Short-Wave Solar Radiation in the Earth’s Atmosphere Part 2 ppsx

20 Solar Radiation in the Atmosphere

π|2 ≤ γ ≤ π) and it is very suitable for the theoretical consideration, as will

be shown further. However, it describes the real phase functions with a large

uncertainty (Vasilyev O and Vasilyev V 1994). Therefore, the using of this

function needs a careful evaluation of the errors. The detailed consideration

of this problem will be presented in Chap. 5.

1.3

Radiative Transfer in the Atmosphere

Within the elementary volume, the enhancing of energy along the length dl

could occur in addition to the extinction of the radiation considered above.

Heat radiation of the atmosphere within the infrared range is an evident exam￾ple of this process, though as will be shown the accounting of energy enhancing

is really important in the short-wave range. Value dEr – the enhancing of energy

– is proportional to the spectral dλ and time dt intervals, to the arc of solid

angle dΩ encircled around the incident direction and to the value of emitting

volume dV = dSdl. Specify the volume emission coefficient ε as a coefficient of

this proportionality:

ε = dEr

dVdΩdλdt . (1.32)

Consider now the elementary volume of medium within the radiation field.

In general case both the extinction and the enhancing of energy of radiation

passing through this volume are taking place (Fig. 1.6). Let I be the radiance

incoming to the volume perpendicular to the side dS and I +dI be the radiance

after passing the volume along the same direction. According to energy defi￾nition in (1.1) incoming energy is equal to E0 = IdSdΩdλdt then the change of

energy after passing the volume is equal to dE = dIdSdΩdλdt. According to the

law of the conservation of energy, this change is equal to the difference between

enhancing dEr and extincting dEe energies. Then, taking into account the def￾initions of the volume emitting coefficient (1.32) and the volume extinction

coefficient, we can define the radiative transfer equation:

dI

dl = −αI + ε . (1.33)

In spite of the simple form, (1.33)is the general transfer equationwith accepting

the coefficients α and ε as variable values. This derivation of the radiative

transfer equation is phenomenological. The rigorous derivation must be done

using the Maxwell equations.

We will move to a consideration of particular cases of transfer (1.33) in

conformity with shortwave solar radiation in the Earth atmosphere. Within the

shortwave spectral range we omit the heat atmospheric radiation against the

solar one and seem to have the relation ε = 0. However, we are taking into

account that the enhancing of emitted energy within the elementary volume

could occur also owing to the scattering of external radiation coming to the

Radiative Transfer in the Atmosphere 21

Fig. 1.6. To the derivation of the radiative transfer equation

volume along the direction of the transfer in (1.33) (i. e. along the direction

normal to the side dS). Specify this direction r0 and scrutinize radiation scat￾tering from direction r with scattering angle γ (Fig. 1.6). Encircling the similar

volume around direction ~r (it is denoted as a dashed line), we are obtaining

energy scattered to direction r0. Then employing precedent value of energy

E0 and definition (1.32), we are obtaining the yield to the emission coefficient

corresponded to direction r:

dε(r) = σ

4πx(γ)I(r)dSdΩdλdtdΩdl

dVdΩdλdt = σ

4π

x(γ)I(r)dΩ .

Then it is necessary to integrate value dε(r) over all directions and it leads to the

integro-differential transfer equation with taking into account the scattering:

dI(r0)

dl = −αI(r0) + σ

4π

4π

x(γ)I(r)dΩ . (1.34)

Consider the geometry of solar radiation spreading throughout the atmosphere

for concretization (1.34) as Fig. 1.7 illustrates. As described above in Sect. 1.1 we

are presenting the atmosphere as a model of the plane-parallel and horizontally

homogeneous layer. The direction of the radiation spreading is characterized

with the zenith angle ϑ and with the azimuth ϕ counted off an arbitrary

direction at a horizontal plane. Set all coefficients in (1.34) depending on the

altitude (it completely corresponds to reality).

Length element dl in the plane-parallel atmosphere is dl = −dz| cos ϑ. The

ground surface at the bottom of the atmosphere is neglected for the present (i. e.

it is accounted that the radiation incoming to the bottom of the atmosphere is

not reflected back to the atmosphere anditis equivalent to the almost absorbing

surface). Within this horizontally homogeneous medium, the radiation field is

also the horizontally homogeneous owing to the shift symmetry (theinvariance

of all conditions of the problem relatively to any horizontal displacement).

22 Solar Radiation in the Atmosphere

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:

dI(z, ϑ, ϕ)

dz cos ϑ = α(z)I(z, ϑ, ϕ)

− σ(z)

4π

2π

0

dϕ

π

0

x(z,γ)I(z, ϑ

, ϕ

)sin ϑ

dϑ

(1.35)

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

ϕ

). It is

easy to express the scattering angle through ϑ, ϕ: 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 triangles7:

cosγ = cos ϑ cos ϑ
+ sin ϑ sin ϑ
cos(ϕ − ϕ

) . (1.36)

To begin with, consider the simplest particular case of transfer (1.35). Neglect

the radiation scattering, i. e. the term with the integral. For atmospheric optics,

7Use in (1.35) of the plane atmosphere model in spite of the real spherical one is an approximation.

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

horizontally homogeneous atmosphere as a result of the averaging of the real atmospheric parameters

over the horizontal coordinate.