# Spectral distribution of the irradiance in a cloud-free atmosphere

## Influence of the turbidity

Figure 7.1 illustrates the influence of the clear atmosphere on the spectral distribution of solar irradiance received on the ground, at wavelengths between 200 and 2300 nm. Two cases of clear atmosphere are presented. One is limpid, i.e., with a low aerosol load. The other is turbid, i.e., with a large aerosol load. The spectral distribution of extraterrestrial irradiance Eqx is also plotted for comparison. The solar zenithal angle is 30°. The spectral distribution gives the irradiance at each wavelength /. The irradiance integrated over an interval [//, *X _{2}]* is given by the surface under the curve between these two wavelengths.

The spectral irradiance received at ground (gray curves) exhibits a shape similar to that of the extraterrestrial irradiance *Eox* with a peak around 500 nm. It is less than that at the top of the atmosphere because of the depletion of the radiation by the atmosphere. The greater the turbidity, the more important the scattering, and the smaller the spectral irradiance on the ground at all wavelengths. The corresponding spectral distributions of the clearness index are drawn in Figure 7.2. If there was no atmospheric attenuation, the clearness index would be equal to 1 at each wavelength. As expected, the clearness index for the clear-sky turbid case is less than or equal to that for the clear-sky limpid case at each wavelength.

The attenuation of solar radiation by the atmosphere is displayed by the difference between the black and gray curves in Figure 7.1, or in an equivalent manner, by the difference between 1 and the corresponding clearness index in gray in Figure 7.2. In the

*Figure 7.1* Typical spectral distributions of the solar irradiance received on horizontal surfaces located one at the top of the atmosphere and the two others at ground in clear-sky conditions respectively limpid and turbid, between 200 and 2300 nm. Solar zenithal angle is 30°. Results from the numerical code libRadtran simulating the radiative transfer in the atmosphere.

*Figure* 7.2 Typical spectral distributions of the clearness index for two cloudless atmospheres respectively limpid and turbid between 200 and 2300 nm. Solar zenithal angle is 30°. Results from the numerical code libRadtran simulating the radiative transfer in the atmosphere.

two cases of turbidity presented, this attenuation is not spectrally constant. Solar radiation at wavelengths *X* less than 280 nm is absorbed by stratospheric ozone (0_{3}), and the spectral irradiance received on the ground is zero. The radiation in the short wavelengths is strongly attenuated by the scattering by the air molecules, with, I recall, a scattering intensity all the greater the shorter the wavelength. This results in a rounding of the spectral distributions of the clearness index between 280 and 750 nm (Figure 7.2).

Absorption lines by dioxygen (0_{2}) and water vapor (H_{2}0) are between 700 and 800nm and cause a very marked attenuation. There are wide and deep deviations from the curve at the top of the atmosphere at wavelengths greater than about 900 nm. Here, is found the influence of the absorption lines, more or less wide, more or less deep, of the main gases such as dioxygen (0_{2}), water vapor (H_{2}0), carbon dioxide (C0_{2}), and methane (CH_{4}). The absorption of radiation by water vapor is so strong between 1350 and 1450nm, and between 1800 and 1950nm that the irradiance at ground and the clearness index are zero. Figures 7.1 and 7.2 are to be compared to Figure 4.2 (Chapter 4) showing a spectral distribution of the transmittance for a moderately dry and clean atmosphere without aerosols. Except aerosols, the same absorption and scattering effects are seen.

Note that the two gray curves are very close to each other at wavelengths greater than about llOOnm, in Figures 7.1 and 7.2. As a first approximation, the attenuation by the atmosphere depends little on the turbidity of the atmosphere at these wavelengths. The situation is very different at wavelengths between 400 and llOOnm, for which the differences between the two gray curves are large. This is due to the importance of the phenomenon of scattering in this interval. There is a greater influence of turbidity on the spectral irradiance and the spectral clearness index because an increase in turbidity mainly corresponds to an increase in the number of scatterers.

## Spectral distribution of the direct and diffuse components

Figure 7.3 exhibits examples of the spectral distributions of the global irradiance received at ground on a horizontal plane and of its direct and diffuse components. The simulation conditions are the same as those in Figure 7.1, i.e., a solar zenithal angle of 30° and two clear atmospheres: a limpid (graph on the left) and a turbid (graph on the right).

When the sky is limpid (graph on the left), the direct component is greater than the diffuse at all wavelengths. The diffuse component is the greatest at wavelengths less than about 750nm, at which scattering by air molecules is important. The spectral distribution of the diffuse component has a peak between 350 and 500 nm, then decreases rapidly. The spectral distribution of the direct component is quite similar to that of the global irradiance. The two do not match at the shortest wavelengths, where the diffuse component is not negligible. They merge at wavelengths greater than 1100nm.

The intensity of the scattering phenomenon increases with the turbidity (graph on the right) because the density of the number of scatterers increases. Consequently, the diffuse component increases with turbidity to the detriment of the direct component since the global irradiance decreases with turbidity (see Figure 7.1). The spectral distribution of the diffuse component has a peak between 350 and 500 nm, then decreases rapidly. The direct component is almost extinguished at very short wavelengths. The two components are equal around 700 nm in this simulation. The direct component predominates at wavelengths greater than 700 nm.

Figure 7.4 shows the spectral distribution of the clearness index, for the direct and diffuse components for the two cloudless atmospheres, between 200 and 2300 nm. I recall that the spectral distributions of the clearness indices but for the global irradiance are plotted in Figure 7.2.

Like the global irradiance, the direct spectral clearness index decreases as the turbidity increases, while on the contrary, the diffuse spectral clearness index increases. This corresponds to the fact that the diffuse component is greater in the turbid case than in the limpid case (Figure 7.3). When the turbidity increases further, for example, in the case of a dust storm, the direct component becomes zero. The radiation then has

*Figure 7.3* Typical spectral distributions of the global irradiance and its components received on a horizontal surface at ground for two cloudless atmospheres: (a) a limpid and (b) a turbid, between 200 and 2300 nm. Solar zenithal angle is 30°. Results from the numerical code libRadtran simulating the radiative transfer in the atmosphere.

*Figure 7.4* Typical spectral distributions of the direct and diffuse clearness indices for two cloudless atmospheres respectively limpid and turbid, between 200 and 2300 nm. Solar zenithal angle is 30°. Results from the numerical code libRadtran simulating the radiative transfer in the atmosphere.

only a diffuse component, which will decrease when the turbidity will further increase. The radiation may become zero. This is what happens in sandstorm episodes, during which darkness settles in gradually and then disappears in the same way.

The spectral distributions of direct and diffuse clearness indices exhibit the same absorption lines as noted above: dioxygen (0_{2}) and water vapor (H_{2}0) between 700 and 800 nm, and at wavelengths greater than about 900 nm, absorption lines more or less wide, more or less deep, of the main gases such as dioxygen (0_{2}), water vapor (H_{2}0), carbon dioxide (C0_{2}), and methane (CH_{4}). As mentioned above, apart from aerosols, the absorption and scattering effects are the same as those present in the spectral distribution of the transmittance, for a moderately dry and clean atmosphere, without aerosols, in Figure 4.2 (Chapter 4).

Apart from these absorption lines, in general, the direct spectral clearness indices increase with the wavelength, while the diffuse indices decrease. Similar to the global irradiance, the direct and diffuse components are absorbed by ozone (0_{3}) at wavelengths *к* less than 280 nm. The radiation in short wavelengths is very attenuated by the scattering by air molecules, the intensity of the scattering increasing when the wavelength decreases. This results in an increase in the direct spectral clearness indices from 280 to 750 nm in a rounded form, and, on the contrary, for the diffuse spectral clearness indices by a peak around 280 nm followed by a decrease.

In Figure 7.2, the two global clearness indices for the limpid and turbid atmospheres are very close to each other at wavelengths greater than about llOOnm: It was concluded that as a first approximation, the spectral distribution of the clearness index depends little on the turbidity of the atmosphere at these wavelengths. Figure 7.4 details this observation. It can be seen that the difference between the two direct spectral clearness indices is compensated by the difference, in opposite directions, between the two diffuse clearness indices.