Formulas Derivation - Short-Wave Solar Radiation in the Earth's Atmosphere - page 290

Substitute to (B.5) the following expansions over powers of small parameter s
for values:

8 s 2 1+ s 6−7.5 g +

+ O ( s 4 ),

3.6
1+ g

= m

1−6 q s +18 q 2 s 2 + O ( s 3 ),
a ∞ =

= l

(B.6)

1−4 s +12 q s 2 + O ( s 3 ),

= 1−3 q s + n 2 s 2 + O ( s 3 ), n

and for functions:
K ( µ = K 0 ( µ )[1 − 3 q s + n 2 ω µ ) s 2 ]+ O ( s 3 ), ) (

(B.7)

µ = µ µ ) s 2 + O ( s 3 ), a ( ) 1−4 K 0 ( ) s + a 2 (

where:

) 3 −0.9)+4 q .

µ = µ µ a 2 ( ) 3 K 0 (

1+ g (1.271

Obtain the expression:

µ 0 )− F ↑ ]8 sK 0 ( µ 0 )(1−3 q s + n 2 w ( µ 0 ) s 2 )(1−3 q s + n 2 s 2 ) [ a (

+(1−6 q s +18 q 2 s 2 )[ a ( µ 0 )− F ↑ ] 2

(B.8)

= F ↓ n (1−6 q s +18 q 2 s 2 )(1 − A +4 As −12 q As 2 ) 2

−8 As (1 − 6 q s +9 δ

2 s 2 +2 n 2 s 2 )(1 − A +4 As −12 q As 2 ).

Accomplishing the multiplication of polynomials and keeping items with the
power of s not exceeding 2 the following is obtained:
[1 − F ↑ −4 K 0 ( µ 0 )(1−6 q s +9 q 2 s 2 + n 2 (1 + w (

µ 0 ) s + a 2 ( µ 0 ) s 2 ]8 sK 0 ( µ 0 )) s 2 )

+(1−6 q s +18 q 2 s 2 )[(1 − F ↑ ) 2 +16 K 0 ( µ 0 ) s 2 +2 a 2 ( µ 0 ) s 2 ]

(B.9)

= F ↓ 2 (1−6 q s +18 q 2 s 2 )((1 − A ) 2 −16 A 2 s 2 −24 q As 2 ).

s +19 q 2 s 2 , keeping the items with the
first and the second power of s , collecting the likewise terms and obtain the
linear equation respected to value s 2 :

δ Divide both parts by polynomial 1 − 3

s +18 q 2 s 2 (1 − F 0 ) 2 +2 s 2 (1 − F 0 ) a 2 ( δ µ 0 )−16 s 2 K 0 ( µ 0 ) 1−3

(B.10)

= F ↓ 2 (1 − A ) 2 −24 q F ↓ 2 A (1 − A ) s 2 −16 F ↓ 2 A 2 s 2 .

Assuming that 1 − F 0 F 0 and F 1 (1 − A ) = = F 1 arethenetfluxesatthetop

(subscript 0) and bottom (subscript 1) of the cloud layer correspondingly, and
assuming that F 1 A F 1 ,obtainforvalue s 2 the following: =

F 0 − F 1

s 2 = . (B.11)

µ 0 )− F ↑ 1 )−2 F 0 a 2 ( µ 0 )−24 q F 1 F 1

16( K 0 (