Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 20

π| ≤ γ ≤ π ) and it is very suitable for the theoretical consideration, as will

be shown further. However, it describes the real phase functions with a large
uncertainty (Vasilyev O and Vasilyev V 1994). Therefore, the using of this
function needs a careful evaluation of the errors. The detailed consideration
of this problem will be presented in Chap. 5.

2

1.3
Radiative Transfer in the Atmosphere

Within the elementary volume, the enhancing of energy along the length dl
could occur in addition to the extinction of the radiation considered above.
Heat radiation of the atmosphere within the infrared range is an evident exam-
ple of this process, though as will be shown the accounting of energy enhancing
is really important in the short-wave range. Value dE r - the enhancing of energy
-isproportionaltothespectral d λ and time dt intervals, to the arc of solid

Ω angle d encircledaroundtheincidentdirectionandtothevalueofemitting

volume dV = ε dSdl .Specify thevolumeemissioncoefficient as a coefficient of

this proportionality:

dE r
dVd

ε = . (1.32)

Ω λ d dt

Consider now the elementary volume of medium within the radiation field.
In general case both the extinction and the enhancing of energy of radiation
passing through this volume are taking place (Fig. 1.6). Let I be the radiance
incoming to the volume perpendicular to the side dS and I + dI be the radiance
afterpassingthevolumealongthesamedirection.Accordingtoenergydefi-
nition in (1.1) incoming energy is equal to E 0

= Ω λ

IdSd d dt then the change of

= Ω λ energy after passing the volume is equal to dE dt .Accordingtothe

law of the conservation of energy, this change is equal to the difference between
enhancing dE r and extincting dE e energies. Then, taking into account the def-
initions of the volume emitting coefficient (1.32) and the volume extinction
coefficient, we can define the radiative transfer equation :

dIdSd d

dI

dl =

α ε − I + . (1.33)

In spite of the simple form, (1.33) is the general transfer equationwith accepting
the coefficients α ε as variable values. This derivation of the radiative

transfer equation is phenomenological. The rigorous derivation must be done
using the Maxwell equations.
We will move to a consideration of particular cases of transfer (1.33) in
conformity with shortwavesolarradiationintheEarthatmosphere . Within the
shortwave spectral range we omit the heat atmospheric radiation against the
solaroneandseemtohavetherelation

and

ε = 0. However, we are taking into

account that the enhancing of emitted energy within the elementary volume
could occur also owing to the scattering of external radiation coming to the