Solar Radiation in the Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 25

and the scattering angle cosine according to (1.36):
χ = µµ + 1−

2 1−

µ µ 2 cos( ϕ ϕ ). − (1.46)

γ For the phase function it is also suitable to pass from scattering angle to its

χ γ = χ cosine with formal substitution arccos .

ω 0 definedby (1.45) is called thesinglescatteringalbedo
or otherwise the probability of the quantum surviving per the single scattering
event. If there is no absorption (

Dimensionless value

κ = 0) then the case is called conservative

ω 0 = scattering , 1. If the scattering is absent then the extinction is caused

only by absorption, σ = ω 0 = 0 and the solution of the transfer equation is

reduced to Beer's Law - (1.41)-(1.43). From consideration of these cases, the
sense of value

0,

ω 0 is following: it defines the part of scattered radiation relatively
to the total extinction, and corresponds to the probability of the quantum to
survive and accepts the quantum absorption as its “death”.
It is necessary to specify the boundary conditions at the topandbottomof the
atmosphere. As it has been mentioned above, solar radiation is characterizing
with values F 0 ,

ϑ 0 , ϕ 0 incomestothetop.Usuallyitisassumed ϕ 0 =

0, i. e. all

µ = ϑ azimuths are counted off the solar azimuth. Additionally specify cos

0 0

S . 9
As has been mentioned above, solar radiation in the Earth's atmosphere
consists of direct and scattered radiation. It is accepted not to include the
direct radiation to the transfer equation and to write the equation only for
thescatteredradiation.Thecalculationofthedirectradiationisaccomplished
using Beer's Law (1.41). Therefore, present the radiance as a sum of direct and
scattered radiance I (

= π

and F 0

τ µ ϕ = τ µ ϕ τ µ ϕ

I ( )+ I (

). From expression for the
direct radiance of the parallel beam (1.10) the following is correct I (0,

, , ) , , , ,

µ ϕ = , )

π τ|µ 0 )

for Beer's Law. Substitute the above sum to (1.44), with taking into account
the validity of (1.37) for direct radiation and properties of the delta function
(Kolmogorov and Fomin 1989). Then introducing the dependence upon value
µ 0 and omitting primes I (

δ µ µ 0 ) δ ϕ τ µ ϕ = π δ µ 0 ) δ ϕ − 0), and it leads to I ( S ( − ( , , ) S ( mu − ( )exp(−

τ µ µ 0 , ϕ , , ), we are obtaining the transfer equation

for scattered radiation .

π

2 1

τ µ µ 0 , ϕ

)+ ω 0 (

τ I ( , , ) )

ϕ µ , ϕ ) d µ

µ = τ µ µ 0 , ϕ τ χ τ µ 0 , − I ( , , d x ( , ) I ( ,

τ π d 4

0 −1

+ ω 0 (

τ )

τ χ 0 ) exp(− τ|µ 0 ) Sx ( , (1.47)

4

9 Specifying

π

S has the following sense. Suppose that radiation equal to radiance S fromall directions
incomes to the top of the atmosphere, and this radiation is called isotropic. Then, according to (1.6)
linking the irradiance and radiance, the incoming to the top irradiance is equal to

π

S .Thus,value S is an
isotropic radiance that corresponds to the same irradiance as a parallel solar beam normally incoming
to the top of the atmosphere is.