5.2.2 Blackbody Radiation
A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation. All blackbodies at a given temperature emit radiation with the same spectrum. The laws of blackbody emission are fundamental for understanding radiative transfer in the atmosphere.
Planck’s law
Planck’s law describes the spectral density of radiative emission for a blackbody at temperature T [K] under thermodynamic equilibrium. It assumes that photons are distributed with frequency ν according to Boltzmann statistics. Under these conditions, the spectral density Bν [W m–2 Hz–1] of the blackbody radiance is given by the Planck function
(5.32)
where k is the Boltzmann constant (1.38 × 10–23 J K–1). One can also express the Planck function for the spectral density Bλ [W m–2 nm–1] as a function of wavelength:
(5.33)
Stefan–Boltzmann law
Integration of the spectral density Bλ over all wavelengths yields the total radiance
(5.34)
where b = 2π4 k4/(15 c2h3). The blackbody emission flux F [W m−2] is obtained by performing a hemispheric integration of the radiance and, since the blackbody radiance is isotropic, we write
The flux varies therefore with the fourth power of the absolute temperature. This expression represents the Stefan–Boltzmann law, and the proportional factor σ = 2π5 k4/(15 c2h3) = 5.67×10−8 Wm−2 K−4 denotes the Stefan–Boltzmann constant.
Wien’s displacement law
By differentiating Bλ(λ, T) with respect to λ, we find that the maximum wavelength of emission λmax is inversely proportional to the blackbody temperature:
(5.35)
The mean temperature of the Sun is 5800 K, so solar radiation peaks in the visible at 0.5 μm. The mean surface temperature of the Earth is 288 K, so terrestrial emission is in the infrared peaking at approximately 10 μm. There is almost no overlap between the solar and terrestrial radiation spectra so that these two types of radiation can be treated separately (see Figure 5.2).
Figure 5.2 Blackbody spectra at 5800 K and 253 K, corresponding to the effective temperatures of the Sun and the Earth.
Kirchhoff’s law
Under thermodynamic equilibrium, the emissivity of a body at a given wavelength (defined as the ratio of the monochromatic emitting intensity to the value given by the Planck function) is equal to its absorptivity (defined as the ratio of the monochromatic absorbed intensity to the value of the Planck function). In the case of a blackbody, the values of the emissivity and absorptivity are equal to 1. Kirchhoff’s law implies that we can compute the emission flux density of any object simply by knowing its surface temperature and its absorption spectrum.
5.2.3 Extra-Terrestrial Solar Spectrum
The spectrum of radiation emitted by the Sun can be approximated as that of a blackbody at a temperature T = 5800 K. The spectral density of the solar flux [W m–2 nm–1] traversing a surface perpendicular to the direction of the solar beam at the top of the Earth’s atmosphere can then be expressed by
Φ∞ , λ(λ, T) = βRBλ(λ, T)
(5.36)
where the dilution factor
(5.37)
accounts for the distance between the Sun and the Earth (d = 1.471 × 108 km at the perihelion and 1.521 × 108 km at the aphelion). Here RSun is the solar radius (6.96 × 105 km). The total solar flux at the top of the Earth’s atmosphere,
Φ∞(T) = βRσT4
(5.38)
is equal to 1380 W m–2 and is called the solar constant.
The observed solar spectrum deviates from the theoretical blackbody curve because it includes contributions from different solar layers at different temperatures. The extra-terrestrial (top of the atmosphere) spectral density of the solar flux is shown in Figure 5.3. Also shown is the spectral irradiance at the surface, which is weaker than at the top of the atmosphere because of atmospheric scattering and absorption. Absorption by ozone, water vapor, and CO2 is responsible for well-defined bands in the spectrum where surface radiation is strongly depleted. A more detailed solar spectrum in the UV region (λ < 400 nm), where sufficient energy is available to dissociate molecules and initiate photochemistry, is shown in Figure 5.4. Radiation in that region of the spectrum interacts strongly with the Earth’s atmosphere through absorption by molecular oxygen and ozone, and through scattering by air molecules. For wavelengths shorter than 300 nm, the radiation reaching the Earth’s surface is orders of magnitude lower than that at the top of the atmosphere.
Figure 5.3 Spectral density of the solar irradiance spectrum at the top of the atmosphere and at sea level over the range 250–2500 nm. The 5800 K blackbody spectrum (thin black line) is shown for comparison. Absorption features by several radiatively active gases are indicated.
Source: Robert A Rodhe, Wikimedia Commons.
Figure 5.4 Solar irradiance spectra at the top of the atmosphere at 119–420 nm (a) and at different altitudes at 200–350 nm (b). The top panel (from Woods et al., 1996) shows the Lyman-alpha line at 121 nm from solar H atoms. The bottom line illustrates the strong absorption of radiation at wavelengths shorter than 300 nm by oxygen and ozone in the Earth’s atmosphere.
5.2.4 Penetration of Solar Radiation in the Atmosphere
Transfer of radiation in the Earth’s atmosphere is sensitive to absorption and scattering by atmospheric molecules, aerosol particles, clouds, and the surface. We examine these different processes here.
Absorption only
We first consider the simple case in which absorption plays the dominant role, with scattering assumed to be negligible. This assumption is often used in the middle and upper atmosphere, where scattering is weak because the atmosphere is thin and there are no clouds. The direct incoming solar beam then represents the dominant component of solar radiation. The actinic flux density qλ(λ; z, θ0) at altitude z and for a given solar zenith angle θ0 is proportional to the radiance associated with that beam. From the Beer–Lambert law, we have
(5.39)
where qλ,∞(λ) is the spectral density of the solar actinic flux at the top of the atmosphere. The exponential attenuation describes the absorption of the solar beam by an absorber with number density n and wavelength-dependent absorption cross-section σabs. The air mass factor T, defined as the ratio of the slant column density to the vertical column density,
(5.40)
accounts for the influence of the solar inclination. If we neglect the effect of the Earth’s curvature and assume a plane-parallel atmosphere, the air mass factor is simply
F(z, θ0) = secθ0
(5.41)
This approximation is generally acceptable if the solar zenith angle θ0 is less than 75°. Otherwise, a more complex approach must be adopted to account for the Earth’s sphericity (see Smith and Smith, 1972; Brasseur and Solomon, 2005).
When several absorbers i are contributing to the attenuation of radiation, the total optical depth is the sum of the optical depth associated with the individual species:
(5.42)
In the middle and upper atmosphere, the optical depth is due primarily to the absorption by ozone and molecular oxygen, so that
τabs(λ, z) = τabs , O3(λ, z) + τabs , O2(λ, z)
(5.43)
The spectral distributions of the absorption cross-sections for O2 and O3 are shown in Figures 5.5 and 5.6. Spectral regions of importance are listed in Table 5.1.
Figure 5.5 Absorption cross-section [cm2 molecule–1] of molecular oxygen between 50 and 250 nm featuring the Schumann–Runge continuum (130–170 nm), the Schumann–Runge bands (175–205 nm), and the Herzberg continuum (200–242 nm). Note the weak absorption cross-section at the wavelength corresponding to the intense solar Lyman-α line. At wavelengths shorter than 102.6 nm, absorption of radiation leads to photo-ionization of O2.
Figure 5.6 Absorption cross-section [cm molecule–1] of ozone between 180 and 750 nm with the Hartley band (200–310 nm), the temperature-dependent Huggins bands (310–
400 nm) and the weak Chappuis bands (beyond 400 nm).
Table 5.1 Spectral regions for atmospheric absorption by O2 and O3
Wavelength Atmospheric absorbers
121.6 nm Solar Lyman-α line, absorbed by O2 in the mesosphere. No absorption by O3.
130–175 nm O2 Schumann–Runge continuum. Absorption by O2 in the thermosphere.
175–205 nm O2 Schumann–Runge bands. Absorption by O2 in the mesosphere and upper stratosphere. Effect of O3 can be neglected in the mesosphere, but is important in the stratosphere.
200–242 nm O2 Herzberg continuum. Absorption of O2 in the stratosphere and weak absorption in the mesosphere. Absorption by the O3 Hartley band is also important (see below).
200–310 nm O3 Hartley band. Absorption by O3 in the stratosphere leading to formation of the O(1D) atom.
310–400 nm O3 Huggins bands. Absorption by O3 in the stratosphere and troposphere leading to formation of the O(3P) atom.
400–850 nm O3 Chappuis bands. Weak absorption by O3 in the troposphere with little attenuation all the way to the surface.
Absorption and scattering
Radiation is scattered by air molecules, aerosols, and clouds. Scattering refers to a change in the direction of incident radiation without loss of energy. The scattering properties of a medium are characterized by the efficiency with which the incoming radiation is scattered (scattering efficiency, defined next) and by the distribution of angles of the scattered radiation relative to the incident beam (scattering phase function, also defined below). The theory of Lorenz (1890) and Mie (1908) describes the interaction between a plane electromagnetic wave and a spherical particle based on the Maxwell equations. Scattering by a sphere is uniform over all azimuth angles so that the scattering phase function is characterized by a single angle Θ [0°, 180°] relative to the incident beam. The scattering properties are determined by the particle size parameter
introduced in Section 3.9.4 where Dp is the particle diameter, and on the refraction index mr defined as the ratio of the speed of light in vacuum to that in the scattering medium. The size parameter determines the scattering regime (Figure 5.7). The refraction index is commonly expressed as a complex number to account for both scattering and absorption:
m = mr − i mi
(5.44)
where the imaginary component mi is a measure of the absorption efficiency. Both mr and mi are wavelength-dependent.
Figure 5.7 Different scattering regimes as a function of radiation wavelength and particle radius.
Reproduced from Petty (2006).
Scattering by air molecules
Scattering of light by air molecules is described by the Rayleigh theory, which can be viewed as the asymptotic case of the Lorenz–Mie theory for a size parameter α ≪ 1. Under these assumptions, the scattering cross-section [cm2 molecule–1] is found to be
(5.45)
where na is the air number density and mr is the real part of the refractive index of air. In this limit there is no dependence on particle size. The correction factor
accounts for the anisotropy of non-spherical molecules with the anisotropy factor for air molecules being δ = 0.035. The factor mr – 1 is approximately proportional to the air density na so that the dependence of the scattering cross-section on air density largely cancels. The dominant feature of the Rayleigh scattering cross-section as described by (5.45) is the λ–4 dependence.
Scattering by aerosols and cloud droplets
Atmospheric scattering due to very large particles such as cloud droplets (α ≫ 1) is described by the laws of geometric optics, which can be regarded as another asymptotic approximation of the electromagnetic theory. In this formulation, the direction of propagation of light rays is modified by local reflection and refraction processes. The case of smaller aerosol particles is more complex as particle dimensions are typically of the same order of magnitude as the radiation wavelength. The Lorenz–Mie theory of light applied to spherical aerosol particles with diameter Dp provides general analytical expressions for the scattering and absorption efficiencies
(5.46)
and
(5.47)
defined as the ratio of the scattering (σscat) and absorption (σabs) cross-sections to the geometric cross-section /4, as a function of size parameter α. For example, the scattering efficiency is expressed by the following expansion
Qscat = c1α[1 + c2α + c3α2 + ….]
(5.48)
where the coefficients ci are provided by the theory as a function of the refraction index. When considering the scattering of visible light by molecules (α ~ 10–3), the dominant contribution is provided by the first-order term
(5.49)
and describes Rayleigh scattering presented earlier.
For aerosol and cloud particles, the dependence of scattering efficiency on particle size becomes important, while the dependence on wavelength is less pronounced than for air molecules. The extinction efficiency Qext is defined as
Qext = Qscat + Qabs
(5.50)
Values of Qscat and Qabs as a function of size parameter α are presented in Figure 5.8 for different values of the refraction index. Scattering is most efficient when the particle radius is equal to the wavelength of incident radiation (α = 2π). It is inefficient for very small particles (α ≪ 1). For very large particles (α ≫ 1), Qscat approaches a diffraction limit. The scattering coefficient for a given aerosol size distribution is obtained by integration of the particle cross-sections over the size distribution weighted by the scattering efficiency:
(5.51)
A similar equation applies to compute βabs.
Figure 5.8 Scattering (a) and absorption (b) efficiencies as a function of the aerosol size parameter α for various amounts of absorption (imaginary part of the refraction index m).
From Frank Evans, University of Colorado, personal communication.
Source function and phase function
When the effects of both absorption and scattering on the solar radiance are taken into account and local emissions are ignored, the source term J that appears in the radiative transfer equation ((5.18) for a plane-parallel atmosphere) must account for light scattered from the direction of the Sun (μ0, φ0) and from all other directions (μ′, φ′). The source term J can be expressed as
(5.52)
where ω(λ) = βscat/βext is again the single scattering albedo. The first term on the right-hand side of (5.52) represents the single scattering of the direct solar radiation (whose irradiance at the top of the atmosphere is Φλ(λ, ∞)), and the second term accounts for multiple scattering. The phase function P(λ, μ, φ, μ′, φ′) defines the probability density that a photon originating from direction (μ′, φ′) is scattered in direction (μ, φ). For spherical particles, the phase function depends only on the scattering angle Θ between the direction of the incident and the scattered radiation. It is often expressed as a function of parameter μs = cos Θ, which is related to the azimuthal and zenithal directions by
μs = cos Θ = μμ ' + (1 − μ2)1/2(1 − μ'2)1/2 cos (φ − φ')
(5.53)
The phase function P is normalized so that
(5.54)
For isotropic scattering, P is constant and equal to 1. The non-isotropy of the scattering process can be expressed by the asymmetry factor
(5.55)
Its value is equal to 1 if all the light is scattered forward, –1 if it is entirely scattered backward, and 0 if scattering is isotropic. The angular distribution of the scattered energy, and hence the asymmetry factor, can be derived from theory. In the most general case, it varies with the value of the size parameter α and with the degree of polarization of light. Light emitted by the Sun is unpolarized, but becomes partially polarized after scattering with molecules and particles in the atmosphere.
For scattering of unpolarized light by gas molecules, Rayleigh’s theory applies (α ≪ 1), and the phase function is found to be
(5.56)
or equiva
lently
(5.57)
In the presence of spherical aerosol particles (α ~ 1), the Lorenz–Mie theory applies and the phase function takes on a complicated form. Derivation is presented in radiative transfer textbooks such as Liou (2002). The phase function is then often described for modeling purposes in terms of associated Legendre polynomials, or more simply by the asymmetry factor. For an ensemble of particles characterized by a size distribution nN(Dp), the mean phase function P(μs) is derived by averaging the size-dependent phase function P(Dp, Θ) or P(Dp, μs) and weighting it by the scattering efficiency Qscat
(5.58)
Sample phase functions P(Dp, Θ) calculated by the Lorenz–Mie theory are shown in Figure 5.9. Scattering by large particles is characterized by a strong forward component with a value for the asymmetry factor of about 0.8.
Figure 5.9 (a) Schematic representation of light scattered by particles of different sizes. (b) Scattering phase function derived from the Lorenz–Mie theory as a function of the scattering angle Θ for different values of the particle size parameter α. In the case of Rayleigh scattering (α ≪ 1), the same amount of light is scattered in the forward and backward directions. Mie scattering tends to favor scattering in the forward direction, especially in the case of large particles.
Modeling of Atmospheric Chemistry Page 28