Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 16

molecules but on the air density fluctuations (thus, the air is considered as
acontinuousmedium)anditispossibletoignorethisviolation(Sivukhin
1980). For the calculation of the radiation field the elementary volume is chosen
so that only one interaction act may happen within the elementary volume.
Such volume is different for particles of different sizes (cloud droplets size is
close to 10-20 µ m, for atmospheric gases molecules (more exactly - density

fluctuations) - the size is about 0.5 10 −3

µ

m). Thus, the diffusive medium is
turned out non continuous. The violation of both conditions could occur when
there are big particles in the air (for example cloud droplets). Actually taking
into account the large size of the droplets (tens and hundreds of microns), there
are a lot of gas molecules and small aerosol particles around these droplets
and the both conditions are violated for them. Therefore, the question about
the applicability of the summarizing rules in the cases mentioned above needs
a special discussion.
Thevolumecoefficientsoftheinteractionbetweenradiationandatmo-
sphere are expressed through the scattering and absorption cross-sections
according to relations (1.22). Thus, the most important problem will be the
theoretical calculation of these cross-sections. The methods of their calcula-
tion are based on the description of the physical processes of the interaction
between radiation and substance (Zuev et al. 1997). However, as we are not
considering them here, the resulting formulas are adduced only, referring the
reader to the cited literature.
Thevolumecoefficientandthephasefunctionofthemolecularscattering
are expressed as follows:

×

δ
6−7

3 ( m 2 −1) 2
n

8
3 π

6+3

σ m = ,

λ δ

4

(1.25)

3
4+2

2
cos γ = δ δ γ x m ( ) [1 + +(1− ) ],

δ

where m istherefractiveindexoftheair, n is the number concentration of
the air molecules, λ δ is the radiation wavelength, is the depolarization factor

δ = (for the air it is equal to 0. 035). The derivation of (1.25) is presented for

example in the topics by Kondratyev (1965) and Goody and Yung (1996) (the
theory of the molecular scattering that is traditional for atmospheric optics)
and in the topic by Sivukhin (1980) (the scattering theory on the fluctuations of
the air density). Using the known thermodynamic relation it is easy to obtain
the number concentration:

P
kT

= n , (1.26)

where P is the air pressure, T is the air temperature, k is the Boltzmann constant.
For assuming the dependence of the air refractive index upon wavelength,
pressure, temperature, and moisture, we are using the semi-empiric relation