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3 . The theory of radiation transfer
Generally, radiation is considered to be the process of electromagnetic waves propagating through a medium (normally a planetary atmosphere). Some of the associated processes like absorption and thermal emission cannot be adequately described by classical theory and require the use of quantum mechanics, which treats radiation as the propagation and interaction of photons. One characteristic property of radiation is its wavelength, A classification of radiation according to wavelength is presented in Fig. 3.1 (from Liou, 1980).
Wavelength , frequency , and wavenumber are related through the relation , where is the speed of light ( m s-1 ). The sensitivity of the human eye is confined to a rather small interval from 0.4 to 0.7 (micrometres) known as the visible region. However, most of the atmospheric exchange of radiative energy occurs from the ultraviolet (from about 0.2 ) to the infrared (around 100 ).
In atmospheric studies, two quantities are generally used, the flux per unit area in W m-2 and the specific intensity or radiance, i.e., the radiative energy per time through unit area into a given solid angle, in W m-2 sr-1.
3.2 Derivation of the monochromatic radiative transfer equation (RTE)
The following discussion concerns the transfer equation for monochromatic radiation in its basic form for a plane-parallel, horizontally homogeneous atmosphere (Chandrasekhar, 1960).
In GCM applications, the radiation field is computed without accounting for polarization effects, and assuming stationarity (no explicit dependence on time). Only the point where radiation is considered, the direction of propagation, and the frequency matter for the problem. Fig. 3.2 presents schematically the different contributions to the specific intensity at a given point P enclosed by an infinitesimally small cyclindrical element of length , of cross-section , and of an orientation expressed in terms of two angles, i.e., , with respect to the -axis and , the angle between the projection of the direction onto the plane and the -axis itself. For a horizontally homogeneous atmospheric slice, the location of point P is given by its height above ground (or any other suitable coordinate). Let us consider the various sources and sinks of radiative energy in this cyclindrical element:
3.2 (a) Extinction
The radiance entering the cylinder at one end will be partially extinguished within the volume, proportionally to the amount of matter encountered, thus contributing a negative increment of radiative energy
3.2 (b) Scattering
Another contribution to a change in radiative energy in the volume is caused by the scattering of radiation from any other direction into the direction of the considered beam, i.e.,
3.2 (c) Emission
The last contribution to the change in the radiative energy in the volume is the thermal emission:
3.2 (d) Total
The total change of radiative energy in the cyclinder can be written as the sum of the individual contributions
If we define as the ratio of the absorption to the extinction coefficient and the single scattering albedo , we finally obtain the monochromatic RTE for a plane parallel, horizontally homogeneous atmosphere in the coordinate system given by ( as
In principle, the RTE derived above allows the complete problem of the radiative transfer in a NWP to be solved if the specific intensities are known for all model layers, i.e., including the surface, all directions and all wavenumbers of the spectrum. But in this present form, the RTE is much too complicated to be used as such in a NWP or climate GCM. We will now consider various simplications linked either to the basic laws of physics or to the very specific circumstances prevailing in the Earth's atmosphere.
3.3 Basic laws
3.3 (a) Planck's law
The energy of an atomic oscillator is quantized, as is any change of energy state. The change of the energy state by one quantum number corresponds to an amount of energy (either radiated or absorbed) of where is the frequency of the atomic oscillator and J s is Planck's constant. For a large sample of atomic oscillators where the distribution of energy levels follows Boltzmann statistics, one can derive Planck's function for the emission of radiant energy by a black body (see details in Liou, 1980)
3.3 (b) Wien's law
The dependence of on for various temperatures is shown in Fig. 3.3 . The dotted line in the diagram marks the wavelength of the maximum intensity. It increases with the temperature of the black body and its variation follows Wien's displacement law. Extreme values of the Planck function are defined by
This translates into .
Wien's displacement law considerably simplifies the radiation transfer problem in the Earth's atmosphere. Atmospheric and surface temperatures are typically in the 200-300 K range and the maximum emission occurs in the 10 to 15 wavelength range. On the other hand, the Sun, the Earth's external source of radiative energy emits most of its energy at wavelengths around about 0.5 corresponding to an equivalent black body at about 6000 K. At the mean Earth-Sun distance of km, the monochromatic radiant intensity received from the Sun over a wide range of the spectrum is much less than the emission by the Earth's atmosphere system at equivalent wavelengths. The cross-over point is approximately at 3.5 . For wavelengths lower than 3.5 , the terrestrial emission is negligible and so is the energy received from the Sun at ToA for wavelengths larger than 3.5 . For this reason, when designing a radiation scheme for application to the Earth's atmosphere, one may separate the transfer problem in two parts, the shortwave and longwave radiative transfer, by dealing independently with the sources and sinks that are important for each part of the spectrum.
3.3 (c) Stefan-Boltzmann law
The total radiant intensity of a black body follows from a spectral integration of the Planck function over all wavelengths, i.e.,
3.3 (d) Kirchhoff's law
The emissivity of a medium is the ratio of the emitted energy to the Planck function at the temperature of the medium for a given wavelength . If the absorptivity is defined as the amount of energy absorbed at that wavelength divided by the appropriate Planck function, Kirchhoff's law states that the two quantites are equal, i.e.,
The Earth's atmosphere is far from being a medium with a uniform temperature distribution. However, the assumption of a thermodynamic equilibrium ( ) is still valid in a local sense, since at least for the lower part of the atmosphere (approximately below 40 km), collision between molecules is a process efficient enough to maintain a local thermodynamic balance.
In Subsection 3.2, the derivation of the RTE and its formal solution were carried out monochromatically. It is now necessary to consider the spectral variations of the various parameters (scattering and extinction coefficients, single scattering albedo and phase function). By contrast with the emission of a black body which displays smooth variations with temperature and wavelength, the absorption, emission and scattering by particles and gases give rise to highly variable spectra of parameters.
3.4 The spectral absorption by gases
Any moving particle has kinetic energy as a result of its motion in space. This energy is equal to and is not quantized. However molecules have other radiative energy types, which can be described from a mechanical model of the molecules. The absorption of radiative energy by gases is an interaction process between molecules and photons and thus obey quantum mechanics laws. Absorption and emission take place when the atoms and molecules undergo transitions from one energy state to another. In contrast to "grey" absorbers such as solid particles and liquids, which absorb radiation fairly uniformly with respect to wavelengths (a consequence of the dense packing of molecules and atoms in these), the absorption and emission by atmospheric gases are highly selective with regards to wavelength, due to the selection rules that govern the transitions.
The gaseous absorption spectrum can be categorized in three parts according to the total (radiative) energy of the molecules:
The molecule may rotate or revolve about an axis through its centre of gravity. The relatively low rotational energy of the molecules is associated with wavelengths in the far infrared (i.e., mm). In this spectral region absorption lines are well separated and are related to the transformation from one discrete rotational state into another.
The atoms of the molecule are bounded by certain forces, but the individual atoms can vibrate about their equilibrium position relative to each other. A combination of rotational and vibrational energy transformations causes absorption lines in the 1 to 20 µm region. The coexistence of both types leads to very complex structures (e.g., the the vibration-rotation band of H2O around 6.3 ) as an enormous number of lines are involved, which partially overlap each other.
Towards shorter wavelengths, the third form of energy transformation, the change of the energy level of electrons, occurs and this change in the electronic levels of energy increases further the complexity of the spectrum (e.g., O2 bands in the ultraviolet).
3.4 (a) Line width
Each spectral line corresponds exactly to one form of energy transformation and should in principle be discontinuous due to the discrete energy levels. However, due to Heisenberg's uncertainty principle (natural broadening), the collision between molecules (Lorentz or pressure broadening), and Doppler effects (resulting from the thermal velocity of atoms and molecules), absorbed and emitted emission is not strictly monochromatic, but is rather associated with spectral lines of finite width. Whereas the natural broadening affects the line width only marginally, the other two have a marked impact on the shape of the spectral lines and therefore on the absorption process itself.
The Doppler broadening occurs due to the thermal agitation within the gas. For a molecule of mass radiating at frequency , with a velocity component in the line of sight following a Maxwell-Boltzmann probability distribution
The pressure broadening is due to collisions between the molecules, which modify their energy levels. The resulting absorption coefficient is
Fig. 3.4 compares typical Lorentz and Doppler line shapes.
3.4 (b) Line intensity
The intensity of a line varies with temperature, due to the variation with of the statistical population of the energy levels of a molecule.
An example of the fine scales involved is given in Fig. 3.5 taken from Fisher (1985). The figure shows the 15 band of CO2 at varying spectral resolution. Even on a wavenumber interval as small as 0.01 cm-1 (the 15 band covers roughly 450 to 800 cm-1), the absorption varies over the full range from total absorption to total transmission. It is this extreme spectral dependency of gaseous absorption that makes an explicit integration of the monochromatic RTE and subsequent spectral integration of fluxes quite a lengthy task, even on the fast super-computers, and renders it impossible for operational use in a NWP model. Very detailed models of the radiation transfer exist, which performs the spectral integration over wavenumber intervals the width of which is typically smaller than the half-width of an individual line. These so-called "line-by-line" calculations can serve as reference for more approximate methods. At present the number of known spectral lines for gases existing in the Earth's atmosphere is of the order of 300,000 (Rothman et al., 1998).
Figs. 3.6 to 3.8 show the spectral dependency of the gaseous absorption coefficient for the three main atmospheric absorbers H2O, CO2 and O3 over the longwave part of the spectrum. The values are derived from an evaluation of spectroscopic data over 5 cm-1 intervals between 0 and 1110 cm-1, 10 cm-1 between 1110 and 2200 cm-1, and 20 cm-1 beyond 2200 cm-1, with the narrow-band model of Morcrette and Fouquart (1985). Similarly, Figs. 3.9 to 3.11 show the temperature dependency of the absorption coefficients for the same absorbers in terms of and , with corresponding to , the absorption coefficient at 250 K.
Fig. 3.12 shows the spectral dependency of the gaseous absorption coefficient for the three main absorbers over the whole range of wavenumbers relevant for atmospheric energy budget. To put the importance of of the absorption lines for the atmospheric transfer in perspective, the normalized Planck function for 6000 K (solar emission) and 255 K (terrestrial emission) are plotted underneath. It is obvious that H2O is the most important absorber in the terrestrial spectrum leaving a transparent region only around 10 . The so called 9.6 band of ozone partially fills this gap. However, as the amount of O3 in the atmosphere is not very large and is concentrated in the stratosphere, the atmosphere is fairly transparent (with respect to gaseous absorption) in this region of the spectrum called the atmospheric window. Absorption effect of trace gases such as methane (CH4), nitrous oxide (N2O) and chlorofluorocarbons (CFC-11 and CFC-12) are mainly felt in this spectral region due to the small background absorption.
Towards longer wavelengths this window is bounded by CO2 absorption (the 15 band) and the H2O rotation band. Towards shorter wavelengths but still in the terrestrial radiation, we find the vibration-rotation of H2O around 6.3 . At very short wavelengths (in the ultraviolet region below 0.25 ), ozone absorbs almost completely the solar radiation penetrating the atmosphere. However, in the so called visible (0.4-0.7 ) and near-infrared (0.7-4 ) regions of the solar spectrum, solar radiation is weakened only by H2O and more modestly by CO2 and other trace gases (NO2, O2, CH4, N2O, not shown in Fig. 3.12 )
3.5 The spectral scattering by particles
A primary electromagnetic wave encountering a particle will excite a secondary wave originating at the particle's location. Such process can occur at all wavelengths covering the whole electromagnetic spectrum. The scattering efficiency will depend on the size, the geometrical shape and on the real part of its complex refractive index, whereas the absorption efficiency will essentially depends on the imaginary part of the refractive index. Particles in the atmosphere are generally at such distances from each other that the radiation originating from individual particles is incoherent (i.e., no interference betwwen scattered radiation from different particles). Therefore the field of scattered radiation due to an ensemble of particles is the sum of the fields from the individual scattering processes. Atmospheric constituents can be separated into different broad categories according to their size: Molecules have a typical radius of 10-4 µm, the so called aerosols range from 0.01 to 10 , cloud particles from 5 to 200 , rain drops and hail particles up to 10-2 m.
The relative intensity of the scattering pattern depends on the so-called Mie parameter where is the radius of the particle (assumed spherical) and is the wavelength.
3.5 (a) Rayleigh scattering
The size of air molecules is small compared to the wavelength of radiation in the Earth's atmosphere. So in a simple model one can assume that molecules interacting with an electromagnetic wave start to vibrate and thereby radiate as linear oscillators. By considering spherical air molecules statistically distributed in a volume, Lord Rayleigh derived (1871) the phase function of such molecules
3.5 (b) Mie scattering
The size of aerosols and or cloud droplets is comparable to the wavelengths at which the radiative energy is dominant in the Earth's atmosphere. Rayleigh scattering is not applicable anymore. Photons encountering a particle off such size will excite secondary waves in various parts of the particles. These waves are coherent and therefore interference causes partial extinction in some directions and enhancement in others. For this reason, scattering by aerosols and cloud particles is far from being isotropic, with a pronounced preference for the forward direction. Under the assumption of spherical particle shape, Mie theory can be used to derive the phase function (see Subsection 3.2 (b)) for a given size distribution of particles. Unfortunately, there is no analytical solution and the result is obtained in the form of an infinite series of Bessel functions. A normalized phase function for a standard aerosol model (after Quenzel, 1985) is shown in Fig. 3.14 .
For detailed calculations involving radiances (e.g., in remote sensing applications) the phase function is usually developed into Legendre polynomials with up to several hundred terms to get the true angular representation
In large-scale atmospheric models, a scattering medium (cloud or aerosols) is usually characterized by its single scattering albedo (see Subsection 3.2 (d)) and its phase function (see Subsection 3.2 (b)) and its optical thickness (see Subsection 3.2 (b)). For very large droplets the laws of geometric optics apply making the calculations very simple. In the Earth's atmosphere a large variety of droplet sizes can be found depending very much on the type of cloud they are embedded in. Stephens (1979) quotes radii ranging from 2.25 for stratus clouds to 7.5 for stratocumulus. Han et al. (1994) derived radii for water clouds from satellite measurements between 6 and 15 . Most radiative transfer schemes in use in GCMs reuiqre only the volume extinction coefficient (related to the optical thickness), the single scattering albedo and the forward scattered fraction of radiation (or asymmetry factor). All these quantities as given by Stephens (1979) are plotted in Figs. 3.15 to 3.18 for various cloud types.
The most complex calculation of the phase function is required for the treatment of ice crystals. The application of Mie theory is not possible since the assumption of a spherical shape is certainly invalidated in this case. Ice crystals take all sorts of shapes ranging from thin needles to complex combinations of hollow prisms forming snowflakes. If a specific shape is assumed for the ice crystals, geometrical optics may be used for particles that are large compared to the wavelength (Takano and Liou, 1989; Ebert and Curry, 1992; Fu and Liou, 1993; Fu et al., 1999, 1988, 1999). However, the results depend very much on the orientation of the crystals relative to the direction of propagation of the radiation. At present, it is almost impossible for a GCM-type radiation scheme to take into account such detailed and complicated dependencies of the optical properties, specially as the GCM is still far from providing the reuiqred information necessary to initiate such calculations.
In the cruder form of parametrization currently used, the ice water loading of the clouds is prognosed by the cloud scheme and thus provides an interactive optical thickness. The effective dimension of the particles is diagnosed from temperature and/or the type of process from which the cloud is originating (large-scale condensation or convection), but the single scattering albedo and asymmetry factor are specified.
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