Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 7

We would like to point out that the definition of the normalized radiative
fluxdivergences(1.8)withtakingintoaccount(1.7)givesthepossibilityof
its theoretical consideration as a continuous function of the altitude after its
writing as a derivation of the net flux b ( z ) = ∂ |∂ z .

When we have defined the intensity and the flux above, we scrutinized the
radiation field, i. e. the situation when radiation spreads on different direc-
tions. Actually, it is possible to amount to nothing more than this definition
because no strictly parallel beam exists owing to the wave properties of light
(Sivukhin 1980). Nevertheless, radiation emitted by some objects could be
often approximated as one directional beam without losses of the accuracy.
Incident solar radiation incoming to the top of the atmosphere is practically
always considered as one-directional radiation in the problems in question.
Actually, it is possible to neglect the angular spread of the solar beam because
of the infinitesimal radiuses of the Earth and the Sun compared with the dis-
tance between them. Thus, we are considering the case of the plane parallel
horizontally homogeneous atmosphere illuminated by a parallel solar beam.
Some difficulties are appearing during the application of the above definitions
to this case because we must attribute certain energy to the one-directional
beam.
The radiance definition corresponding to (1.1) is not applicable in this case
because it does not show the dependence of energy dE upon solid angle d

F ( z )

Ω
[formally following (1.1) we would get the zero intensity]. As for the irradiance
definition (1.3), it is applicable. Thus, it makes sense to examine the irradiance
of the strictly one-directional beams. Then the dependence of energy dE upon
the area of the surfaces dS projection in (1.3) appears for differently orientated
surfaces dS , which gives the follows:
F ( ϑ = ϑ ) F 0 cos , (1.9)

ϑ where F 0 is the irradiance for the perpendicular incident beam, F ( )isthe

ϑ irradiance for the incident angle .

The incident flux F 0 is of fundamental importance for atmospheric optics
and energetics. This flux is radiation energy incoming to the top of the at-
mosphere per unit area, per unit intervals of the wavelength and time in the
case of the average distance between the Sun and the Earth, and it is called
a spectral solar constant .Figure1.3illustratesthe solar constant F 0 as a function
of wavelength. Concerning the radiance of the parallel incident beam, we can
define it formally using (1.5). Actually, for accomplishing (1.5) and (1.9), it is
necessary to assume the following:
I ( ϑ ϕ = δ ϑ ϑ 0 ) δ ϕ ϕ 0 ) , , ) F 0 ( − ( − (1.10)

ϕ 0 are the
solar zenith angle and the azimuth angle which are determining the direction
of the incident parallel beam. Remember that the delta function is defined as:
b

δ ϑ 0 , where () is the delta function (Kolmogorov and Fomin 1999),

δ = f ( x ) ( x − x 0 ) dx f ( x 0 ).

a