Theoretical Base of Solar Irradiance and Radiance Calculation in the Earth Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 65

µ behavior of function i ( ) relative to the phase function shape and absorption

in the medium has been studied in the topic by Yanovitskij (1997). The expan-
sion for function i ( µ ) has been derived in the paper by Yanovitskij (1972) in the

case of weak true absorption, which is presented here in terms of parameter s :

+3 1− g 2 +2P 2 (

µ )

µ = µ s 2 i ( ) 1+3 s

1+ g

+ 9(1 − 1.5 g )

s 3 + O ( s 4 ). (2.41)

µ µ

1+ g (1 + g )(1 + g + g 2 ) + 3.6

10.8 P 3 ( )

µ +

µ = Functions P i ( 1,2,...areLegendrepolynomialsofpower i .

The diffused irradiance in relative units of

)for i

π S within the optically cloud

layer is described with the following:

µ 0 ) exp(−2 k τ 0 ) K (

)) − i ↑ l exp(− k (

F ↓ ( µ 0 , τ τ 0 ) =

τ 0 ) [ i ↓ exp( k (

τ 0 − τ τ 0 − τ , ))] ,

1− ll exp(−2 k

τ 0 )
1− ll exp(−2 k

µ 0 ) exp(− k K (

F ↑ ( µ 0 , τ τ 0 ) =

τ 0 ) [ i ↑ exp( k (

τ 0 − τ )) − i ↓ l exp(− k ( τ 0 − τ , ))] ,

(2.42)

where

1 1

i ↓ = µ µ µ i ↑ = µ µ µ 2 i ( ) d , 2 i (− ) d .

0 0

Expansions for values i ↓ and i ↑ have been derived in the topic by Minin (1988)
after integrating (2.41):

3 s 3 2−3 g +

+ O ( s 4 ) .

2 s +3 s 2 1.5 − g 2
1+ g

0.8
1+ g

i ↓↑ = 1 ± ± (2.43)

It is also convenient to describe the internal radiation field with the values
of internal albedo b ( τ i ) = F ↑ ( τ i ) | F ↓ ( τ i ) and net flux F ( τ i ) = F ↓ ( τ i )− F ↑ ( τ i ),

according to Minin (1988) and Ivanov (1976)

µ 0 ) exp(− k τ 4 sK ( )

τ 0 ) [1 + l exp(− k (

τ µ 0 ) = F ↓ ( τ µ 0 )− F ↑ ( τ µ 0 ) = τ 0 − τ F ( , , , ))]

1− ll exp(−2 k

b ∞ − l exp(−2 k (

F ↑ ( τ µ 0 ) τ 0 − τ , ))

µ 0 ) =

τ = b ( ) .

1− b ∞ l exp(−2 k ( τ τ 0 − τ

F ↓ ( ,

))

(2.44)

Va l u e b ∞ and function b ( τ ) are called the internal albedo of the infinite atmo-

sphere and the internal albedo of the atmosphere of the large optical thickness
correspondingly, moreover b ∞ = 1−4 s + s 2 and the values of function b ( τ )

couldbeobtainedfromtheobservationsorfromthecalculationsofthesemi-
spherical irradiances at level τ .