Radiative Processes in the Climate System - Climate Dynamics - page 84

If I l >> B l , so emission into the path of the beam is negligible, the solution
to Eq. 4.26 is

− ks

I = I (0) e , (4.27)

λ

where I l (0) is the intensity of the beam at s  0 (see Fig. 4.13 ). According to Eq.
4.27, the intensity of the beam decreases exponentially with path length due
to molecular absorption. If there are N ABS absorbers per unit volume, and each
has a cross section denoted by A ABS and an absorptivity of 1, then

(4.28)

kNA
ABSABS

 .

Note that k has units of m −1 .
Optical depth , t, is a dimensionless quantity that measures the opacity of
the atmosphere (or any medium), or the penetration depth of a beam of radia-
tion into the medium. It is defined as

/ (4.29) t ks ,

so that

−

(4.30) I = I (0) e

λ λ

for the case of beam attenuation by molecular absorption. If t  0, the medium
is completely transparent to the radiation, and () λ λ When t  1, the

intensity of the beam is reduced to about one-third (1/ e ) of its original intensity,
since, from Eq. 4.30,

Is I 0 ().

I (0)

λ

I = . 0.3 0) I (4.31)

λ

e

λ

when t  1.
A similar approach allows us to express the loss of energy from a beam due
to scattering. If there are N SCAT scatterers, each with area A SCAT , then the attenu-
ation of the beam by scattering is

dI

λ

=− f NAI

SCAT λ

, (4.32)

SCAT SCAT

ds

where the factor f SCAT is the fraction of the scattering that is not forward scat-
tering. For example, f SCAT will be larger for the Rayleigh scattering of shorter
wavelengths than for Mie scattering.
Assuming that the density of absorbers, emitters, and scatterers in the
volume is sufficiently low so there is no significant interference among these
processes, the various influences on the beam intensity can be linearly superim-
posed to form the equation of transfer

dI

λ

=−− kB I f NAI

SCAT

. (4.33)

_ i

λ λ SCAT SCAT λ

ds