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3 . The theory of radiation transfer
3.1 Terminology
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 planeparallel,
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 crosssection , 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
where is the monochromatic extinction coefficient
(units m^{1}) and is the solid angle differential element.
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.,

,


where is the monochromatic scattering coefficient,
is the solid angle differential element for the
originating beam, is the normalized phase function, which represents the probability
for a photon incoming from direction to be scattered in direction . Since scattered radiation may originate from any direction, we
have to integrate over all possible angle combinations of and .

.


The direct (unscattered) solar beam is generally considered
separately and not contained in the intensity of the direction considered.
Thus scattering of the solar beam into the direction of interest is as well
separate from the scattering of diffuse radiation

,


where is the specific intensity of incident solar
radiation, and define the direction of the incident solar radiation at ToA, is the cosine of the solar zenith angle, and if is the height of the top of the atmosphere, is the optical thickness defined as

.


3.2 (c) Emission
The last contribution to the change in the
radiative energy in the volume is the thermal emission:

,


where is the monochromatic absorption coefficient,
and is the monochromatic Planck function.
3.2 (d) Total
The total change of radiative energy in
the cyclinder can be written as the sum of the individual contributions
Replacing the length dl of the cylindrical element by the
geometrical relation where , it follows from the expression of the optical thickness that
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)

,


where is the wavelength of emission, is Boltzmann constant, J K^{1}, is the velocity of light in a vacuum,
and is the absolute temperature of the black body (in
K).
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

.


With and , and defining and , the previous condition can be rewritten as

.


After some manipulation (see, e.g., Liou, 1980), we obtain a transcendental
equation ,which can be shown to give provided that .
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 200300 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 EarthSun 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 crossover 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) StefanBoltzmann law
The total radiant intensity of a black body
follows from a spectral integration of the Planck function over all wavelengths,
i.e.,

.


Using the same substitutions as when deriving Wien's displacement
law, we get

.


The integral term has the value , so we finally obtain StefanBoltzmann law

,


where W m^{2} K^{4} is Stefan's
constant. The total emitted flux given by this equation already implies
that the angular integration over the total radiant intensity is performed
assuming an isotropic emission by the black body (which is verified), leading
to the factor .
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.,

.


A black body is defined by its ability to absorb all the
incoming radiation at a given wavelength and therefore to emit isotropically
according to Planck function; then for a black body . In contrast, grey bodies absorb and emit only partially with values
of and being less than unity.
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.
Figure 3.1 . The electromagnetic
spectrum
Figure 3.2 .Processes contributing
to the radiative transfer in a cylindrical volume
Figure 3.3 . The dependence of
the Planck function on wavelength and temperature. Wavelengths of maximum
emssion are connected by a dotted line
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 vibrationrotation band of H_{2}O 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., O_{2} 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 MaxwellBoltzmann probability distribution

,


the absorption coefficient of such a Doppler broadened line
is

,


with .
The pressure broadening is due to collisions
between the molecules, which modify their energy levels. The resulting absorption
coefficient is

,


with making it proportional to the frequency of
collisions.
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.

,


where is the energy of the lower state of the transition,
is an exponent the value of
which depends on the shape of the molecule (1 for CO_{2}, 3/2 for
H_{2}O, 5/2 for O_{3}) and is the reference temperature at which the line intensities are known
(and compiled, usually 296 K).
An example of the fine scales involved is
given in Fig. 3.5 taken from
Fisher (1985). The figure
shows the 15 band of CO_{2} 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 supercomputers,
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
halfwidth of an individual line. These socalled "linebyline" 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
H_{2}O, CO_{2} and O_{3} 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 narrowband 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
H_{2}O 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
O_{3} 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 (CH_{4}),
nitrous oxide (N_{2}O) and chlorofluorocarbons (CFC11 and CFC12)
are mainly felt in this spectral region due to the small background absorption.
Towards longer wavelengths this window is
bounded by CO_{2} absorption (the 15 band) and the H_{2}O rotation band. Towards shorter wavelengths
but still in the terrestrial radiation, we find the vibrationrotation of
H_{2}O 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 (0.40.7 ) and nearinfrared (0.74 ) regions of the solar spectrum, solar radiation is weakened
only by H_{2}O and more modestly by CO_{2} and other trace
gases (NO_{2}, O_{2}, CH_{4}, N_{2}O, not
shown in Fig. 3.12 )
Figure 3.4 . Transmission in the
15 µm band of CO_{2} at various spectral resolutions (after
Fisher, 1985)
Figure 3.5 . Lorentz and Doppler
line shapes for similar intensity and line width (after Liou,
1980)
Figure 3.6 . The absorption coefficient
for H_{2}O at 250 K as a function
of the wavenumber. The abscissa is given by the fraction of the black
body function between 0 and (in cm^{1}) divided by the integral of between 0 and 2500 cm^{1}.
Figure 3.7 . As in Fig.
3.5 , but for CO_{2.}
Figure 3.8 . As in Fig.
3.6 , but for O_{3}.
Figure 3.9 . Effect of the temperature
of the absorption coefficient for H_{2}O. The upper curve is for
300 K, the lower one for 200 K.
Figure 3.10 . As in Fig.
3.8 , but for CO_{2}.
Figure 3.11 . As in Fig.
3.8 , but for O_{3}.
Figure 3.12 . Absorption coefficients
of H_{2}O, O_{3} and uniformly mixed gases (indexed as
CO_{2}) at different wavelengths and the normalized Planck functions
at 6000 K and 255 K (bottom panel).
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 socalled 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
Rayleigh scattering being conservative ( ), the integral over the phase function is unity, i.e.,

.


It is completely symmetric ( ) (see Fig. 3.13 ),
and the scattering coefficient, i.e., the probability of a photon being
scattered in a volume is for Rayleigh scattering proportional to the density
of air and inversely proportional to the fourth power of the wavelength.
The visible consequence of this spectral dependency is the blueness of the
sky, caused by the preferential scattering of shorter wavelengths in a clear
atmosphere. The rapid decrease of the Rayleigh scattering coefficient with
wavelength simplifies the transfer problem for the Earth's atmosphere, since
Rayleigh scattering can be completely neglected for the terrestrial part
of the spectrum.
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

.


When dealing with the impact of scatterers on fluxes, only
the very few terms necessary for an adequate description of the hemispherically
integrated fluxes are required and one can use some analytic formula such
as HenyeyGreenstein function
where is the asymmetry factor, the first moment of
the expansion

.


When , all the energy is backscattered, all energy appears in the forward direction, and corresponds to an equipartition between the forward and backward
spaces defined by a plane perpendicular to the incoming direction
In largescale 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 GCMtype 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 (largescale condensation or convection),
but the single scattering albedo and asymmetry factor are specified.
Figure 3.13 . Rayleigh phase function
for a molecule located at the origin of the polar coordinate system. The
probability of scattering in a certain direction is proportional to the
length of the arrow for the outgoing beam.
Figure 3.14 . Normalized phase
function in polar coordinates with logarithmic axis for a standard aerosol
model. Note that the scattering is strongly dominated by the forward peak.
Figure 3.15 . The droplet distribution
of eight cloud models (after Stephens, 1979) [top left]
Figure 3.16 . Wavelength dependency
of the total extinction coefficient [top right]
Figure 3.17 . Wavelength dependency
of the single scattering albedo [bottom left]
Figure 3.18 . Wavelength dependency
of the asymmetry factor [bottom right]
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