5.1.1 Solar Radiation

The Sun consists of about three-quarters of hydrogen and one quarter of helium. Helium is formed by a fusion reaction between hydrogen atoms, which generates energy. This energy escapes from the Sun as electromagnetic radiation. This solar radiation comprises photons that carry energy, which is a function of wavelength. The radiation frequency, ν(s⁻¹), associated with a photon is related to the wavelength of the radiation, λ (m), via the speed of light, c (m s⁻¹):

ν=cλ(5.1)

The energy of a photon, E_p (J), of wavelength λ is:

Ep=h ν=h cλ(5.2)

where h is the Planck constant (h = 6.626 × 10⁻³⁴ J s). Therefore, the energy of a photon increases as its wavelength decreases.

The electromagnetic spectrum covers a wide range of wavelengths. Visible light ranges from about 400 to 700 nm, with violet light at about 400 nm, blue light at about 450 nm, green light at about 550 nm, and red light at about 650 nm. Below 400 nm is ultraviolet (UV) radiation. At wavelengths far below UV radiation are X rays (from about 0.001 to 10 nm) and gamma rays (from about 0.1 to 1 pm). Above 700 nm is infrared (IR) radiation. At greater wavelengths are radar wavelengths (from about 1 mm to 1 m), as well as television and radio wavelengths. Since the energy of radiation decreases with increasing wavelength, UV radiation has more energy than IR radiation.

Solar radiation covers a range of wavelengths. The Sun emits radiation, which corresponds approximately to that of a black body at a temperature of about 5,800 K. In comparison, IR radiation reemitted by the Earth corresponds to that of a black body at about 300 K. (The term “black body” corresponds to the assumption that this body absorbs all radiation completely and does not reflect it; however, the temperature of this body leads to radiation that is maximum at a wavelength that depends on the temperature of this body and, therefore, it is not actually black.)

Solar radiation is partially scattered and absorbed by gases present in the atmosphere and, consequently, some of the radiation does not reach the Earth’s surface. Oxygen and nitrogen molecules absorb solar radiation below 100 nm. Between 100 and 200 nm, molecular oxygen strongly absorbs solar radiation. Between 200 and 280 nm, ozone (a product of the oxygen photolysis, see Chapter 7) also absorbs radiation very effectively. Therefore, solar radiation reaching the Earth’s surface corresponds mostly to wavelengths greater than 280 nm. Figure 5.1 shows the solar radiation spectrum at the top of the atmosphere, as well as at sea level, i.e., after scattering and absorption by atmospheric gases. Scattering of solar radiation by cloud droplets is also taken into account in this figure.

Figure 5.1. Solar radiation spectrum reaching the Earth. The solid black line corresponds to the theoretical irradiance of a black body at 5,772 K located at the distance of the Sun and calculated at the top of the Earth’s atmosphere. The light gray shaded area corresponds to solar radiation at the top of the Earth’s atmosphere and the dark gray shaded area corresponds to solar radiation reaching the Earth’s surface after scattering and absorption by atmospheric gases and reflection by clouds. Gases absorbing radiation are indicated at their major absorption wavelengths.

Source of the data: NREL (2017); the black body irradiance was calculated using Planck’s law.

Triatomic gases such as water vapor (H₂O), carbon dioxide (CO₂), and ozone (O₃) form dipoles, which interact with the electromagnetic radiation at IR wavelengths. These molecules are, therefore, greenhouse gases, because they do not absorb much solar radiation, which is mostly in the UV and visible light range (except ozone, which absorbs some UV light, see Chapter 7), but they absorb IR light, which is emitted by the Earth. For example, CO₂ shows its maximum absorption at about 15 μm, which corresponds to the wavelength range near the maximum IR emission by the Earth’s surface.

5.1.2 Radiance and Irradiance

Radiance (or intensity), I, represents the radiative energy per unit time (W or J s⁻¹) per unit surface area (m⁻²), and per unit solid angle (sr⁻¹). It can be integrated over all wavelengths (W m⁻² sr⁻¹) or expressed monochromatically (spectral radiance) per unit wavelength (W m⁻² sr⁻¹ nm⁻¹).

Irradiance, E_e, represents the radiative energy flux going through a surface. Therefore, it corresponds to the component of the radiance that is perpendicular to the surface and integrated over all solid angles of the hemisphere corresponding to the side of the surface where the source is located. If the angle between the direction of the radiation and the direction perpendicular to the surface is θ_r:

Ee(λ)=∫ΩI(λ) cos(θr) dΩ(5.3)

where Ω is the direction of the incoming radiation. Irradiance may be integrated over all wavelengths (W m⁻²) or expressed monochromatically per unit wavelength (monochromatic or spectral irradiance in W m⁻² nm⁻¹).

5.1.3 Radiative Budget of the Earth’s Atmosphere

The spectral radiance of a black body is governed by Planck’s law:

I=2 h c2λ−5(exp(hckB λ T)−1)(5.4)

where k_B is the Boltzmann constant (1.381 × 10⁻²³ J K⁻¹) and T is the temperature in K. Therefore, radiance increases as the temperature of the black body increases, for all wavelengths. The wavelength corresponding to the maximum radiance, λ_max, is approximately given by Wien’s law:

λmax=h c(5 kBT)=2.9×106T(5.5)

where λ is in nm and T in K. Therefore, the greater the temperature, the smaller the wavelength corresponding to the maximum radiance. The estimation of the effective temperature of the Sun’s surface (photosphere) is on the order of 5,770 to 5,780 K. For a temperature of 5,772 K (NASA, 2017), the wavelength corresponding to the maximum radiance of the Sun is calculated as follows:

λmax=2.9×1065772≈500nm

This wavelength corresponds to a blue-green light. However, blue light is preferentially scattered by the molecules of the Earth’s atmosphere and the Sun’s appearance is yellow or even red when the Sun is near the horizon (see the discussion of atmospheric visibility in Section 5.2). Outside the atmosphere, the Sun’s appearance is white because its radiation spectrum covers all visible wavelengths. The full spectrum of the colors of solar radiation is clearly visible in a rainbow, because the water droplets decompose this radiation spectrum due to refraction of the incoming radiation, which modifies the direction of the photon trajectories depending on their wavelength.

The Stefan-Boltzmann law relates the irradiance of a black body to its temperature. It is derived from Planck’s law by integrating over all wavelengths:

where σ_SB is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴).

Thus, the irradiance of the Sun at its source is:

At the distance of the Earth’s orbit, this irradiance is smaller, since it has been dispersed over a greater sphere, which has a radius equal to the distance between the Sun and the Earth. This irradiance of solar radiation at the Earth’s orbit is called the solar constant, E_S:

ES=6.29×107 (radius of the Sundistance Earth–Sun)2(5.7)

Thus:

where the distances are expressed in km. This value corresponds to that obtained by irradiance measurements performed via the Solar Radiation and Climate Experiment (SORCE) NASA satellite (Kopp and Lean, 2011). The radiative energy intercepted by the Earth corresponds to the radiative energy flux (irradiance) integrated over the Earth’s cross-section, i.e., E_s π r_T², where r_T is the Earth’s radius (6,371 km). Therefore, the average radiative flux reaching the top of the Earth’s atmosphere is equal to this value divided by the Earth’s surface area, π r_T² E_S / (4 π r_T²), i.e., E_S / 4, which corresponds to about 340 W m⁻².

However, part of the solar radiation is reflected toward space by clouds, atmospheric particles, and the Earth’s surface. This fraction, called albedo, is on the order of 30 %. Therefore, the solar radiative flux absorbed by the Earth and its atmosphere, E_e,E, is estimated to be on average about 235 W m⁻² and about 105 W m⁻² are reflected toward space.

If the Earth is considered to be a black body and if one assumes that the radiative flux received by the Earth (235 W m⁻²) is at equilibrium with that reemitted by the Earth, then the temperature at the Earth’s surface is estimated from the Stefan-Boltzmann law:

T=(Ee,EσSB) 14=(2355.67×10−8) 14=254 K = −19 °C(5.8)

The average temperature at the Earth’s surface is actually greater, since it is on average 288 K (15 °C). The difference between these two temperatures is due mostly to the absorption of part of the radiative energy reemitted by the Earth by some gases present in the Earth’s atmosphere. These gases are called greenhouse gases (GHG). The radiative budget of the Earth must, therefore, take into account the effect of these GHG, as well as some other important processes.

First, the radiative energy reflected to space may be decomposed into a fraction reflected by the atmosphere (clouds and particles), which is about 75 W m⁻², and a fraction reflected by the Earth’s surface (in particular by snow and ice, which have a strong albedo), which is about 30 W m⁻². The fraction absorbed may be decomposed into a fraction absorbed by the atmosphere (mostly in the UV), which is 67 W m⁻², and a fraction absorbed by the Earth’s surface, which is 168 W m⁻².

Radiation emitted by the Earth is located in the IR range. Therefore, this radiation is partially absorbed by GHG (see Figure 5.2). Natural GHG are water vapor (H₂O) and CO₂. In addition, anthropogenic activities lead to greater CO₂ atmospheric concentrations, since it is a product of combustion. Other GHG that are due to anthropogenic activities include methane (CH₄), nitrous oxide (N₂O), and ozone (O₃). The relative contributions of these GHG to the absorption of IR radiation are presented in Chapter 14. All GHG absorb about 350 W m⁻². This amount corresponds to about 90 % of the IR radiative flux emitted by the Earth’s surface, which is 390 W m⁻²; therefore, 40 W m⁻² are directly emitted into space through a window of the IR spectrum where there is little absorption by GHG.

Figure 5.2. Radiation spectrum emitted by the Earth in the infrared with the absorption bands of some greenhouse gases (CO₂, H₂O, CH₄, and O₃). The solid line corresponds to a simulation with the MODTRAN radiative transfer model (Spectral Sciences, Inc. and U.S. Air Force Research Laboratory; http://modtran.spectral.com) for a standard U.S. atmosphere at an altitude of 50 km. The dashed line corresponds to the radiation flux of a black body at 288 K at the same altitude. The atmospheric windows through which the Earth’s radiation flux escapes to space appear clearly between 8 and 9 μm and between 11 and 13 μm.

In addition, the Earth’s surface emits heat (1) via water evaporation, which releases heat during its subsequent condensation in the atmosphere leading to cloud and fog formation (latent heat flux), and (2) via heat fluxes due to atmospheric turbulence (sensible heat flux); see Chapter 4 for the description of the physical processes corresponding to these heat fluxes. These two fluxes correspond to about 102 W m⁻² (78 and 24 W m⁻², respectively).

Therefore, the total amount of energy absorbed by the atmosphere is 519 W m⁻², which includes 67 W m⁻² via absorption in the UV, 350 W m⁻² via absorption in the IR, and 102 W m⁻² via the latent and sensible heat fluxes. At equilibrium, the atmosphere reemits this energy: 62 % (324 W m⁻²) toward the Earth’s surface and 38 % (195 W m⁻²) toward space. The fact that the most important fraction is emitted toward the Earth’s surface results from the vertical distribution of the atmospheric density, which is greater near the Earth’s surface.

In summary, the radiative flux at the top of the atmosphere is 340 W m⁻², the atmosphere absorbs and reemits 519 W m⁻², and the Earth receives and reemits 522 W m⁻², of which 30 W m⁻² correspond to the Earth’s albedo. Table 5.1 summarizes the different components of these energy fluxes in terms of their sources and associated processes.

The radiative energy reemitted by the Earth’s surface as IR radiation is 390 W m⁻². Applying the black body formula for this irradiance (see Equation 5.8) leads to a temperature of 288 K (i.e., 15 °C), which is consistent with the mean temperature at the Earth’s surface. The wavelength corresponding to the maximum radiation emitted by the Earth may be calculated by Wien’s law (Equation 5.5): it is 10 μm, which corresponds to the infrared range.

More solar radiation is absorbed at the equator than at the poles, because of the zenith angle (i.e., the angle between the direction of the direct solar radiation and the direction perpendicular to the surface). The Earth reemits also more IR radiation at the low latitudes (tropical regions) than at the high latitudes (polar regions). However, the gradient from the equator to the poles is less pronounced for reemission than for absorption. This difference is due (1) to a greater reflectance (i.e., greater albedo) of the Earth’s surface at the poles than in the tropical regions and (2) to the presence of more water vapor (greater humidity) in the tropical regions and the associated clouds and latent heat release aloft. The energy budget leads, therefore, to a net gain in the tropical regions and a net loss in the polar regions (see Figure 5.3). These energy differences must, therefore, lead to a transfer of energy from the equator to the poles, in order to maintain global equilibrium for the Earth. This energy transfer corresponds to the Hadley cells in the tropical and sub-tropical regions and the polar cells near the poles. In addition, the ocean currents transfer also some heat from the equator toward regions at higher latitudes. Finally, the water vapor emitted in the tropical regions is transported toward mid-latitude regions, which corresponds to a latent heat flux from the low toward the higher latitudes.

Figure 5.3. Schematic representation of the meridional profile of the Earth’s energy budget.

5.2.1 Absorption of Atmospheric Radiation

Some gases and particles absorb atmospheric radiation. The amount of light absorbed is proportional to the absorption efficiency of the absorbing species, k_a (m² g⁻¹), its concentration, C (g m⁻³), and the distance traveled by the radiation, dx. The absorption coefficient (m⁻¹) is defined as the product of the absorption efficiency and the concentration:

Assuming a uniform concentration of the absorbing species, the change in the radiance may be calculated as a function of the distance traveled by the radiation according to Beer-Lambert’s law: