From deepocean.net (a) and Frank Evans, University of Colorado, personal communication (b).
Solution of the radiative transfer equation
The solution of the radiative transfer equation (5.18) for the upward radiance at a vertical level corresponding to an optical depth of τ is
(5.59)
where τs is the optical depth at the surface. For the downward radiance
(5.60)
Finally, for μ = 0, the horizontal radiance is
Lλ(λ, τ, μ = 0, φ) = J(λ, τ, μ = 0, φ)
(5.61)
Different numerical methods are available to obtain approximate solutions to the integro-differential radiative transfer equation in an absorbing and scattering atmosphere. These include, for example, iterative Gauss, successive order, discrete ordinate, two-stream, and Monte-Carlo methods. See Lenoble (1977) and Liou (2002) for more details.
In the successive order method, the solution is obtained by solving iteratively (n = 0, 1, …)
(5.62)
with the zeroth order radiance given by the Beer–Lambert law applied to the incoming direct solar flux
(5.63)
Here, δ(x – x′) represents the Dirac function, which is equal to one for x = x′ and zero otherwise. The final radiance is the sum of the different components
(5.64)
In the discrete ordinates method (Chandrasekhar, 1950; Stamnes et al., 1988), the radiance is expanded as a Fourier series about the cosine of the azimuthal angle,
(5.65)
while the phase function is expanded into associated Legendre polynomials:
(5.66)
The radiative transfer equation (5.18) in three variables (τ, μ, φ) is replaced by N + 1 uncoupled integro-differential equations for Lm(τ, μ) (m = 0, 1, 2, … N) in two variables (τ; μ). The integration over μ is replaced by an accurate Gaussian quadrature formula at 2N + 1 values of μi (i = –N, …, –1, 0, 1, …, N), chosen to be the 2N roots of the Legendre polynomial P2N(μ).
In the two-stream method, the phase function (5.66) is expanded in terms of Legendre polynomials, with only the two first terms of the expansion being retained (N = 1). The radiative transfer equation for hemispherically averaged radiances in a plane-parallel atmosphere is approximated by replacing the integrals over the zenith angles that characterize the source function J (see (5.52)) by a Gaussian summation with only two quadrature points, corresponding to ascending and descending directions, respectively. The diffuse radiance is therefore divided into an upward-propagating and a downward-propagating component, and two coupled equations, one for each stream, must be solved. In the Eddington method, both the phase function and the radiance are expanded in terms of Legendre polynomials and only the two first terms of this expansion are retained, with T
L(λ, τ, μ) = L0(λ, τ) + L1(λ, τ)μ (−1 ≤ μ ≤ 1)
(5.67)
Here g is the asymmetry factor (see 5.55). Coefficients and are derived by solving two coupled differential equations established by inserting the above expression into the azimuthally averaged transfer equation for diffuse radiation. The two-stream and Eddington approximations provide computationally efficient methods and are commonly used. They are inaccurate in the presence of clouds, where photons are repeatedly scattered by cloud droplets in a predominantly forward direction. The delta-Eddington approximation addresses this issue by adjusting the phase function to account for the strong forward contribution in the multiple scattering process (Joseph et al., 1976). In this case, the phase function for the fraction of the scattered light that resides in the forward peak is expressed by a delta function.
Albedo
Radiation reflected by the surface must be included in radiance calculations through a lower boundary condition of the type
Lλ(μ > 0, φ, z = 0, ) = A Lλ(μ < 0, φ, z = 0)
(5.68)
where the surface albedo varies with surface type (Figure 5.10), wavelength, and the incident and reflected zenith and azimuthal angles. It is frequently assumed that the albedo is isotropic (the surface is then called Lambertian) but this is often not precise enough for retrieval of atmospheric parameters such as aerosol optical depth from satellite measurements of solar backscatter. In those cases one needs to describe the full angular dependence of the surface albedo, known as the bidirectional reflectance distribution function (BRDF).
Figure 5.10 Land surface albedo at 470 nm. Data based on observations by the Moderate Resolution Imaging Spectroradiometer (MODIS) in June 2001. Areas where no data are available are shown in gray.
Image from MODIS Atmosphere support group incl. E. Moody, NASA Goddard Space Flight Center and C. Schaaf, Boston University and National Aeronautics and Space Administration (NASA).
5.2.5 Emission and Absorption of Terrestrial Radiation
In the infrared, at wavelengths larger than approximately 3.5 μm, the radiation field is determined primarily by radiative emission from the Earth’s surface and the atmosphere. The contribution of solar radiation is small, and scattering by air molecules can be neglected. Because of the limited overlap between solar (shortwave) and terrestrial (longwave) radiation, a clear distinction can be made in the approach that is adopted to solve the radiative transfer equation. In the case of longwave radiation, the spatial distribution of the radiance is derived by integrating the radiative transfer equation (5.18) in which the source function is represented by the Planck function Bλ(λ, T). This formulation requires that collisions be sufficiently frequent so that the energy levels of the molecules are populated according to the Boltzmann distribution, a condition called local thermodynamic equilibrium (LTE). Local thermodynamic equilibrium is met in the lower atmosphere but breaks down above 60–90 km altitude, depending on the spectral band. In the limit of no scattering and for LTE conditions, radiative transfer is described by Schwarzschild’s equation
(5.69)
Assuming azimuthal symmetry (no dependence of the radiance on angle φ), and providing the boundary conditions Lλ(λ, zsurface, μ > 0) = Bλ(λ, Ts) at the surface and Lλ(λ, ∞, μ < 0) = 0 at the top of the atmosphere, the upward and downward components of the radiance at altitude z and for the zenith direction μ are obtained by integration of (5.69)
(5.70)
and
(5.71)
Here Ts represents the temperature at the Earth’s surface and
(5.72)
denotes the transmission function between altitudes z′ and z for an inclination μ and wavelength λ. As stated in Section 5.2, βabs(λ, z) represents the absorption coefficient [m–1] proportional to the concentration of the absorbers and to their wavelength-dependent absorption cross-sections. The transmission function was previously introduced in equation (5.7).
Absorption of radiation by molecules in the IR involves transitions between vibrational and rotational energy levels of the molecule, in contrast to absorption in the UV which involves transitions between electronic levels. Vibrational transitions generally require λ < 20 μm, while rotational transitions require λ > 20 μm. Combined vibrational–rotational transitions create fine structure in the absorption spectrum. The radiation emitted by the Earth is mainly in the wavelength range λ < 20 μm, so that absorption by molecules generally involves vibrational transitions. Molecules absorbing in that range reduce the flux of terrestrial radiation escaping to space and are called greenhouse gases. A selection rule of quantum mechanics is that vibrational transitions are allowed only if they change the dipole moment of the molecule. All molecules with an asymmetric distribution of charge (H2O, O3, N2O, CO, chlorofluorocarbons) or the ability to acquire a distribution of charge by stretching or flexing (CO2, CH4) are greenhouse gases. Homonuclear diatomic molecules and single atoms (N2, O2, Ar) are not greenhouse gases. A peculiarity of the Earth’s atmosphere is that the dominant constituents are not greenhouse gases. Figure 5.11 shows the atmospheric absorption of terrestrial radiation in the IR from 1 to 16 μm. The strongest bands are the 15 μm and 4.3 μm CO2 bands, the 9.6 μ
m ozone band, the 6.3 μm water band, the 7.66 μm methane band, the 7.78 μm and 17 μm N2O bands, and the 4.67 μm CO band.
Figure 5.11 Vertical atmospheric transmission (absorption) of infrared radiation from the surface to the top of the atmosphere represented as a function of wavelength (1–16 μm) for different radiatively active gases. The different absorption bands and an aggregate spectrum are represented.
Adapted from Shaw (1953).
Calculation of the IR radiance by (5.70) and (5.71) requires quantitative knowledge of the absorption spectra for the different radiatively active trace gases. Detailed radiative transfer models calculate the radiance under different conditions and derive the corresponding atmospheric heating rates. Line-by-line models with very high spectral resolution can account for individual absorption lines. Absorption lines have an extremely narrow natural width determined by the uncertainty principle of quantum mechanics, but are broadened in the atmosphere by thermal motions of the molecules (pressure-independent Doppler broadening) and also in the lower atmosphere by collisions between molecules (pressure-dependent collisional broadening). Line-by-line models are too computationally costly for application in atmospheric models, and parameterizations must be developed. Narrow-band and broad-band models have been developed to simplify the calculation of the mean transmittance over specified spectral intervals. Some of these models assume that the same line repeats itself periodically as a function of wavenumber (e.g., regular model of Elsasser, 1938), while others assume a particular statistical distribution of the line positions and intensities within each spectral interval (e.g., random model of Goody, 1952).
The correlated k-distribution method (Goody et al., 1989; Fu and Liou, 1992) is a computationally efficient algorithm used to calculate the average transmission over a given frequency (or wavenumber) interval Δν without having to perform a tedious spectral integration that accounts for the complexity of the rapidly varying absorption coefficient k(ν) inside this interval. This is accomplished by replacing the spectral integration required for the calculation of the mean transmission (expressed here for the case of a homogeneous atmosphere, see (5.13)
(5.73)
by an integration in the k space
(5.74)
Here the function f(k)dk represents the fraction of the interval Δν in which the mass absorption cross-section k(ν) has a value between k and k + dk. According to this expression, f(k) is the inverse Laplace transform of the transmission function. If
(5.75)
represents the cumulative probability distribution, which is a monotonically increasing smooth function, (5.74) can be expressed as
(5.76)
Since k(g) is also a smoothly varying function, the mean transmission over a spectral interval Δν can be estimated by using a simple and efficient numerical quadrature (with a small number of k-intervals) to calculate the integral. Liou (2002) and Goody (1995) discuss the method in the more realistic case of a non-homogeneous atmosphere in which parameter k varies with temperature and pressure.
5.3 Gas Phase Chemistry
5.3.1 Photolysis
When a photon absorbed by a molecule exceeds in energy one of the chemical bonds of that molecule, it can cause cleavage of the bond and break the molecule into two fragments. This process is called photolysis. For example, molecular oxygen has a chemical bond between its O atoms corresponding to the energy of a 242 nm photon. It follows that only photons of greater energy (shorter wavelength) can drive O2 photolysis:
O2 + hν (λ ≤ 242 nm) → O + O
(5.77)
The rate at which O2 is photolyzed is given by
(5.78)
where JO2 is the photolysis frequency [generally expressed in s–1]. In the general case of photolysis of a molecule A, we have
(5.79)
The photolysis frequency of A is derived by spectral integration of the product of three quantities: the wavelength-dependent absorption cross-section σA(λ), the quantum yield εA(λ) and the local solar actinic flux qλ
(5.80)
where λmax is the wavelength corresponding to the energy threshold for dissociation of the molecule. The absorption cross-section [cm2 molecule–1], called σabs in Section 5.2, represents the ability of a molecule to absorb a photon at a particular wavelength, and the quantum yield represents the probability that this absorption will lead to photolysis.
Numerical integration of (5.80) over spectral intervals Δλ is straightforward when the absorption spectrum is a continuum. In certain cases, however, the absorption spectrum exhibits complex structures of discrete bands with many narrow spectral lines. Examples are the Schumann–Runge bands (175–205 nm) of molecular oxygen and several bands of nitric oxide (e.g., the δ-bands) in the same spectral area. In this case, rather than reducing by several orders of magnitude the size of the spectral interval used in the numerical integration, parameterizations are adopted to use effective values of the absorption cross-section averaged over large spectral intervals (see Box 5.1).
Box 5.1 Photolysis in the Schumann–Runge Bands
The Schumann–Runge bands (175–205 nm) feature high-frequency variations in the absorption cross-sections of molecular oxygen (see Figure 5.5). These high-frequency variations complicate the calculation of photolysis rates by numerical integration over the wavelength spectrum. Accounting for the Schumann–Runge bands is important for computing photolysis frequencies in the upper stratosphere and mesosphere. Scattering is negligible at those altitudes so that the actinic flux is defined by attenuation of the direct beam by O2 and O3. The photolysis frequency of a molecule A at altitude z can be calculated as:
where σA (Δλk) is the mean absorption cross-section over the wavelength interval Δλk, qk,∞(Δλk) = ∫Δλk qλ,∞(λ) dλ is the mean top-of-atmosphere actinic flux over that interval, and are the effective O2 and O3 transmission functions from the top of the atmosphere averaged over Δλk, and θ0 is the solar zenith angle. We have assumed a quantum yield of unity for simplicity of notation. The effective O2 transmission function is defined by
where is the actual transmission function accounting for the fine absorption structure.
For A ≡ O2, the photolysis frequency is given by
where the effective O2 cross-section for wavelength interval Δλk is defined as
Rather than performing a computationally expensive line-by-line calculation of the effective parameters and , these parameters can be fitted as a function of the O2 slant column density
(see e.g., Kockarts, 1994).
Gijs et al. (1997) adopt the following expression
where T(N) is the temperature [K] at the altitude where the column is equal to N. Coefficients A and B are expressed as a function of N, using Chebyshev polynomial fits. The effective transmission is then derived by noting that
Other methods to parameterize these effective coefficients have been developed by Fang et al. (1974), Minschwaner et al. (1993), Zhu et al. (1999), and others. Minschwaner and Siskind (1993) have derived fast methods to calculate the photolysis frequency of nitric oxide using a similar approach. Chabrillat and Kockarts (1997) propose a parameterization for the calculation of photolysis frequencies in the spectral range close to the Lyman (121 nm), where both the solar flux and the O2 absorption cross-section vary rapidly over a narrow spectral interval.
5.3.2 Elementary Chemical Kinetics
Gas-phase chemical reactions can be classified for kinetic purposes as unimolecular, bimolecular, and termolecular (or three-body). A unimolecular reaction involves the dissociation of a molecule by photons (photolysis) or heat (thermolysis). It has the general form
A + hν → C + D (photolysis)
(5.81)
A + M → C + D + M (thermolysis)
(5.82)
where A is the reactant and C and D are its dissociation products. M is an inert molecule, such as N2 or O2, that transfers energy to the reactant by collision. The rate of reaction is given by
(5.83)
where k is a ra
te constant (equivalently called rate coefficient) for the reaction, usually given in [s–1]. For photolysis reactions, k is the photolysis frequency and the J notation of (5.79) is commonly used.
A bimolecular reaction has the general form
A + B → C + D
(5.84)
and the rate of reaction is given by
(5.85)
where the rate constant k is usually given in units of [cm3 molecule–1 s–1]. The reaction rate k[A][B] is proportional to the number of collisions ZAB per unit time between A and B. This collision frequency can be derived from the gas kinetics theory; it is proportional to the collision cross-section π(rA + rB)2 and the thermal velocity
(5.86)
if r and m are the molecular radii and masses of A and B. Thus
ZAB = π(rA + rB)2vth[A] [B]
(5.87)
Reaction of A and B involves formation of an activated complex AB* that breaks down either to the original reactants A and B or to the products C and D. The minimum energy needed to form the activated complex is called the activation energy Ea [J mol–1]. Gas kinetics theory shows that the fraction of collisions with an energy larger than Ea is proportional to exp[–Ea/T], so that the reaction rate coefficient can be written
Modeling of Atmospheric Chemistry Page 29