming from the atmosphere, not the
surface, so the radiative balance for the entire system is

S = σ T

4

T = 254K . (4.14)

A & ABS A

The surface emits longwave radiation at a much higher temperature than that
of the atmosphere, which is at the radiative equilibrium temperature.
This simple model illustrates the essence of the greenhouse effect. The at-
mosphere absorbs longwave radiation emitted by the surface and reradiates
that energy at the atmospheric temperature. This reradiation directs addi-
tional energy back to the surface. Thus, the surface has two heat sources—the
solar radiation and the longwave back radiation from the atmosphere. The
surface is much warmer as a result. Meanwhile, the OLR from the earth
system originates in the atmosphere, so the atmospheric temperature is main-
tained at the value that balances the solar radiation absorbed by the system,
that is, T E .

GREENHOUSE CASE III

A slightly more sophisticated, and realistic, assumption is that the atmosphere
absorbs only a fraction, f , of the longwave radiation emitted from the surface.