Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 22

Fig. 1.7. Geometry of propagation of solar radiation in the plane parallel atmosphere

Thus, the radiance is a function of only three coordinates: altitude z and two
angles, defining direction ( ϑ ϕ , ). Hence, (1.34) could be written as:

ϑ ϕ dI ( z , , )

ϑ = α ϑ ϕ cos ( z ) I ( z , , )

dz

π π 2

(1.35)

− σ

( z )
4

ϕ γ ϑ , ϕ ) sin ϑ d ϑ d x ( z , ) I ( z ,

π

0 0

γ ϑ ϕ ϑ ϕ ). It is where scattering angle is an angle between directions ( , )and(

ϑ ϕ easy to express the scattering angle through : to consider the scalar product

of the orts in the Cartesian coordinate system and then pass to the spherical
coordinates. This procedure yields the following relation known as the Cosine
law for the spheroid triangles 7 :

,

γ = ϑ ϑ + sin ϑ ϑ cos( ϕ ϕ ) . cos cos cos sin − (1.36)

To begin with, consider the simplest particular case of transfer (1.35). Neglect
the radiation scattering, i. e. the termwith the integral. For atmospheric optics,

7 Usein(1.35)oftheplaneatmospheremodelinspiteoftherealsphericaloneisanapproximation.
It has been shown, that it is possible to neglect the sphericity of the atmosphere with a rather good
accuracy if the angle of solar elevation is more than 10 ◦ . Then the refraction (the distortion) of the
solar beams, which has been neglected during the deriving of the transfer equation is not essential.
Mark that the horizontal homogeneity is not evident. This property is usually substantiated with
the great extension of the horizontal heterogeneities compared with the vertical ones. However, this
condition could be invalid for the atmospheric aerosols. It is more correct to interpret the model of the
horizontallyhomogeneousatmosphereasaresultoftheaveragingoftherealatmosphericparameters
over the horizontal coordinate.