The laws of extinction

air column

Absorption and diffusion is different at different altitudes due to the fact that the composition of the air is not homogeneous.

Ozone
Oxygen

CO2
Water vapor
aerosols

Molecular

Molecular + aerosol

Aerosol

Absorption

Scattering

mass of the atmosphere (m) and altitude of the Sun (ho)

Let’s inspect the area A1A2A

At high altitude of the Sun ho, this area can be approximated as rectangular triangle. A1A is hypotenuse, A1A2 and A2A are legs.

Optical mass of the atmosphere is equal to cosecant of the Sun altitude.

of the Earth’s surface and Sun beam refraction must be accounted for at ho>30°

Optical mass of the atmosphere is quick to rise with decrease of the Sun altitude

m=1

m=35,4

ho=0°

ho=90°

of solar radiation extinction in the atmosphere

Dimensionless magnitude

Solar radiation extinction index or OPTICAL DEPTH
of the atmosphere

means that value Iλ is quick to increase. Before sunset value m is quick to increase. It means that value Iλ is quick to decrease.
Near the noon hours the solar radiation flux is slow to change its value.

In case the air density is constant (the beam spreads horizontally and a short distance)

Volume extinction index

obtain that

Optical depth of the atmosphere ( ) is the magnitude inverse to that optical mass, which attenuates the radiation flux in e times.

more notion transmission coefficient (коэф.прозрач.)of the atmosphere (Pλ).

Adopting m=1, i. e. the Sun is in zenith

Transmission coefficient is the fraction of the solar radiation flux, which reaches the Earth surface as the Sun is in zenith.

Since

the atmosphere, the more the value of for a given wavelength, and the less the transmission coefficient.

Important: Transmission coefficient of monochromatic flux depends on physical state of the vertical air column and does not depend on altitude of the Sun.
Transmission coefficient is also a function of wavelength. For the ideal atmosphere (no water vapor, no admixtures), this coefficient increases with the increase of the wavelength since the scattering of the shorter wavelengths is more intensive than the longer ones.

know total (integrated) flux of SR but monochromatic.
Due to very intricate dependence of the transmission coefficient on wavelength, it is not easy to take this integral. The only way to make necessary calculation is to use average values of P and
We may do it because in this case transmission coefficient P also shows the fraction of SR that reaches Earth’s surface when the Sun is in zenith:
However, in this case P value depends on the optical mass m.

being attenuated, but it also change its spectral composition

Maximum emittance is shifted to the longer wave side as the Sun altitude decreases.

The shorter wave beams suffer the largest extinction. Thus, passing through every new layer, the SR becomes more and more enriched with longer wave radiation.

as a sum of three items.
is the optical depth of the ideal atmosphere.
is the optical depth formed by variable constituents (CO2, H2O )
is the optical depth formed by aerosols.
is turbidity factor.
As we know

Comparing the formulas suggests how many masses of ideal atmosphere are needed to get the SR extinction produced by one mass of the real atmosphere.

transmission coefficient does.
ATF does not depend on m value as much as the transmission coefficient does.

ATF depends on physical properties of air masses

Air mass is a huge air body characterized by homogeneous distribution of the air properties such as temperature, humidity, transparency etc.

as a bundle of parallel rays is called DSR.
Fluxes of I and depends on the following factors:
Solar constant.
Distance between the Earth and the Sun.
Physical state of the atmosphere over the point.
Altitude of the Sun.
Values of I and I’ have well-defined diurnal and annual variations. Maximal values is observed at the local noon. They are also influenced by turbidity of the atmosphere. They increase with increasing altitude of a locality (in this case optical mass decreases). It is why in mountain areas these quantities are larger than over planes.

it completely blocks the DSR.
The DSR fluxes falling on slanted surfaces are different of those falling on horizontal surfaces

of area in a unit of time is named SCATTERED RADIATION FLUX (i).
It depends on
Altitude of the Sun
Transparency of the atmosphere
Cloudiness
The DR flux reaches its maximum value at medium and high level clouds. At some cases it can be 2-3 times more intensive than the clear sky does.
The maximal value of DR is observed at local noon when the Sun altitude is the highest for the given day.

In a certain condition, contrary to the DSR, cloudiness makes DR stronger. However, some interior clouds (St, Sc at ho<15°)can not do that.

coefficient

For ideal atmosphere

For real atmosphere

There are some other formulas

“c” is parameter describing the atmosphere transparency.

From these formulas it follows:
At c=const , the DR flux is proportional to I . The Sun altitude increases (m value decreases), DR grows up.
The ratio i/I depends upon c value only. For ho=10..75
The ratio i/I grows up when the Sun altitude and c value decrease