4
Radiative Transfer1
With Qiang Fu
Department of Atmospheric Sciences
University of Washington
Radiative transfer is a branch of atmospheric physics.
Like a nation composed of many different ethnic
groups, radiative transfer has a rich, but sometimes
confusing language that reflects its diverse heritage,
which derives from quantum physics, astronomy, climatology, and electrical engineering. Solving radiative
transfer problems requires consideration of the
geometry and spectral distribution of radiation, both
of which are straightforward in principle, but can be
quite involved in real world situations.
This chapter introduces the fundamentals of radiative transfer in planetary atmospheres. The first two
sections describe the electromagnetic spectrum and
define the terms that are used to quantify the field of
radiation. The third section reviews physical laws
relating to blackbody radiation and concludes with a
qualitative discussion of the so-called “greenhouse
effect.” The fourth section describes the processes by
which gases and particles absorb and scatter radiation. The fifth section presents an elementary quantitative treatment of radiative transfer in planetary
atmospheres, with subsections on radiative heating
rates and remote sensing. The final section describes
the radiation balance at the top of the atmosphere as
determined from measurements made by satelliteborne sensors.
4.1 The Spectrum of Radiation
Electromagnetic radiation may be viewed as an
ensemble of waves propagating at the speed of light
(c* 2.998 108 m s1 through a vacuum). As for
any wave with a known speed of propagation, frequency , wavelength , and wave number (i.e., the
1
number of waves per unit length in the direction of
propagation) are interdependent. Wave number is the
reciprocal of wavelength
1
(4.1)
c* c*
(4.2)
and
Small variations in the speed of light within air
give rise to mirages, as well as the annoying distortions that limit the resolution of ground-based telescopes, and the difference between the speed of light
in air and water produces a number of interesting
optical phenomena such as rainbows.
Wavelength, frequency, and wave number are used
alternatively in characterizing radiation. Wavelength,
which is in some sense the easiest of these measures
to visualize, is used most widely in elementary texts
on the subject and in communicating between radiation specialists and scientists in other fields; wave
number and frequency tend to be preferred by those
working in the field of radiative transfer because they
are proportional to the quantity of energy carried
by photons, as discussed in Section 4.4. Wavelength is
used exclusively in other chapters of this book, but in
this chapter , , and are used in accordance
with current practice in the field. Here we express
wavelength exclusively in units of micrometers (m),
but in the literature it is also often expressed in
nanometers (nm).
Because radiative transfer in planetary atmospheres
involves an ensemble of waves with a continuum of
We thank Qiang Fu for his guidance in preparing this chapter.
113
Radiative Transfer
114
0.01
0.1
1
microwave
infrared
visible
near infrared
ultraviolet
x-ray
wavelengths and frequencies, the energy that it carries
can be partitioned into the contributions from various
wavelength (or frequency or wave number) bands. For
example, in atmospheric science the term shortwave2
( 4 m) refers to the wavelength band that
carries most of the energy associated with solar radiation and longwave ( 4 m) refers to the band that
encompasses most of the terrestrial (Earth-emitted)
radiation.
In the radiative transfer literature, the spectrum is
typically divided into the regions shown in Fig. 4.1.
The relatively narrow visible region, which extends
from wavelengths of 0.39 to 0.76 m, is defined by
the range of wavelengths that the human eye is capable of sensing. Subranges of the visible region are discernible as colors: violet on the short wavelength end
and red on the long wavelength end. The term monochromatic denotes a single color (i.e., one specific frequency or wavelength).
The visible region of the spectrum is flanked by
ultraviolet (above violet in terms of frequency) and
infrared (below red) regions. The near infrared
region, which extends from the boundary of the visible up to 4 m, is dominated by solar radiation,
whereas the remainder of the infrared region is dominated by terrestrial (i.e., Earth emitted) radiation:
hence, the near infrared region is included in the
term shortwave radiation. Microwave radiation is not
important in the Earth’s energy balance but it is
widely used in remote sensing because it is capable
of penetrating through clouds.
10
100
103
wavelength Fig. 4.1 The electromagnetic spectrum.
2
104
105 μ m
1
10 cm
4.2 Quantitative Description
of Radiation
The energy transferred by electromagnetic radiation in
a specific direction passing through a unit area (normal
to the direction considered) per unit time at a specific
wavelength (or wave number) is called monochromatic intensity (or spectral intensity or monochromatic
radiance) and is denoted by the symbol I (or I).
Monochromatic intensity is expressed in units of watts
per square meter per unit arc of solid angle,3 per unit
wavelength (or per unit wave number or frequency) in
the electromagnetic spectrum.
The integral of the monochromatic intensity over
some finite range of the electromagnetic spectrum is
called the intensity (or radiance) I, which has units of
W m2 sr1
I
2
1
Id 2
1
I d
(4.3)
For quantifying the energy emitted by a laser, the
interval from 1 to 2 (or 1 to 2) is very narrow,
whereas for describing the Earth’s energy balance,
it encompasses the entire electromagnetic spectrum. Separate integrations are often carried out for
the shortwave and longwave parts of the spectrum
corresponding, respectively, to the wavelength
ranges of incoming solar radiation and outgoing terrestrial radiation. Hence, the intensity is the area
under some finite segment of the the spectrum of
monochromatic intensity (i.e., the plot of I as a
function of , or I as a function of , as illustrated
in Fig. 4.2).
Although I and I both bear the name monochromatic intensity, they are expressed in different units.
The shapes of the associated spectra tend to be
somewhat different in appearance, as will be apparent in several of the figures later in this chapter. In
Exercise 4.13, the student is invited to prove that
I 2I
(4.4)
The term shortwave as used in this book is not to be confused with the region of the electromagnetic spectrum exploited in shortwave
radio reception, which involves wavelengths on the order of 100 m, well beyond the range of Fig. 4.1.
3 The unit of solid angle is the dimensionless steradian (denoted by the symbol $) defined as the area subtended by the solid angle
on the unit sphere. Alternatively, on a sphere of radius r, $ r2. Exercise 4.1 shows that a hemisphere corresponds to a solid angle of 2
steradians.
4.2 Quantitative Description of Radiation 115
The“spectrum”
θ
I
or
F
δθ
δφ
1 2
Wavelength Fig. 4.2 The curve represents a hypothetical spectrum of
monochromatic intensity I or monochromatic flux density F
as a function of wavelength . The shaded area represents the
intensity I or flux density F of radiation with wavelengths ranging from 1 to 2.
The monochromatic flux density (or monochromatic irradiance) F is a measure of the rate of energy
transfer per unit area by radiation with a given wavelength through a plane surface with a specified orientation in three-dimensional space. If the radiation
impinges on a plane surface from one direction
(e.g., upon the horizontal plane from above) the flux
density is said to be incident upon that surface, in
which case
F 2
I cos d%
(4.5)
The limit on the bottom of the integral operator indicates that the integration extends over the entire
hemisphere of solid angles lying above the plane, d%
represents an elemental arc of solid angle, and is the
angle between the incident radiation and the direction
normal to dA. The factor cos represents the spreading and resulting dilution of radiation with a slanted
orientation relative to the surface. Monochromatic
flux density F has units of W m2 m1. Analogous
quantities can be defined for the wave number and
frequency spectra.
Exercise 4.1 By means of a formal integration over
solid angle, calculate the arc of solid angle subtended
by the sky when viewed from a point on a horizontal
surface.
Solution: The required integration is performed
using a spherical coordinate system centered on a
point on the surface, with the pole pointing straight
upward toward the zenith, where is called the
Fig. 4.3 Relationship between intensity and flux density. is
the angle between the incident radiation and the normal to
the surface. For the case of radiation incident upon a horizontal surface from above, is called the zenith angle. is
referred to as the azimuth angle.
zenith angle and the azimuth angle, as defined in
Fig. 4.3. The required arc of solid angle is given by
2
d% 2
2
0 0
sin d d 2
2
0
sin d 2
■
Combining (4.3) and (4.5), we obtain an expression for the flux density (or irradiance) of radiation
incident upon a plane surface
F
2
I cos d% 2
1
2
I cos d% d
(4.6)
Flux density, the rate at which radiant energy passes
through a unit area on a plane surface, is expressed
in units of watts per square meter. The following two
exercises illustrate the relation between intensity and
flux density.
Exercise 4.2 The flux density Fs of solar radiation
incident upon a horizontal surface at the top of
the Earth’s atmosphere at zero zenith angle is 1368 W
m2. Estimate the intensity of solar radiation. Assume
that solar radiation is isotropic (i.e., that every point
on the “surface” of the sun emits radiation with the
same intensity in all directions, as indicated in Fig. 4.4).
For reference, the radius of the sun Rs is 7.00 108 m
and the Earth–sun distance d is 1.50 1011 m.
Solution: Let Is be the intensity of solar radiation. If
the solar radiation is isotropic, then from (4.5) the
116
Radiative Transfer
and
δω
Is R
Ir
Fr
r
FR
Er
R
ER
Fig. 4.4 Relationships involving intensity I, flux density F, and
flux E of isotropic radiation emitted from a spherical source
with radius r, indicated by the blue shading, and incident upon
a much larger sphere of radius R, concentric with the source.
Thin arrows denote intensity and thick arrows denote flux density. Fluxes ER Er and intensities IR Ir. Flux density F
decreases with the square of the distance from the source.
flux density of solar radiation at the top of the Earth’s
atmosphere is
Fs %
The intensity of radiation is constant along ray
paths through space and is thus independent of distance from its source, in this case, the sun. The corresponding flux density is directly proportional to the
arc solid angle subtended by the sun, which is
inversely proportional to the square of the distance
from the sun. It follows that flux density varies
inversely with the square of the distance from the
sun, i.e.,
F d 2
Es Fs 4d2 const.
and because the zenith angle, in this case, is zero,
Fs Is %
Exercise 4.3 Radiation is emitted from a plane
surface with a uniform intensity in all directions.
What is the flux density of the emitted radiation?
Solution:
2
I cos d% 2
0
2
0
I cos sin d d (4.8)
2
0
cos sin d
I [(sin2(2) sin2(0)]
I
%
R2s
2 2d2
Although the geometrical setting of this exercise
is quite specific, the result applies generally to
isotropic radiation, as illustrated, for example, in
Exercise 4.31.
■
7.00 10
1.50
10 Performing the integrations over wavelength
and solid angle in reverse order (with solid angle
first) yields the monochromatic flux density F
as an intermediate by-product. The relationships
from which
2
F
2I
The fraction of the hemisphere of solid angle (i.e.,
“the sky”) that is occupied by the sun is the same as
the fraction of the area of the hemisphere of radius d,
centered on the Earth, i.e., occupied by the sun, i.e.,
s
(4.7)
This so-called inverse square law also follows from
the fact that the flux of solar radiation Es (i.e., the
flux density Fs multiplied by the area of spheres, concentric with the sun, through which it passes as it
radiates outward) is independent of distance from
the Sun, i.e.,
Fs Is cos %
Rd ■
Is cos d%
where % is the arc of solid angle subtended by the
sun in the sky. Because % is very small, we can
ignore the variations in cos in the integration. With
this so-called parallel beam approximation, the integral reduces to
% Fs
1368 W m 2
% 6.84 10 5 sr
2.00 107 W m2 sr1
8 2
11
6.84 105 sr
4.3 Blackbody Radiation
117
discussed in this section can be summarized in
terms of an expression for the flux of radiation
emitted by, incident upon, or passing through a
surface A
E
2
A
2
1
I ( , ) d cos d% dA (4.9)
which is expressed in units of watts (W).
4.3 Blackbody Radiation
Fig. 4.5 Radiation entering a cavity with a very small aperture and reflecting off the interior walls. [Adapted from
K. N. Liou, An Introduction to Atmospheric Radiation, Academic
Press, p. 10, Copyright (2002), with permission from Elsevier.]
7000 K
70
B (MW m–2 μ m–1 sr –1)
A blackbody4 is a surface that completely absorbs
all incident radiation. Examples include certain
substances such as coal and a small aperture of a
much larger cavity. The entrances of most caves
appear nearly black, even though the interior walls
may be quite reflective, because only a very small
fraction of the sunlight that enters is reflected back
through the entrance: most of the light that enters
the cave is absorbed in multiple reflections off the
walls. The narrower the entrance and the more
complex the interior geometry of the cave, the
smaller the fraction of the incident light that is
returned back through it, and the blacker the
appearance of the cave when viewed from outside
(Fig. 4.5).
60
50
6000 K
40
30
5000 K
20
10
0
4.3.1 The Planck5 Function
It has been determined experimentally that the intensity of radiation emitted by a blackbody is given by
B(T) c1 5
c
(e 2T 1)
(4.10)
where c1 3.74 1016 W m2 and c2 1.45 102 m K. The quest for a theoretical justification
of this empirical relationship led to the development of the theory of quantum physics. It is
observed and has been verified theoretically that
blackbody radiation is isotropic. When B(T) is
0
0.5
1.0
1.5
2.0
(μ m)
Fig. 4.6 Emission spectra for blackbodies with absolute temperatures as indicated, plotted as a function of wavelength on
a linear scale. The three-dimensional surface formed by the
ensemble of such spectra is the Planck function. [Adapted from
R. G. Fleagle and J. A. Businger, An Introduction to Atmospheric
Physics, Academic Press, p. 137, Copyright (1963) with permission from Elsevier.]
plotted as a function of wavelength on a linear
scale, the resulting spectrum of monochromatic
intensity exhibits the shape shown in Fig. 4.6, with
a sharp short wavelength cutoff, a steep rise to a
4 The term body in this context refers to a coherent mass of material with a uniform temperature and composition. A body may be a
gaseous medium, as long as it has well-defined interfaces with the surrounding objects, media, or vacuum, across which the intensity of the
incident and emitted radiation can be defined. For example, it could be a layer of gas of a specified thickness or the surface of a mass of
solid material.
5 Max Planck (1858–1947) German physicist. Professor of physics at the University of Kiel and University of Berlin. Studied under
Helmholtz and Kirchhoff. Played an important role in the development of quantum theory. Awarded the Nobel Prize in 1918.
118
Radiative Transfer
maximum, and a more gentle drop off toward
longer wavelengths.
4.3.2 Wien’s Displacement Law
Differentiating (4.10) and setting the derivative
equal to zero (Exercise 4.24) yield the wavelength of
peak emission for a blackbody at temperature T
Exercise 4.4 Use Wien’s displacement law to compute the “color temperature” of the sun, for which
the wavelength of maximum solar emission is observed
to be 0.475 m.
Solution:
T
2897
m T
(4.11)
where m is expressed in micrometers and T in degrees
kelvin. On the basis of (4.11), which is known as
Wien’s6 displacement law, it is possible to estimate the
temperature of a radiation source if we know its emission spectrum, as illustrated in the following example.
2897
2897
6100 K.
m
0.475
■
Wien’s displacement law explains why solar radiation is concentrated in the visible (0.4–0.7 m) and
near infrared (0.7–4 m) regions of the spectrum,
whereas radiation emitted by planets and their atmospheres is largely confined to the infrared (4 m), as
shown in the top panel of Fig. 4.7.
B (normalized)
(a) Blackbody
curves
5780 K
Absorption (%)
0.1 0.15 0.2 0.3
100
80
60
40
20
0
100
80
60
40
20
0
0.5
255 K
1
1.5 2
3
5
Wavelength μ m
10
15 20
30
50
100
(b) 11 km
(c) Ground
level
O2
O3
O2 H2O
CO2 CO2 H2O O3
H2O
H2O H2O
N2O CO2
CH4
CH
H2O (rotation)
4
Fig. 4.7 (a) Blackbody spectra representative of the sun (left) and the Earth (right). The wavelength scale is logarithmic rather
than linear as in Fig. 4.6, and the ordinate has been multiplied by wavelength in order to retain the proportionality between
areas under the curve and intensity. In addition, the intensity scales for the two curves have been scaled to make the areas under
the two curves the same; (b) Spectrum of monochromatic absorptivity of the part of the atmosphere that lies above the 11-km
level; (c) spectrum of monochromatic absorptivity of the entire atmosphere. [From R. M. Goody and Y. L. Yung, Atmospheric
Radiation: Theoretical Basis, 2nd ed., Oxford University Press (1995), p. 4. By permission of Oxford University Press, Inc.]
6 Wilhelm Wien (1864–1925) German physicist. Received the Nobel Prize in 1911 for the discovery (in 1893) of the displacement law
named after him. Also made the first rough determination of the wavelength of x-rays.
4.3 Blackbody Radiation
4.3.3 The Stefan–Boltzmann Law
119
Therefore, the equivalent blackbody temperature is
The blackbody flux density (or irradiance) obtained
by integrating the Planck function B over all wavelengths is given by the Stefan7–Boltzmann law
F T4
6.28 107 14
5.67 10 8
(1108 1012)14
5.77 103
5770 K
TE (4.12)
where , the Stefan–Boltzmann constant, is equal
to 5.67 108 W m2 K4. Given measurements of
the flux density F from any black or nonblack body,
(4.12) can be solved for the equivalent blackbody
temperature (also referred to as the effective emission temperature) TE; that is, the temperature
TE that a blackbody would need to be at in order
to emit radiation at the measured rate F. If the
body in question emits as a blackbody, its actual
temperature and its equivalent blackbody temperature will be the same. Applications of the
Stefan–Boltzmann law and the concept of equivalent blackbody temperature are illustrated in the
following exercises.
Exercise 4.5 Calculate the equivalent blackbody
temperature TE of the solar photosphere (i.e., the
outermost visible layer of the sun) based on the
following information. The flux density of solar
radiation reaching the Earth, Fs, is 1368 W m2. The
Earth–sun distance d is 1.50 1011 m and the radius
of the solar photosphere Rs is 7.00 108 m.
■
That this value is slightly lower than the sun’s color
temperature estimated in the previous exercise is evidence that the spectrum of the sun’s emission differs
slightly from the blackbody spectrum prescribed by
Planck’s law (4.10).
Exercise 4.6 Calculate the equivalent blackbody
temperature of the Earth as depicted in Fig. 4.8,
assuming a planetary albedo (i.e., the fraction of the
incident solar radiation that is reflected back into
space without absorption) of 0.30. Assume that the
Earth is in radiative equilibrium; i.e., that it experiences no net energy gain or loss due to radiative
transfer.
Solution: Let Fs be the flux density of solar radiation incident upon the Earth (1368 W m2); FE the
flux density of longwave radiation emitted by
the Earth, RE the radius of the Earth, as shown in
Fig. 4.8; A the planetary albedo of the Earth (0.30);
Solution: We first calculate the flux density at
the top of the layer, making use of the inverse square
law (4.7).
Fphotosphere Fs
Fphotosphere 1,368 2
Rs
d
FS
FE
1.50
7.00 10 1011 2
8
6.28 107 W m 2
From the Stefan–Boltzmann law
TE4 6.28 107 W m 2
Fig. 4.8 Radiation balance of the Earth. Parallel beam solar
radiation incident on the Earth’s orbit, indicated by the
thin red arrows, is intercepted over an area R2E and outgoing
(blackbody) terrestrial radiation, indicated by the wide red
arrows, is emitted over the area 4R2E .
7 Joseph Stefan (1835–1893) Austrian physicist. Professor of physics at the University of Vienna. Originated the theory of the diffusion
of gases as well as carrying out fundamental work on the theory of radiation.
Radiative Transfer
120
and TE the Earth’s equivalent blackbody temperature. From the Stefan–Boltzmann law (4.12)
FE TE4 (1 A)Fs (1 0.30) 1368
4
4
239.4 W m2
Solving for TE, we obtain
TE √
4
FE
239.4
5.67 108
14
255 K
■
which implies an equivalent blackbody temperature
TE 279 K. When the satellite passes into the
Earth’s shadow, it will no longer be in radiative equilibrium. The flux density of solar radiation abruptly
drops to zero while the emitted radiation, which is
determined by the temperature of the satellite,
drops gradually as the satellite cools by emitting
radiation. The instant that the satellite passes into
the shadow, its temperature is still equal to TE, so we
can write
mc
Analogous estimates for some of the other planets in
the solar system are shown in Table 4.1.
Exercise 4.7 A small blackbody satellite is orbiting
the Earth at a distance far enough away so that the flux
density of Earth radiation is negligible, compared to
that of solar radiation. Suppose that the satellite suddenly passes into the Earth’s shadow. At what rate will
it initially cool? The satellite has a mass m 103 kg
and a specific heat c 103 J (kg K)1: it is spherical
with a radius r 1 m, and temperature is uniform over
its surface.
Solution: Consideration of the energy balance, as
in the previous problem, but with an albedo of zero
yields an emission F 342 W m2 from the satellite,
Table 4.1 Flux density of solar radiation Fs, planetary albedo A,
and equivalent blackbody temperature TE of some of the
planets based on the assumption that they are in radiative
equilibrium. Astronomical units are multiples of Earth–sun
distance
Distance
from suna
Fs (W m2)
A
TE (K)
Mercury
0.39
8994
0.06
439
Venus
0.72
2639
0.78
225
Earth
1.00
1368
0.30
255
Mars
1.52
592
0.17
216
Jupiter
5.18
51
0.45
105
Planet
a
Astronomical units are multiples of Earth–Sun distance.
8
dT
4r2TE4
dt
Solving, we obtain dTdt 4.30 103 K s1 or
15.5 K hr1.
■
4.3.4 Radiative Properties of Nonblack
Materials
Unlike blackbodies, which absorb all incident radiation, nonblack bodies such as gaseous media can also
reflect and transmit radiation. However, their behavior can nonetheless be understood by applying the
radiation laws derived for blackbodies. For this purpose it is useful to define the (monochromatic) emissivity , i.e., the ratio of the monochromatic intensity
of the radiation emitted by the body to the corresponding blackbody radiation
I(emitted)
B(T)
(4.13)
and the (monochromatic) absorptivity, reflectivity,
and transmissivity8 the fractions of the incident monochromatic intensity that a body absorbs, reflects, and
transmits, i.e.,
I(absorbed)
I (reflected)
, R I(incident)
I(incident)
and T &(transmitted)
I(incident)
(4.14)
Absorptivity, emissivity, reflectivity, and transmissivity may refer either to intensities or to flux densities, depending on the context in
which they are used. In this subsection they refer to intensities. Also widely used in the literature is the term transmittance, which is synonymous with transmissivity. In contrast, emittance and reflectance sometimes refer to actual intensities or flux densities. Here we use only
the terms ending in ..ivity, which always refer to the dimensionless ratios defined in (4.13) and (4.14).
4.3 Blackbody Radiation
4.3.5 Kirchhoff’s Law
Kirchhoff’s9 law states that
(4.15)
To understand the basis for this relationship, consider
an object (or medium) fully enclosed within a cavity
with interior walls that radiates as a blackbody.
Every point on the surface of the object is exchanging radiation with the walls of the cavity. The object
and the walls of the cavity are in radiative equilibrium. Hence the second law of thermodynamics
requires that they be at the same temperature. The
radiation incident on the object at a particular wavelength coming from a particular direction is given
by the Planck function (4.10). The fraction of that
incident radiation is absorbed. Because the system is
in radiative equilibrium, the same monochromatic
intensity must be returned from the body at each
wavelength and along each ray path. Because the
object and the walls of the cavity are at the same
temperature, it follows that at all wavelengths its
emissivity must be the same as its absorptivity.
Because absorptivity and emissivity are intrinsic
properties of matter, Kirchhoff postulated that the
equality should hold even when the object is
removed from the cavity and placed in a field of radiation that is not isotropic, and with which it is not
necessarily in radiative equilibrium. Additional
rationalizations of Kirchhoff’s law are provided in
Exercises 4.35–4.38.
Kirchhoff’s law is applicable to gases, provided that
the frequency of molecular collisions is much larger
than the frequency with which molecules absorb and
emit radiation in the vicinity of the electromagnetic
spectrum near the wavelength of interest. When this
condition is satisfied, the gas is said to be in local thermodynamic equilibrium (LTE). In the Earth’s atmosphere LTE prevails below altitudes of 60 km.
4.3.6 The Greenhouse Effect
Solar and terrestrial radiation occupy different ranges
of the electromagnetic spectrum that we have been
referring to as shortwave and longwave. For reasons
explained in the next section, water vapor, carbon
dioxide, and other gases whose molecules have electric
121
dipole moments absorb radiation more strongly in the
longwave part of the spectrum occupied by outgoing
terrestrial radiation than the shortwave part occupied
by incoming solar radiation. This distinction is reflected
in the transmissivity spectra of the atmosphere shown
in the lower part of Fig. 4.7. Hence, incoming solar
radiation passes through the atmosphere quite freely,
whereas terrestrial radiation emitted from the Earth’s
surface is absorbed and reemitted in its upward passage through the atmosphere. The following highly
simplified exercise shows how the presence of such
greenhouse gases in a planetary atmosphere tends to
warm the surface of the planet.
Exercise 4.8 The Earth-like planet considered in
Exercise 4.6 has an atmosphere consisting of multiple isothermal layers, each of which is transparent to
shortwave radiation and completely opaque to longwave radiation. The layers and the surface of the
planet are in radiative equilibrium. How is the surface temperature of the planet affected by the presence of this atmosphere?
Solution: Begin by considering an atmosphere composed of a single isothermal layer. Because the layer is
opaque in the longwave part of the spectrum, the
equivalent blackbody temperature of the planet corresponds to the temperature of the atmosphere. Hence,
the atmosphere must emit F units radiation to space
as a blackbody to balance the F units of incoming
solar radiation transmitted downward through the top
of the atmosphere. Because the layer is isothermal, it
also emits F units of radiation in the downward direction. Hence, the downward radiation at the surface of
the planet is F units of incident solar radiation plus F
units of longwave radiation emitted from the atmosphere, a total of 2F units, which must be balanced by
an upward emission of 2F units of longwave radiation
from the surface. Hence, from the Stefan–Boltzmann
law (4.12) the temperature of the surface of the planet
is 303 K, i.e., 48 K higher than it would be in the
absence of an atmosphere, and the temperature of the
atmosphere is the same as the temperature of the surface of the planet calculated in Exercise 4.6.
If a second isothermal, opaque layer is added, as
illustrated in Fig. 4.9, the flux density of downward
radiation incident upon the lower layer will be 2F
9 Gustav Kirchhoff (1824–1887) German physicist; professor at the University of Breslau. In addition to his work in radiation, he made
fundamental discoveries in electricity and spectroscopy. Discovered cesium and iridium.
122
Radiative Transfer
F
F
F
2F
F
F
3F
2F
Fig. 4.9 Radiation balance for a planetary atmosphere that is
transparent to solar radiation and consists of two isothermal
layers that are opaque to planetary radiation. Thin downward
arrows represent the flux of F units of shortwave solar radiation
transmitted downward through the atmosphere. Thicker arrows
represent the emission of longwave radiation from the surface
of the planet and from each of the layers. For radiative equilibrium the net radiation passing through the Earth’s surface and
the top of each of the layers must be equal to zero.
(F units of solar radiation plus F units of longwave
radiation emitted by the upper layer). To balance the
incident radiation, the lower layer must emit 2F units
of longwave radiation. Because the layer is isothermal, it also emits 2F units of radiation in the downward radiation. Hence, the downward radiation at
the surface of the planet is F units of incident solar
radiation plus 2F units of longwave radiation emitted
from the atmosphere, a total of 3F units, which must
be balanced by an upward emission of 3F units of
longwave radiation from the surface.
■
By induction, the aforementioned analysis can be
extended to an N-layer atmosphere. The emissions
from the atmospheric layers, working downward
from the top, are F, 2F, 3F . . . NF and the corresponding radiative equilibrium temperatures are 303,
335 . . . . [(N 1)F ]14 K. The geometric thickness
of opaque layers decreases approximately exponentially as one descends through the atmosphere due to
the increasing density of the absorbing medium with
depth. Hence, the radiative equilibrium lapse rate
steepens with increasing depth. In effect, radiative
transfer becomes less and less efficient at removing
the energy absorbed at the surface of the planet due
to the increasing blocking effect of greenhouse gases.
Once the radiative equilibrium lapse rate exceeds the
adiabatic lapse rate (Eq. 3.53), convection becomes
the primary mode of energy transfer.
That the global mean surface temperature of the
Earth is 289 K rather than the equivalent blackbody
temperature 255 K, as calculated in Exercise 4.6, is
attributable to the greenhouse effect. Were it not for
the upward transfer of latent and sensible heat by
fluid motions within the Earth’s atmosphere, the disparity would be even larger.
To perform more realistic radiative transfer calculations, it will be necessary to consider the dependence of absorptivity upon the wavelength of the
radiation. It is evident from the bottom part of
Fig. 4.7 that the wavelength dependence is quite pronounced, with well-defined absorption bands identified with specific gaseous constituents, interspersed
with windows in which the atmosphere is relatively
transparent. As shown in the next section, the
wavelength dependence of the absorptivity is even
more complicated than the low-resolution absorption
spectra in Fig. 4.7 would lead us to believe.
4.4 Physics of Scattering
and Absorption and Emission
The scattering and absorption of radiation by gas
molecules and aerosols all contribute to the extinction of the solar and terrestrial radiation passing
through the atmosphere. Each of these contributions
is linearly proportional to (1) the intensity of the
radiation at that point along the ray path, (2) the
local concentration of the gases and/or particles
that are responsible for the absorption and scattering, and (3) the effectiveness of the absorbers or
scatterers.
Let us consider the fate of a beam of radiation
passing through an arbitrarily thin layer of the
atmosphere along a specific path, as depicted in Fig.
4.10. For each kind of gas molecule and particle that
the beam encounters, its monochromatic intensity is
decreased by the increment
dI IKN ds
(4.16)
where N is the number of particles per unit volume
of air, is the areal cross section of each particle,
K is the (dimensionless) scattering or absorption
efficiency, and ds is the differential path length
along the ray path of the incident radiation. An
extinction efficiency, which represents the combined
effects of scattering and absorption in depleting the
intensity of radiation passing through the layer, can
be defined in a similar manner. The product KnNs is
called the scattering, absorption, or extinction cross
4.4 Physics of Scattering and Absorption and Emission
section. In the case of a gaseous atmospheric
constituent, it is sometimes convenient to express
the rate of scattering or absorption in the form
104
Raindrops
s
tic
103
p
co
i
K(extinction) K(scattering)
K(absorption)
ig
yle
Solar
radiation
10–3
Terrestrial
radiation
10–1
1
e
att
c
hs
1
10
Ra
Cloud
droplets
g
rin
M
es
ibl
g
gli
ng
eri
tt
ca
Ne
102
103
(μ m)
104
Smoke,
dust, haze
Air
molecules
105
Fig. 4.11 Size parameter x as a function of wavelength ()
of the incident radiation and particle radius r.
basis of theory, as a function of a dimensionless size
parameter
(4.18)
4.4.1 Scattering by Air Molecules
and Particles
At any given place and time, particles including
aerosols with a wide variety of shapes and sizes, as
well as cloud droplets and ice crystals, may be present. Nonetheless it is instructive to consider the
case of scattering by a spherical particle of radius r,
for which the scattering, absorption, or extinction
efficiency K in (4.16) can be prescribed on the
I
θ
G
10
10–2
Drizzle
ng
eri
att 1
c
s
ie x =
m
eo
Weather
radar
(4.17)
where is the density of the air, r is the mass of the
absorbing gas per unit mass of air, and k is the mass
absorption coefficient, which has units of m2 kg1.
In the aforementioned expressions the products
NK and rk are volume scattering, absorption, or
extinction coefficients, depending on the context,
and have units of m1. The contributions of the
various species of gases and particles are additive
(i.e., KN (K)1 N11 (K)2 N22 . . . .), as
are the contributions of scattering and absorption
to the extinction of the incident beam of radiation; i.e.,
etr
102
r (μ m)
dI I rk ds
123
ds = sec θ dz
x
2r
and a complex index of refraction of the particles
(m mr imi), whose real part mr is the ratio of
the speed of light in a vacuum to the speed at
which light travels when it is passing through the
particle. Figure 4.11 shows the range of size parameters for various kinds of particles in the atmosphere and radiation in various wavelength ranges.
For the scattering of radiation in the visible part of
the spectrum, x ranges from much less than 1 for
air molecules to 1 for haze and smoke particles to
1 for raindrops.
Particles with x 1 are relatively ineffective at
scattering radiation. Within this so-called Rayleigh
scattering regime the expression for the scattering
efficiency is of the form
K 4
dz
I – dI
Fig. 4.10 Extinction of incident parallel beam solar radiation as it passes through an infinitesimally thin atmospheric
layer containing absorbing gases and/or aerosols.
(4.19)
(4.20)
and the scattering is divided evenly between the
forward and backward hemispheres, as indicated in
Fig. 4.12a. For values of the size parameter comparable to or greater than 1 the scattered radiation is
directed mainly into the forward hemisphere, as indicated in subsequent panels.
Figure 4.13 shows K as a function of size parameter for particles with mr 1.5 and a range of values
of mi. Consider just the top curve that corresponds
Radiative Transfer
124
(a)
the geometric optics regime, the oscillatory behavior
is less prominent and K 2.
(b)
Incident Beam
Exercise 4.9 Estimate the relative efficiencies with
which red light ( 0.64m) and blue light
( 0.47m) are scattered by air molecules.
(c)
Solution:
3.45
Forward
0.64
K(blue)
K(red)
0.47
Fig. 4.12 Schematic showing the angular distribution of the
radiation at visible (0.5 m) wavelength scattered by spherical
particles with radii of (a) 104 m, (b) 0.1 m, and (c) 1 m.
The forward scattering for the 1-m aerosol is extremely large
and is scaled for presentation purposes. [Adapted from
K. N. Liou, An Introduction to Atmospheric Radiation, Academic
Press, p. 7, Copyright (2002), with permission from Elsevier.]
5
Scattering efficiency K
mi = 0
4
mi = 0.01
3
mi = 0.1
2
4
■
Hence, the preponderance of blue in light scattered by
air molecules, as evidenced by the blueness of the sky
on days when the air is relatively free from aerosols.
Figure 4.14 shows an example of the coloring of
the sky and sunlit objects imparted by Rayleigh scattering. The photograph was taken just after sunrise.
Blue sky is visible overhead, while objects in the
foreground, including the aerosol layer, are illuminated by sunlight in which the shorter wavelengths
(bluer colors) have been depleted by scattering along
its long, oblique path through the atmosphere.
Ground-based weather radars and remote sensing
of rainfall from instruments carried aboard satellites
exploit the size strong dependence of scattering
efficiency K upon size parameter x for microwave
radiation in the 1- to 10-cm wavelength range incident upon clouds with droplet radii on the order of
millimeters. In contrast to infrared radiation, which
mi = 1
1
0
From (4.20)
1
5
10
Size parameter x
50
100
Fig. 4.13 Scattering efficiency K as a function of size
parameter x, plotted on a logarithmic scale, for four different
refractive indices with mr 1.5 and mi ranging from 0 to 1, as
indicated. [From K. N. Liou, An Introduction to Atmospheric
Radiation, Academic Press, p. 191, Copyright (2002), with
permission from Elsevier.]
to mi 0 (no absorption). For 0.1 ' x ( 50, referred
to as the Mie10 scattering regime, K exhibits a
damped oscillatory behavior, with a mean around a
value of 2, and for x ) 50, the range referred to as
Fig. 4.14 Photograph of the Great Wall of China, taken just
after sunrise.
10 Gustav Mie (1868–1957) German physicist. Carried out fundamental studies on the theory of electromagnetic scattering and kinetic
theory.
4.4 Physics of Scattering and Absorption and Emission
is strongly absorbed by clouds, microwave radiation
passes through clouds with droplets ranging up to
hundreds of micrometers in radius with little or no
scattering. Radiation backscattered from the pulsed
radar signal by the larger drops reveals the regions of
heavy precipitation.
Scattering of parallel beam solar radiation and
moonlight gives rise to a number of distinctive optical effects, that include
125
Solar ra
ys
• rainbows (Figs. 4.15 and 4.16) formed by the
refraction and internal reflection of sunlight in
water drops (usually raindrops). The bow
appears as a circle, rarely complete, centered on
the antisolar point, which lies on the line from
the sun passing through the eye of the observer.
The colors of the rainbow from its outer to its
inner circumference are red, orange, yellow,
green, blue, indigo, and violet. Rainbows are
usually seen in falling rain with drop diameters
of a few millimeters. Because the drops are much
larger than the wavelength of the light, the
classical theory of geometric optics provides a
reasonable description for the primary bow.
• bright haloes produced by the refraction of light
by hexagonal, prismshaped ice crystals in high,
thin cirrostratus cloud decks, the most common
of which are at angles of 22° and 46°, as shown in
Fig. 4.17 and the schematic Fig. 4.18.11
Fig. 4.16 A solar ray is refracted and internally reflected
(once) and refracted again by a raindrop to form the (primary) rainbow. The rainbow ray is the brightest and has the
smallest angle of deviation of all the rays that encounter raindrop undergo these optical processes. Like a prism, refraction
by the raindrop disperses visible light into its component
colors forming the rainbow color band. The secondary
rainbow is produced by double reflection within raindrops,
which appears about 8 degrees above the primary rainbow,
with the order of the colors reversed. The rainbow is a mosaic
produced by passage of light through the circular cross
section of myriad raindrops.
Fig. 4.15 Primary rainbow with a weaker secondary rainbow
above it and supernumerary bows below it. [Photograph
courtesy of Joanna Gurstelle.]
Fig. 4.17 Haloes of 22° and 46° (faint) formed in a thin
cloud consisting of ice crystals. [Photograph courtesy of
Alistair Fraser.]
Solar ra
ys
42°
11 The angular radius of the 22° halo is roughly the same as that subtended at the eye by the distance between the top of the thumb and
the little finger when the fingers are spread wide apart and held at arms length. (The reader is warned not to stare at the sun since this can
cause eye damage.)
126
Radiative Transfer
Minimum angle
of deviation 46°
Refracting
angle 90°
46° halo
Cloud of hexagonal ice columns
Light from sun
46°
22°
22° halo
Refracting
angle 60°
Minimum angle
of deviation 22°
Observer
Fig. 4.18
Refraction of light in hexagonal ice crystals to produce the 22° and 46° haloes.
• coronas produced by the diffraction of light in
water droplets in low or (sometimes) middle
cloud decks. They consist of colored rings at
an angular radius of less than 15° from the
light sources, e.g., the sun or moon. If the
cloud droplets are fairly uniform in size,
several sequences of rings may be seen: the
spacing of the rings depends upon droplet
size. In each sequence the inside ring is violet
or blue and the outside ring is red. An example
is shown in Fig. 4.19.
4.4.2 Absorption by Particles
The absorption of radiation by particles is of
interest in its own right, and affects scattering as
well. The theoretical framework discussed in the
previous subsection (commonly referred to as
Mie theory) predicts both the scattering and the
absorption of radiation by homogeneous spherical
particles, where the real part of the refractive
index relates to the scattering and the imaginary
part relates to the absorption. This brief section
mentions just a few of the consequences of absorption. In the plot of K versus x (Fig. 4.13) the presence of absorption tends to damp the oscillatory
behavior in the Mie regime. In the limit of x 1,
Fig. 4.19 A corona around the sun produced by the diffraction of light in cloud droplets. [Photograph courtesy of Harald
Edens.]
the extinction coefficient always approaches 2 but,
in accordance with (4.13) the scattering coefficient,
ranges from as low as 1 in the case of strong absorption to as high as 2 for negligible absorption. In the
longwave part of the electromagnetic spectrum,
cloud droplets absorb radiation so strongly that
even a relatively thin cloud layer of clouds behaves
as a blackbody, absorbing virtually all the radiation
incident from above and below.
4.4 Physics of Scattering and Absorption and Emission
4.4.3 Absorption and Emission
by Gas Molecules
N2, O2, N, O
O2
150
Height (km)
Whenever radiation interacts with matter it is
absorbed, scattered, or emitted in discrete packets
called photons. Each photon contains energy
E hv
127
(4.21)
50
where h is Planck’s constant (6.626 1034 J s). Hence,
the energy carried by a photon is inversely proportional to the wavelength of the radiation.
0
a. Absorption continua
Extreme ultraviolet radiation with wavelengths
(0.1 m, emitted by hot, rarefied gases in the sun’s
outer atmosphere, is sufficiently energetic to strip
electrons from atoms, a process referred to as photoionization. Solar radiation in this wavelength range,
which accounts for only around 3 millionths of the
sun’s total output, is absorbed in the ionosphere, at
altitudes of 90 km and above, giving rise to sufficient
numbers of free electrons to affect the propagation
of radio waves.
Radiation at wavelengths up to 0.24 m is sufficiently energetic to break O2 molecules apart into
oxygen atoms, a process referred to as photodissociation. The oxygen atoms liberated in this reaction
are instrumental in the production of ozone (O3), as
explained in Section 5.7.1. Ozone, in turn, is dissociated by solar radiation with wavelengths extending
up to 0.31 m, almost to the threshold of visible
wavelengths. This reaction absorbs virtually all of
the 2% of the sun’s potentially lethal ultraviolet
radiation. The ranges of heights and wavelengths of
the primary photoionization and photodissociation
reactions in the Earth’s atmosphere are shown in
Fig. 4.20.
Photons that carry sufficient energy to produce
these reactions are absorbed and any excess energy is
imparted to the kinetic energy of the molecules, raising the temperature of the gas. Since the energy
required to liberate electrons and/or break molecular
bonds is very large, the so-called absorption continua
associated with these reactions are confined to the
x-ray and ultraviolet regions of the spectrum. Most of
the solar radiation with wavelengths longer than
0.31 m penetrates to the Earth’s surface.
O3
100
Lyman α
0
0.05
0.1
0.15
0.2
Wavelength (μ m)
0.25
0.3
Fig. 4.20 Depth of penetration of solar ultraviolet radiation
in the Earth’s atmosphere for overhead sun and an average
ozone profile. [Adapted from K. N. Liou, An Introduction to
Atmospheric Radiation, Academic Press, p. 78, Copyright (2002),
with permission from Elsevier.]
b. Absorption lines
Radiation at visible and infrared wavelengths does
not possess sufficient energy to produce photoionization or photodissociation, but under certain conditions appreciable absorption can nonetheless occur.
To understand the processes that are responsible for
absorption at these longer wavelengths, it is necessary to consider other kinds of changes in the state of
a gas molecule. The internal energy of a gas molecule
can be written in the form
E Eo Ev Er Et
(4.22)
where Eo is the energy level of the orbits of the electrons in the atoms, Ev and Er refer to the energy levels corresponding to the vibrational and rotational
state of the molecule, and Et is the translational
energy associated with the random molecular
motions. In discussing the first law of thermodynamics in Chapter 3, we considered only changes Et, but
in dealing with radiative transfer it is necessary to
consider changes in the other components of the
internal energy as well.
Quantum mechanics predicts that only certain
configurations of electron orbits are permitted
within each atom, and only certain vibrational frequencies and amplitudes and only certain rotation
rates are permitted for a given molecular species.
Each possible combination of electron orbits, vibration, and rotation is characterized by its own
128
Radiative Transfer
energy level, which represents the sum of the three
kinds of energy. (The translational component of
the energy is not quantized in this manner.) A molecule may undergo a transition to a higher energy
level by absorbing electromagnetic radiation and it
may drop to a lower level by emitting radiation.
Absorption and emission can occur only in association with discrete changes in energy level E. The
frequency of the absorbed or emitted radiation is
related to the change in energy level through the
relation
E hv
(4.23)
Absorptivity at visible and longer wavelengths can
thus be described in terms of a line spectrum consisting of extremely narrow absorption lines separated
by much wider gaps in which the gas is virtually
transparent to incident radiation.
The changes in state of molecules that give rise to
these absorption lines may involve orbital, vibrational, or rotational transitions or combinations
thereof. Orbital transitions are associated with
absorption lines in the ultraviolet and visible part
of the spectrum; vibrational changes with nearinfrared and infrared wavelengths; and rotational
lines, which involve the smallest changes in energy,
with infrared and microwave radiation. The absorption spectra of the dominant species O2 and N2
exhibit a sparse population of absorption lines
because these molecular species do not possess
an electric dipole, even when they are vibrating.
In contrast, so-called “greenhouse gases” (notably
H2O, CO2, O3, and trace species such as such as
CH4, N2O, CO, and the chlorofluorocarbons)
exhibit myriads of closely spaced absorption lines
in the infrared region of the spectrum that are
due to pure rotational or simultaneous vibrational–
rotational transitions.
energy levels, but this “natural broadening” is
inconsequential in comparison to the broadening
attributable to the motions and collisions of the gas
molecules, i.e.,
• Doppler broadening: the Doppler shifting of
frequencies at which the gas molecules experience
the incident radiation by virtue of their random
motions toward or away from the source of the
radiation, and
• pressure broadening: (also referred to as
collision broadening) associated with
molecular collisions.
The absorption spectra in the vicinity of pressureand Doppler-broadened absorption lines can be
represented by
k Sf( 0)
(4.24)
(4.25)
where
S
0
The absorption lines of molecules are of finite width
due to the inherent uncertainty in quantizing their
k d
is the line intensity, 0 is the wave number on
which the line is centered, and f is the so-called
shape factor or line profile. The shape factor for
Doppler broadening is inferred from the
Maxwell12–Boltzmann distribution of the velocity
of the molecules in a gas, which has the shape of
the familiar Gaussian probability distribution. It is
of the form
f
1
0
exp D√
D
2
(4.26)
where
D c. Broadening of absorption lines
0 2kT
c* m
12
(4.27)
In this expression the so-called half-width of the
line (i.e., the distance between the center of the line
12 James Clerk Maxwell (1831–1879) Scottish physicist. Often rated as second only to Newton in terms of his contributions to physics.
First Cavendish Professor of physics at Cambridge University. He showed that light is an electromagnetic wave, made one of the first color
photographs, and made major contributions to thermodynamics and the kinetic theory of gases.
4.4 Physics of Scattering and Absorption and Emission
f
[( 0)2 2]
p
TN
(4.29)
which is proportional to the frequency of molecular
collisions. The exponent N ranges from 12 to 1
depending on the molecular species.
Shapes of absorption lines of the same strength
and half-width, but broadened by these two distinctly different processes, are contrasted in
Fig. 4.21. The “wings” of the absorption lines shaped
by pressure broadening extend out farther from the
center of the line than those shaped by Doppler
broadening. For a water vapor line at 400 cm1 and
a temperature of 300 K, the Doppler line width is
7 104 cm1. A typical water vapor line width for
air at the same temperature at the Earth’s surface is
100 times wider due to the presence of pressure
broadening.14 Below 20 km, pressure broadening
is the dominant factor in determining the width of
absorption lines, whereas above 50 km, where
molecular collisions are much less frequent,
Doppler broadening is the dominant factor. In the
intermediate layer between 20 and 50 km, the line
shape is a convolution of the Doppler and Lorentz
shapes.
13
Lorentz
(4.28)
In this expression the half-width of the line is determined by
Doppler
kν
and the points at which the amplitude is equal to half
the peak amplitude) is D√ln 2, m is the mass of the
molecule and k is the Boltzmann’s constant (1.381 1023 J K1 molecule1).
The shape factor for pressure broadening, commonly referred to as the Lorentz13 line shape, is
given by
129
–5
–4
–3
–2
–1
0
1
2
3
4
5
(ν − ν0)/α
Fig. 4.21 Contrasting absorption line shapes associated
with Doppler broadening and pressure broadening. Areas
under the two profiles, indicative of the line intensity S, are
the same. [Courtesy of Qiang Fu.]
Laboratory measurements of absorption spectra
exist for only a very limited sampling of pressures
and temperatures. However, through the use of theoretically derived absorption line information, adjusted
empirically to improve the fit with existing measurements, atmospheric physicists and climate modelers
are able to calculate the absorption spectra for each
of the radiatively important atmospheric gases for
any specified thermodynamic conditions.15 An example showing the excellent agreement between
observed and theoretically derived absorption spectra is shown in Fig. 4.22. Note the narrowness of the
lines, even when the effects of Doppler and pressure
broadening are taken into account. The greatest
uncertainties in theoretically derived absorption
spectra are in the so-called “continua,” where
the superposition of the outermost parts of the wings
of many different lines in nearby line clusters produces weak but in some cases significant absorption.
Hendrick Antoon Lorentz (1853–1928) Dutch physicist. Won Nobel prize for physics in 1902 for his theory of electromagnetic
radiation, which gave rise to the special theory of relativity. Refined Maxwell’s theory of electromagnetic radiation so it better explained
the reflection and refraction of light.
14 The dominance of pressure broadening is mainly due to the fact that, for typical temperatures and pressures in the lower atmosphere,
D. The difference in line shape also contributes, but it is of secondary importance.
15 The most comprehensive archive of these theoretically derived absorption line information, the high-resolution transmission molecular absorption (HITRAN) data base, contains absorption lines for many different gases. This data base contains line intensities at reference
temperature, the wave numbers at which the lines are centered, the pressure half-widths at reference temperature and pressure, and the
lower energy levels for over a million absorption lines.
130
Radiative Transfer
beam or “pencil” of radiation has been attenuated
due to absorption and/or scattering and how
much remains undepleted. Integrating (4.17) with
ds sec dz yields
ln I ln I sec z
k rdz
(4.30)
Taking the antilog of both sides we obtain
I & e! sec IT
where
! z
k rdz
(4.31)
(4.32)
and
T e! sec Fig. 4.22 Comparisons of observed and calculated transmissivity spectra. (a) Spectral range 1931 to 1939 cm1 for carbon dioxide and (b) Spectral range 1156 to 1164 cm1 for
ozone and nitrous oxide. The upper plot in each panel is the
observed spectrum and the lower plot is the calculated spectrum. [From R. M. Goody, R. M. and Y. L. Yung, Atmospheric
Radiation, 2nd ed., Oxford University Press (1995), p.120. By
permission of Oxford University press, Inc.]
4.5 Radiative Transfer
in Planetary Atmospheres
4.5.1 Beer’s Law
Equations (4.17) and/or (4.16) may be integrated
from the top of the atmosphere (z ) down to any
level (z) to determine what fraction of the incident
16
(4.33)
is the transmissivity of the layer.
This set of relationships and definitions, collectively referred to here as Beer’s law, but also known
as Bouguer’s law, and Lambert’s law,16,17,18 states
that the monochromatic intensity I decreases
monotonically with path length as the radiation
passes through the layer. The dimensionless quantity !, referred to as the normal optical depth or
optical thickness depending on the context in
which it is used, is a measure of the cumulative
depletion that a beam of radiation directed straight
downward (zenith angle 0) would experience
in passing through the layer. It follows that,
in the absence of scattering, the monochromatic
absorptivity
1 T 1 e! sec (4.34)
approaches unity exponentially with increasing optical depth. Optical depths for the scattering and
extinction of radiation passing through a medium
containing aerosols or cloud droplets can be defined
in a similar manner.
August Beer (1825–1863). German physicist, noted for his work on optics.
Pierre Bouguer (1698–1758). Taught by his father. Awarded the Grand Prix of the Academie des Sciences for studies of naval
architecture, observing the stars at sea, and for observations of the magnetic declination at sea. First to attempt to measure the density
of the Earth, using the deflection of a plumb line due to the attraction of a mountain. Sometimes known as the “father of photometry.”
He compared the brightness of the moon to that of a “standard” candle flame (1725). Bouguer’s law was published in 1729.
18 Johann Heinrich Lambert (1728–1777) Swiss–German mathematician, astronomer, physicist, and philosopher. Son of a tailor,
Lambert was largely self-educated. Proved that is an irrational number (1728). Made the first systematic investigation of hyperbolic
functions. Carried out many investigations of heat and light.
17
4.5 Radiative Transfer in Planetary Atmospheres 131
Solution: The mass of the absorbing gas that the
beam of radiation encounters along its slant path
length is given by
zT
u sec zB
rdz
(4.35)
where zB and zT are the heights of the bottom and top
of the layer. Substituting, sec 2, 0.1 kg m3,
r 1, and a layer thickness of 100 m, we obtain
u 2 0.1 kg m3 100 m
20 kg m2
Since k can be assumed to be uniform within through
the layer, Eq. (4.33) can be rewritten as
T e! eku
0.05
0.1 τ
0.2
∂I
∂z
ρ I
ρ
1.0
2.0
3.0
5.0
Linear scale
Fig. 4.23 Vertical profiles of the monochromatic intensity of
incident radiation, the rate of absorption of incident radiation
per unit height, air density and optical depth, for k and r
independent of height.
Exercise 4.44 at the end of this chapter that they
decrease most rapidly around the level where ! 1,
commonly referred to as the level of unit optical
depth. This result can be understood by considering
the shape of the vertical profile of the absorption
rate dIdz, which is shown in Fig. 4.23 together with
profiles of I and . We recall from (4.17) that if r, the
mixing ratio of the absorbing gas, and k, the mass
absorption coefficient, are both independent
of height,
dI
(I )
dz
and (4.34) as
1 T 1 eku
where
! k sec
zT
zB
rdz ku
(4.36)
is the slant path optical thickness. Substituting for k
and u in the aforementioned equation yields
!
T
I
z (linear scale)
Exercise 4.10 Parallel beam radiation is passing
through a layer 100 m thick, containing an absorbing gas with an average density of 0.1 kg m3.
The beam is directed at an angle of 60° relative
to the normal to the layer. Calculate the optical
thickness, transmissivity, and absorptivity of the
layer at wavelengths 1, 2, and 3, for which the
mass absorption coefficients are 103, 101, and
1 m2 kg1.
1
0.02
0.98
0.02
2
2
0.135
0.865
3
20
2 109
1.00
■
I and T decrease monotonically with increasing
geometric depth in the atmosphere. For downward
directed radiation (sec 1), it is shown in the
The scale for optical depth is shown at the right-hand
side of Fig. 4.23. Well above the level of unit optical
depth, the incoming beam is virtually undepleted, but
the density is so low that there are too few molecules
to produce appreciable amounts of absorption per
unit path length. Well below the level of unit optical
depth, there is no shortage of molecules, but there is
very little radiation left to absorb.
The larger the value of the absorption coefficient k and the larger the secant of the zenith
angle, the smaller the density required to produce
significant amounts of absorption and the higher
the level of unit optical depth. For small values of
k, the radiation may reach the bottom of the
atmosphere long before it reaches the level of unit
optical depth. It is shown in Exercise 4.47 that for
overhead parallel beam radiation incident upon an
optically thick atmosphere, 80% of the energy is
absorbed at levels between ! 0.2 and ! 4.0,
132
Radiative Transfer
which corresponds to a geometric depth of three
scale heights.
The level of slant path unit optical depth is
strongly dependent on the solar zenith angle. It is
4.1
lowest in the atmosphere when the sun is directly
overhead and it rises sharply as the sun drops close
to the horizon. This dependence is exploited in
remote sensing, as discussed in Box 4.1.
Indirect Determination of the Solar Spectrum
Before the advent of satellites, Beer’s law in the
form (4.30) was used to infer the emission spectrum
of the sun on the basis of ground based measurements. Over the course of a single day with clear
sky and good visibility, the incident solar radiation
was measured at different time of day, yielding
data like that shown in Fig 4.24. Over the relatively
short interval in which the measurements were
taken, I and the normal optical depth are
assumed to be constant so that ln I is proportional
to sec . This assumption is borne out by the almost
perfect alignment of the data points in Fig. 4.24.
= 0.58 μ m
Measurements are inherently limited to the range
sec 1, but the linear fit to the data can be
extrapolated back to sec 0 to estimate ln I.
An instrument called the sunphotometer is used
to make instantaneous measurements of the optical thickness due to scattering and absorption by
aerosols (called the aerosol optical depth) of the
radiation from the sun (or moon) in its transit
through the atmosphere to the location of the
measurement.19 For example, if the sunphotometer
is located on the ground, it measures the total column aerosol optical depth. A sunphotometer
uses photodiodes and appropriate narrow-band
interference filters to measure I1 at zenith angle 1
and I2 at zenith angle 2. It follows from (4.30)
that
ln I
ln
= 0.40 μ m
0
2
4
6
sec θ
8
10
Fig. 4.24 Monochromatic intensity of solar radiation
measured at the ground as a function of solar zenith angle
under clear, stable conditions at Tucson, Arizona on 12
December 1970. [From J. Appl. Meteor., 12, 376 (1973).]
I1
! (sec 2 sec 1)
I2
(4.37)
from which ! can be derived. To isolate attenuation due to aerosols alone, 1 and 2 must be
chosen so as not to coincide with molecular
atmospheric absorption lines or bands and the
influence of Rayleigh scattering by air molecules
must be taken into account. Because particles of
different sizes attenuate light differently at different wavelengths, variations of ! with can be
used to infer particle size spectra.
Sunphotometers can be calibrated using socalled Langley20 plots like the one in Fig. 4.24,
Continued on next page
19 Bouguer made visual estimates of the diminution of moonlight passing through the atmosphere. His measurements, made in 1725 in
Britanny, showed much cleaner air than now.
20 Samuel Pierpont Langley (1834–1906) American astronomer, physicist, and aeronautics pioneer. Built the first successful
heavier-than-air, flying machine. This unmanned machine, weighing 9.7 kg and propelled by a steam engine, flew 1280 m over the
Potomac River in 1896. His chief scientific interest was solar activity and its effects on the weather. Invented the bolometer to study
radiation, into the infrared, from the sun. First to provide a clear explanation of how birds soar and glide without moving their wings.
His first manned aircraft, catapulted off a houseboat in 1903, never flew but crashed “like a handful of wet mortar” into the Potomac.
Nine days later the Wright brothers successfully flew the first manned aircraft. Langley died 3 years later, some say broken by the
ridicule that the press treated his flying attempts. The NASA Langley Research Center is named after him.
4.5 Radiative Transfer in Planetary Atmospheres 133
4.1 Continued
In this subsection, Beer’s law and the concept of
optical depth were derived from equations in which
height is used as a vertical coordinate. If atmospheric
pressure or some function of pressure had been used
as a vertical coordinate, the profiles in Fig. 4.23
would be quite different in appearance. For example,
in Exercise 4.45 the reader is invited to show that in
an isothermal atmosphere, the rate of absorption per
unit increment of pressure (i.e., per unit mass) is
largest, not at the level of unit depth, but at the top
of the atmosphere, where the optical depth is much
less than 1 and the incident radiation is virtually
undepleted. Hence, Beer’s law should not be interpreted as indicating that layers of the atmosphere
that are far removed from the level of unit optical
depth are unaffected by the incident radiation.
4.5.2 Reflection and Absorption by a Layer
of the Atmosphere
The conservation of energy requires that for radiation
incident on a layer of aerosols or clouds
f Rf Tf 1
(4.38)
A
4
B
4
0.380 μ m
0.606 μ m
1.558 μ m
3
3
2
0
0.2
0.4
0.6
Aerosol
optical depth
Altitude [km]
from which both I and ! can be determined.
The derived values of I can then be compared
with tabulated values of solar radiation at the top
of the atmosphere at each filter wavelength to
estimate the attenuation. Such calibrations are
best done in an atmosphere that is temporally
invariant and horizontally homogeneous (within
50 km of the observer), because the Langley
method assumes these conditions hold during
measurements at different zenith angles. A
favorite location for calibrating sunphotometers
is atop Mauna Loa, Hawaii.
Networks of sunphotometers are deployed
worldwide for monitoring atmospheric aerosol.
Sunphotometers may also be mounted on aircraft, as was done in acquiring data for Fig. 4.25;
in this case, the aerosol optical depths of layers of
the atmosphere can be determined by subtracting the column optical depth at the top of a layer
from that at the base of the layer. Small handheld sunphotometers are also available.
2
1
1
0
0.8
0
0
0.1
0.2
0.3
0.4
Aerosol
extinction [k m–1]
Fig. 4.25 Aerosol optical depth and aerosol extinction
coefficient measured with an airborne sunphotometer south
of Korea. The enhanced extinction coefficients between
2.5–3 km were due to dust from Asia. [Courtesy of
Gregory Schmidt, NASA–Ames Research Center.]
where f, Rf, and Tf are the flux absorptivity, flux
reflectivity, and flux transmissivity of the layer, i.e.,
the fractions of the incident flux density of solar radiation that are absorbed, transmitted, and reflected.
The concept of scattering and absorption efficiencies was introduced in the previous section. The combined effects of scattering and absorption in reducing
the intensity are referred to as extinction, as defined
by (4.18). The incident radiation may be scattered
more than once in its passage through a layer, with
each successive scattering event increasing the diversity of ray paths. In the absence of absorption, what
started out as parallel beam radiation would (after a
sufficient number of scattering events) be converted
to isotropic radiation. So-called multiple scattering
also greatly increases the path length of the incident
radiation in its passage through the layer.
Three basic parameters are used to characterize
the optical properties of aerosols, cloud droplets, and
ice crystals:
• The volume extinction coefficient NK
(extinction), a measure of the overall importance
of the particles in removing radiation of the
incident beam.
134
Radiative Transfer
• The single scattering albedo, defined as
%0() K(scattering)
(4.39)
K(scattering) K(absorption)
a measure of the relative importance of
scattering and absorption. Values of single
scattering albedo range from 1.0 for
nonabsorbing particles to below 0.5 for strongly
absorbing particles.
• The asymmetry parameter
( ) 1
2
1
1
P(cos ) cos dcos (4.40)
where P(cos ) is the normalized angular
distribution of the scattered radiation (the
so-called scattering phase function) and is the
angle between the incident radiation and the
scattered radiation. The asymmetry factor is zero
for isotropic radiation and ranges from 1 to
1, with positive values indicative of a
predominance of forward scattering. It is evident
from Fig. 4.12 that forward scattering tends to
predominate: typical values of g range from 0.5
for aerosols to 0.80 for ice crystals and 0.85 for
cloud droplets. Because of the predominance of
forward scattering, a cloud of a given optical
depth consisting of spherical cloud droplets
reflects much less solar radiation back to space
than a “cloud” of isotropic scatterers having the
same optical depth.
Whether the presence of a particular kind of
aerosol serves to increase or decrease the planetary
albedo depends on the interplay among all three
parameters just discussed. If the other parameters
are held fixed, the higher the value of the single scattering albedo, the higher the fraction of the incident
radiation that will be backscattered to space.
However, even if the single scattering albedo is close
to unity, the enhancement of the Earth’s albedo due
to the presence of the aerosol layer may be quite
small if the sun is nearly overhead and the scattering
is strongly biased toward the forward direction.
However, if the optical thickness of the aerosol layer
is sufficiently large, the incident radiation will be subject to multiple scattering events, each of which
increases the fraction of the incident radiation that is
reflected back to space. Whether a layer of aerosols
contributes to or detracts from the planetary albedo
also depends on the albedo of the underlying surface
or cloud layer. If the underlying surface were completely black, then any backscattering whatsoever
would contribute to increasing the planetary albedo,
and if the underlying surface were white, any
absorption whatsoever would detract from the
albedo.
The optical properties of cloud layers are determined by similar considerations, but the geometry of
the radiation is quite different because of the much
larger volume scattering coefficients. Even though
the scattering of radiation by spherical cloud droplets
is predominantly in the forward direction, multiple
scattering of parallel beam solar radiation yields a
distribution of intensity that is much more isotropic
than the distribution in cloud-free air. This is why
clouds are the dominant contributor to the planetary
albedo. Due to the multiplicity of scattering events
within clouds, even minute concentrations of black
carbon and other absorbing substances in the
droplets can produce appreciable amounts of absorption, reducing the albedo of the cloud layer and serving as a heat source within the cloud layer, which
may cause the cloud to evaporate. The penetration of
sunlight into a deep cloud layer decreases with zenith
angle and the cloud albedo increases with zenith
angle.
4.5.3 Absorption and Emission of Infrared
Radiation
The two previous subsections dealt with the scattering and absorption of radiation in planetary atmospheres in the absence of emission. The relationships
discussed in those subsections are applicable to the
transfer of solar radiation through planetary atmospheres. The remainder of this section is concerned
with the absorption and emission of infrared radiation in the absence of scattering. This simplified
treatment is justified by the fact that the wavelength
of infrared radiation is very long in comparison to
the circumference of the air molecules so the scattering efficiency is negligible.
a. Schwarzschild’s equation
First we will derive the equation that governs the
transfer of infrared radiation through a gaseous
medium. Rewriting (4.17), the rate of change of
the monochromatic intensity of outgoing terrestrial
4.5 Radiative Transfer in Planetary Atmospheres 135
radiation along the path length ds due to the absorption within the layer is
dI(absorption) Ik rds I
where is the absorptivity of the layer. The corresponding rate of change due to the emission of
radiation is
dI(emission) B(T )
Invoking Kirchhoff’s law (4.15) and summing the two
expressions, we obtain Schwarzschild’s21 equation
dI (I B(T))k rds
(4.41)
By analogy with Beer’s law [Eqs. (4.30)–(4.33)], it
follows that as radiation passes through an isothermal layer, its monochromatic intensity exponentially approaches that of blackbody radiation
corresponding to the temperature of the layer. By
the time the radiation has passed through an optical thickness of 1, the departure sB(T) Is has
decreased by the factor 1e. It can also be argued,
on the basis of Exercise (4.44), that the monochromatic intensity of the radiation emitted to space is
emitted from levels of the atmosphere near the
level of unit optical depth. For ! 1, the emissivity is so small that the emission from the atmosphere is negligible. For ! 1, the transmissivity of
the overlying layer is so small that only a minute
fraction of the monochromatic intensity emitted
from these depths escapes from the top of the
atmosphere.
As in the case of absorbed solar radiation, optical
depth is strongly dependent on zenith angle. This
dependence is exploited in the design of satelliteborne limb scanning radiometers, which monitor
radiation emitted along very long, oblique path
lengths through the atmosphere for which the level
of unit optical depth is encountered at very low air
density. With these instruments it is possible to sample the radiation emitted from much higher levels in
the atmosphere than is possible with nadir (directly
downward) viewing instruments.
21
The level of unit optical depth of emitted radiation
is strongly dependent upon the wavelength of the
radiation. At the centers of absorption lines it is
encountered much higher in the atmosphere than in
the gaps between the lines. In the gaps between the
major absorption bands the atmosphere absorbs and
emits radiation so weakly that even at sea level ! is
less than 1. Within these so-called windows of the
electromagnetic spectrum, an appreciable fraction of
the radiation emitted from the Earth’s surface
escapes directly to space without absorption during
its passage through the atmosphere. The level of unit
optical depth is also dependent upon the vertical
profiles of the concentrations r of the various greenhouse gases. For example, throughout much of the
infrared region of the spectrum, the level of unit
optical depth is encountered near 300 hPa, which
corresponds to the top of the layer in which sufficient
water vapor is present to render even relatively thin
layers of the atmosphere opaque.
The integral form of Schwarzschild’s equation is
obtained by integrating (4.41) along a ray path from
0 to s1, depicted in Fig. 4.26, as shown in Exercise
4.51. The monochromatic intensity of the radiation
reaching s1 is
I(s1) I0e!(s1, 0)
s1
0
krB[T(s)]e!(s1, s)ds (4.42)
The first term on the right-hand side represents the
monochromatic intensity from s 0 that reaches s1,
s
s 1,
τ (
)
s1
s
0
Fig. 4.26 Radiation passing through an absorbing medium
along a ray path from 0 to S1. The ray path may be directed
either upward or downward. [From K. N. Liou, An Introduction to
Atmospheric Radiation, Academic Press, p. 30, Copyright (2002),
with permission from Elsevier.]
Karl Schwarzschild (1873–1916) German astronomer and physicist. Held positions at the Universites of Göttingen and Potsdam and
in the Berlin Academy. Used photogrammetric methods to document emissions from stars and correctly attributed changes in emission of
variable stars to changes in their temperature. Formulated the first exact solutions to Einstein’s gravitational equations. Although he was
not convinced of its physical relevance at the time, his work provided the theoretical basis for understanding black holes.
136
Radiative Transfer
accounting for the depletion along the way, and the
second term represents the monochromatic intensity
of the radiation emitted by the gas along the path
from s 0 to s s1 that reaches s1. The factor e!
in both terms represents the transmissivity of the
radiation along the path from its source to its destination: in the first term the path extends from 0 to s1,
whereas in the second term it extends from s to s1.
b. The plane-parallel approximation
Many atmospheric radiative transfer calculations
can be simplified through use of the plane-parallel
approximation in which temperature and the densities of the various atmospheric constituents are
assumed to be functions of height (or pressure) only.
With this simplification, the flux density passing
through a given atmospheric level is given by
c
Fb
(!) c
Ib
(! , cos ) cos d%
2
(4.43)
where ! , the normal optical depth as defined in
(4.32), is used as a vertical coordinate and the arrows
(pq) denote that both upward and downward fluxes
are taken into account. Since this section focuses on
infrared radiative transfer, we use wave number ()
notation to be consistent with the literature in this
subfield. Integrating over azimuth angle and denoting cos by , we obtain
c
Fb
(!) 2
1
0
c
Ib
(! , ) d
(4.44)
The monochromatic intensity in this expression may
be broken down into three components:
• Upward emission from the Earth’s surface that
reaches level ! without being absorbed.
• Upward emission from the underlying
atmospheric layer.
• Downward emission from the overlying layer.
The expressions used in evaluating these terms are
analogous to Beer’s law, with the intensity transmissivity T replaced by the flux transmissivity
Tf 2
22
1
0
e! d
(4.45)
For many purposes it is sufficient to estimate the flux
transmissivity from the approximate formula
Tf e! (4.46)
where the “average” or “effective zenith angle” is
1
sec 53 1.66
(4.47)
In other words, the flux transmissivity of a layer is
equivalent to the intensity transmissivity of parallel
beam radiation passing through it with a zenith
angle 53°. The factor 1 is widely used in
radiative transfer calculations and is referred to as
the diffusivity factor. Applying the concept of a
diffusivity factor (4.44) can be integrated over to
obtain
c
bc
Fb
(!) I (! , )
(4.48)
c. Integration over wave number22
To calculate the flux density F, it is necessary to integrate the monochromatic flux density F over wave
number, where the integrand involves the product of
the slowly varying Planck function B(T) and a rapidly varying flux transmissivity Tf . For this purpose it
is convenient to separate the smooth B dependence
from the much more complex and Tf dependences
by breaking up the wave number spectrum into intervals narrow enough so that B(T) within each
interval can be regarded as a function of T only.
Hence, B may be taken outside the integral so that
the integrands reduce to expressions for the flux
transmissivities Tf of various layers of the atmosphere. These wave number-integrated flux transmissivities T for the respective wave number intervals
are the basic building blocks in infrared radiative
transfer calculations.
Integration of equations such as (4.44) over wave
number, when performed in the conventional manner,
requires resolution of myriads of absorption lines with
half-widths ranging from 101 cm1 near the Earth’s
surface to 103 cm1 in the upper atmosphere. A
single line-by-line integration over the entire infrared
This subsection consists of more advanced material that can be skipped without loss of continuity.
4.5 Radiative Transfer in Planetary Atmospheres 137
region of the spectrum requires a summation over
roughly a million points. The transmissivity T needs
to be calculated for each infinitesimal wavelength
interval d, and these values then need to be summed.
Here we describe an alternative method of performing the integration in which this computationally intensive summation over wave number is
greatly simplified. In effect, the same summation is
performed, but instead of adding the individual values of T in order of increasing wave number, they
are ordered in terms of increasing value of k, i.e.,
T 1
ek udv 1
0
ek()ud
(4.49)
In this expression (k) is the cumulative probability
density function; i.e., the fraction of the individual
values of k that lie within the broad frequency interval with numerical values smaller than k.23 Unlike
k, k() is a monotonic, smooth function, which can
be inverted, as illustrated in Fig. 4.27.
The integration is performed by grouping the
ranked data into M bins, each spanning width .
The transmissivity is estimated from the summation
*
T ek(j)uj
Because the function (k) is monotonic and very
smooth, it can be closely approximated by the discrete function j(k) with only 10 bins, thereby
enabling a reduction of several orders of magnitude
in the number of calculations required to evaluate
the transmissivity.
As noted in Section 4.4.3, k is a function of temperature and pressure, both of which vary along the path
of the radiation. This dependence can be accounted
for by assuming that within each spectral interval 1
ek dzdv 1
0
ek(, p,T)dzd (4.51)
2.0
Absorption coefficient (log k ν)
Absorption coefficient (log k ν)
that is, may replace as an independent variable in
the transmittance calculations. If the wave number
spectrum k is independent of height within the layer
for which the integration is performed, then (4.51) is
exact to within the degree of approximation inherent
in estimating the integral, and the layer is said to be
homogeneous. If k varies significantly with height in
2.0
1.0
0
–1.0
1.0
0
–1.0
–2.0
–2.0
980
(4.50)
j1
1000
1020
1040
1060
Wavenumber (cm–1)
1080
1100
0
.1
.2
.3
.4
.5
.6 .7
Cumulative probability g
.8
.9
1.0
Fig. 4.27 Illustration of the use of the k distribution method for integrating over wave number. (Left) Log k [in units of
(cm atm)1] plotted as a function of wave number. (The log scale is used to capture the wide range of values of k.) (Right) on the x axis plotted as a function of log k on the y axis. For example, within the frequency range shown in the left panel, k is
less than 1 (i.e., log 0) over 70% of the interval . Hence (reading from the scale in the right-hand panel) (k 1) 0.7.
[Courtesy of Qiang Fu.]
23 The derivative of (k) with respect to k (represented by the slope of g plotted as a function of k) is the corresponding probability
density function h(k) within the broad frequency interval .
138
Radiative Transfer
response to vertical gradients of temperature and
pressure within the layer, then (4.51) is only an
approximation, but it has proven to be a very good
one because the dependence of k on T and p tends
to be quite similar in different wavelength bands d.
When applied to such a vertically inhomogeneous
layer, this method of estimating the transmissivity is
thus referred to as the correlated k method.
4.5.4 Vertical Profiles of Radiative
Heating Rate
(a)
The radiatively induced time rate of change of
temperature due to the absorption or emission of
radiation within an atmospheric layer is given by
dF(z)
dT
cp dt
dz
(4.52)
where F Fq Fp is the net flux and is the total
density of the air, including the radiatively inactive
constituents. The contribution per unit wave number
interval at wave number is
dTdt 1 dF (z)
cp dz
1 d
cp dz
1
cp
2
cp
(4.53)
I d%
4
2
cp
1
p
k r (I B)d
(4.54)
Here we describe an approximation of (4.54) that is
widely used in estimating infrared radiative heating
rates.
A thin, isothermal layer of air cools by emitting
infrared radiation to space, from which it receives no
(infrared) radiation in return. Such a layer also
exchanges radiation with the layers above and below
it, and it may exchange radiation with the Earth’s surface as well, as illustrated in Fig. 4.28. If the lapse rate
24
dTdt 2c k rB (z)e
dT
dt
dI
d
1 ds
1
were isothermal, these two-way exchanges would
exactly cancel and the only net effect of infrared
radiative transfer would be cooling to space. In the
real atmosphere there often exists a sufficient degree
of cancellation in these exchanges that they can be
ignored in calculating the flux of radiation emitted
from the layer. With this assumption (4.54) reduces to
0
! d
(4.55)
which can be integrated over solid angle (as shown in
Exercise 4.53 at the end of the chapter) to obtain
1
1
Fig. 4.28 Radiation geometry in cooling to space. (a) The
complete longwave radiative balance for the layer in question
(not drawn to scale) and (b) the simplified balance based on
the assumption of cooling to space.
dI
d%
4 ds
(b)
where cos and ds dz. Substituting for dIds
from Schwarzschild’s equation (4.41), we obtain
dT
dt
space
The effects of tropospheric ozone are not included in this plot.
e! k rB (z)
cp
(4.56)
where 1.66 is the diffusivity factor, as defined in
Eqs. (4.46) and (4.47). Based on the form of (4.56), it
can be inferred that cp(dTdt) is greatest at and
near the level where !v , which corresponds to the
level of unit optical depth for flux density. Using this
so-called cooling to space approximation, it is possible
to estimate the radiative heating rates for water vapor
and carbon dioxide surprisingly well, without the
need for more detailed radiative transfer calculations.
Vertical profiles of radiative heating rates for the
atmosphere’s three most important radiatively active
greenhouse gases, H2O, CO2, and O3, are shown in
Fig. 4.29. In the troposphere, all three constituents
produce radiative cooling in the longwave part of the
spectrum.24 Water vapor is the dominant contributor,
4.5 Radiative Transfer in Planetary Atmospheres 139
2.3
Shortwave
Longwave
40
30
20
100
longwave shortwave
1000
–4
–2
0
2
4
Rate of Temperature Change (°C/day)
Height (km)
Pressure (hPa)
10
10
CO2
H2O
O3
6
Fig. 4.29 Vertical profiles of the time rate of change of
temperature due to the absorption of solar radiation (solid
curves) and the transfer of infrared radiation (dashed
curves) by water vapor (blue), carbon dioxide (black), and
ozone (red). The heavy black solid curve represents the combined effects of the three gases. [Adapted from S. Manabe
and R. F. Strickler, J. Atmos. Sci., 21, p. 373 (1964).]
but its influence decreases with height in parallel
with the decline in water vapor mixing ratio. The
cooling by tropospheric water vapor is partially offset by the absorption of solar radiation at nearinfrared wavelengths by water vapor molecules.
However, the troposphere experiences net radiative
cooling due to the presence of greenhouse gases.
In contrast to the troposphere, the stratosphere is
very close to radiative equilibrium (i.e., a net radiative heating rate of zero). Longwave cooling to space
by CO2, H2O, and O3 almost exactly balances the
radiative heating due to the absorption of solar radiation in the ultraviolet region of the spectrum by
ozone molecules. The most important contributor to
the longwave cooling at stratospheric levels is CO2.
The vertical distribution of heating in cloud
layers is depicted schematically in Fig. 4.30. During
the daytime, heating rates due to the absorption of
solar radiation by ice crystals and cloud droplets
range from a few °C day1 in high cirrostratus
cloud layers up to a few tens of °C day1 near the
tops of dense stratus cloud layers. The emission of
infrared radiation from the tops of low and middle
cloud decks results in cooling rates ranging up to
50 °C day1 averaged over the 24-h day. If the base
of the cloud layer is much colder than the Earth’s
Fig. 4.30 Schematic of vertical profiles of heating in cloud
layers at various heights in the atmosphere as indicated.
Orange shading indicates warming and blue shading indicates
cooling. Effects of shortwave radiation are represented on the
left, and effects of longwave radiation on the right.
surface (e.g., as in middle and high cloud decks in
the tropics), the infrared radiation emitted from the
Earth’s surface and absorbed near the base of the
layer can produce substantial heating as well. Hence,
the overall effect of infrared radiative transfer is to
increase the lapse rates within cloud layers, promoting convection. During the daytime this tendency is
counteracted by the shortwave heating near the top
of the cloud layer.
4.5.5 Passive Remote Sensing by Satellites
Monitoring of radiation emitted by and reflected
from the Earth system by satellite-borne radiometers provides a wealth of information on weather
and climate. Fields that are currently routinely monitored from space include temperature, cloud cover,
cloud droplet concentrations and sizes, rainfall
rates, humidity, surface wind speed and direction,
concentrations of trace constituents and aerosols,
and lightning. Many of these applications are illustrated in the figures that appear in the various chapters of this book. This section focuses on just a few
of the many applications of remote sensing in
atmospheric science.
a. Satellite imagery
Many of the figures in this book are based on high
(horizontal)-resolution satellite imagery in one of
three discrete wavelength bands (or channels) identified in the bottom part of Fig. 4.6.
Radiative Transfer
• Visible imagery is based on reflected solar
radiation and is therefore available only during
the daylight hours. Because the atmosphere’s
gaseous constituents are nearly transparent at
these wavelengths, visible imagery is used mainly
for mapping the distributions of clouds and
aerosols.
• Infrared imagery corresponds to a spectral
window at a wavelength of 10.7 m, in which
radiation emitted from the Earth’s surface and
cloud tops penetrates through cloud-free air with
little absorption. Intensities in this channel (or
radiances, as they are more commonly referred
to in the remote sensing literature) are indicative
of the temperatures of the surfaces from which
the radiation is emitted as inferred from the
Planck function (4.10). In contrast to visible
imagery, high clouds are usually clearly
distinguishable from low clouds in infrared
imagery by virtue of their lower temperatures.25
• Water vapor imagery exploits a region of the
spectrum that encompasses a dense complex of
absorption lines associated with
vibrational–rotational transitions of water vapor
molecules near 6.7 m. In regions that are free
of high clouds, the emission in this channel is
largely determined by the vertical profile of
humidity in the middle or upper troposphere,
at the level where the infrared cooling rate in
Fig. 4.29 drops off sharply with height. The more
moist the air at these levels, the higher (and
colder) the level of unit optical depth and the
weaker the emitted radiation.26
b. Remote temperature sensing
By comparing the monochromatic radiances in a number of different channels in which the atmosphere
exhibits different absorptivities in the infrared or
microwave regions of the spectrum, it is possible to
make inferences about the distribution of temperature
along the path of the emitted radiation. With the
assumption of plane parallel radiation, the radiance
emitted by gas molecules along a slanting path through
the atmosphere can be interpreted in terms of a local
25
vertical temperature profile or “sounding.” The physical basis for remote temperature sensing is that (1)
most of the radiation reaching the satellite in any given
channel is emitted from near the level of unit optical
depth for that channel and (2) wave number ranges
with higher absorptivities are generally associated with
higher levels of unit optical depth, and vice versa.
Figure 4.31 gives a qualitative impression of how
remote temperature sensing works. The smooth
curves are radiance (or intensity) spectra as inferred
from the Planck function (4.10) for various blackbody temperatures, plotted as a function of wave
number. The jagged curve is an example of the emission spectrum along a cloud-free path through the
atmosphere, as sensed by a satellite-borne instrument called an infrared interferometer spectrometer.
The monochromatic radiance at any given wave
number can be identified with an equivalent blackbody temperature for that wave number simply by
noting which of the Planck function spectra B(, T)
passes through that point on the plot. These wavelength-dependent equivalent blackbody temperatures are referred to as brightness temperatures.
Intensity (W m–2 sr –1/cm–1)
140
0.20
0.16
300 K
0.12
275
0.08
0.04
0
200
175
0
200
∞
50
400
600
20
250
225
800 1000 1200 1400 1600 1800 2000
Wavenumber (cm–1)
12.5 10
8
Wavelength (μ m)
5
Fig. 4.31 Monochromatic intensity of radiation emitted
from a point on the Earth measured by an infrared interferometer spectrometer carried aboard a spacecraft. The gray curves
are blackbody spectra computed from Eq. (4.10). The monochromatic intensity of the radiation at any given wavelength is
indicative of the temperature of the layer of the atmosphere
from which that radiation is emitted (i.e., the level of unit
optical depth for that wavelength), which can be inferred
from the blackbody spectra. [Adapted from K. N. Liou, An
Introduction to Atmospheric Radiation, Academic Press, p. 117,
Copyright (2002), with permission from Elsevier.]
Higher radiances, indicative of higher equivalent blackbody temperatures, are often rendered in darker shades of gray so that high
clouds appear brighter than low clouds.
26 As in infrared imagery, higher radiances in the water vapor channel are rendered in darker shades of gray, which, in this case, should
be interpreted as a drier (more transparent) upper troposphere.
4.5 Radiative Transfer in Planetary Atmospheres 141
The highest brightness temperature for the spectrum in Fig. 4.31 is just above 290 K, which corresponds closely to the temperature of the Earth’s
surface at the particular place and time that the
sounding was taken. This maximum value is realized
within a broad “window” covering most of the range
from 800 to 1200 cm1; the same window in the
infrared region of the spectrum that was mentioned
in the discussion of Fig. 4.7. This feature is revealed
more clearly by the more finely resolved emission
spectrum shown in Fig. 4.32, in which the ordinate is
brightness temperature. Values as low as 220 K in
Fig. 4.31 and 207 K in Fig. 4.32 are realized near the
center of a broad CO2 absorption band extending
from around 570 to 770 cm1. The radiation in much
of this range is emitted from around the tropopause
level. Intermediate values of brightness temperature,
observable at various points along the spectrum, tend
to be associated with intermediate absorptivities for
which the levels of unit optical depth are encountered in the troposphere.
Throughout most of the emission spectrum shown
in Figs. 4.31 and 4.32 absorption bands appear
inverted; i.e., higher atmospheric absorptivity is associated with lower brightness temperature, indicative of
temperatures decreasing with height. A notable exception is the spike at 667 cm1, which corresponds to a
Brightness temperature (K)
300
CFC11 CFC12
CO2
H2O O3
band made up of particularly strong CO2 absorption
lines. This feature is more clearly evident near the left
end of the more finely resolved spectrum in Fig. 4.32.
Within this narrow range of the spectrum the radiation reaching the satellite is emitted from levels high
in the stratosphere, where the air is substantially
warmer than at the tropopause level. Such upward
pointing spikes in emission spectra of upwelling radiation are indicative of temperature inversions.
If the radiation reaching the satellite in each channel were emitted from a discrete level of the atmosphere, precisely at the level of unit optical depth
for that channel, remote temperature sensing would
be straightforward. One could simply monitor the
monochromatic intensities in a set of channels whose
optical depths match the levels at which temperatures are desired and use the measurements in the
respective channels to infer the temperatures.
Vertical resolution would be limited only by the
number of available channels. In the real world,
retrieval of the temperatures from the radiances is
not quite so simple. The vertical resolution that is
practically achievable is inherently limited by the fact
that the radiation in each channel is emitted, not
from a single level in the atmosphere, but from a
deep layer on the order of three scale heights in
depth (Exercise 4.47).
CO
H2O
CO2
CH4
N2O
N2O
N2O
CH4
H2O
280
260
240
220
200
600
800 1000 1200 1400 1600 1800 2000 2200 2400 2600
–1
Wavenumber (cm )
17 15 1312 11 10
9
8
7
6
5
4
3.5
Wavelength (μ m)
Fig. 4.32 As in Fig. 4.31, but a more finely resolved top of atmosphere spectrum sensed by an instrument aboard the NASA
ER-2 aircraft flying over the Gulf of Mexico at the 20-km level. Note that in this plot the ordinate is brightness temperature rather
than monochromatic intensity. [From K. N. Liou, An Introduction to Atmospheric Radiation, Academic Press, p. 122 (2002). Courtesy
of H. -L. Allen Huang and David Tobin.]
Radiative Transfer
Applying (4.42) to radiation reaching the satellite
from directly below, we can write
Ii Bi (Ts)e !i * 0
B [T(z)]e !(z)k rdz (4.57)
0
wi Bi [T(z)]dz
wi e ! i(z)kir
where I is the monochromatic radiance sensed by
the satellite at wave number , Ts is the temperature
of the underlying surface, !* is the optical thickness
of the entire atmosphere, and !(z) is the optical
thickness of the layer extending from level z to the
top of the atmosphere. Integrating over the narrow
range of wave numbers encompassed by the ith
channel on the satellite sensor, we obtain
(4.58)
(4.59)
called the weighting function, represents the contribution from a layer of unit thickness located at level
z to the radiance sensed by the satellite. The weighting function can also be expressed as the vertical
derivative of the transmittance of the overlying layer
(see Exercise 4.55).
Figure 4.33 shows transmittances and weighting
functions for six channels that were used in one of
0.12
Intensity (W m–2 sr –1/cm–1)
where Ii is the radiance and
I B (Ts)e ! * 300
6
5
0.08
275
4
1
2
3
Temperature (K)
142
250
0.04
0
500
600
700
Wavenumber (cm–1)
800
225
0.1
0.1
1
1
1
10
20
Pressure (hPa)
10
1
2
50
50
2
100
3
100
3
200
200
300
400
500
300
400
500
5
700
850
1000
6
700
850
1000
4
5
6
0
0.01
0.02
0.03
Weighting
0.04
4
0
0.2
0.4
0.6
0.8
Transmissivity
1.0
Fig. 4.33 (Top) Intensities or radiances (scale at left) and equivalent blackbody temperatures (scale at right) in the vicinity of the
15-m CO2 absorption band, as observed from the satellite. Arrows denote the spectral bands or “channels” sampled by the
satellite-borne vertical temperature profiling radiometer (VTPR). The weighting functions and transmittances for each of the channels are shown below. [Adapted from K. N. Liou, An Introduction to atmospheric radiation, Academic Press, pp. 389–390, Copyright
(2002), with permission from Elsevier.]
4.5 Radiative Transfer in Planetary Atmospheres 143
the early instruments used for operational remote
temperatures sensing. The locations of the channels
with respect to the emission spectrum, shown in
the inset, span a range of absorptances within the
broad CO2 absorption band centered near 15 m
( 600–750 cm1). The channels are numbered in
order of decreasing absorptance. The weighting functions for channels 1 and 2 peak in the stratosphere.
The weighting function for channel 3, which exhibits
the lowest radiance in the spectrum, straddles the
tropopause level, but because of the substantial depth
of its weighting function, its brightness temperature
is substantially higher than the tropopause temperature. The weighting function for channel 4 peaks in
the middle troposphere and those for channels 5
and 6 peak at or near the Earth’s surface. Note the
high degree of overlap (or redundancy) between
the weighting functions, especially those for channels
5 and 6.
Well-mixed trace gases are well suited for remote
temperature sensing because it can be assured that
the variations in the radiances from one sounding to
another are mainly due to differences in the vertical
profile of B rather than to differences in the vertical profiles of the concentrations of the absorbing
constituents. In the foregoing example, the differences in absorptivity among the six channels are
dominated by features in the absorption spectrum
of CO2.
Over much of the globe the interpretation of
satellite-sensed radiation in the infrared region
of the spectrum is complicated by the presence of
cloud layers. To circumvent this problem, several
operational instruments have been designed that
make use of radiation emitted by oxygen molecules
in the microwave region of the spectrum at frequencies around 55 GHz, in which clouds are nearly
transparent. In interpreting microwave radiances it
is necessary to take into account variations in the
emissivity of the Earth’s surface, which are much
larger than those for infrared radiation. Emissivities
over land range from close to 1 for vegetated or dry
surfaces to below 0.8 for wet surfaces. Emissivities
over water average 0.5 and vary with surface
roughness. Remote sensing in the microwave region
of the spectrum is also used to infer the distributions of precipitation and surface winds over the
oceans.
27
c. Retrieval of temperatures from radiances
There are two fundamentally different approaches to
retrieving a temperature sounding from the radiances in a suite of channels on a satellite instrument:
one involves solving the radiative transfer equation
and the other is based on a statistical analysis of the
relationships between the radiances in the various
channels and the temperatures at various atmospheric levels along the path of the radiation. Both
approaches are subject to the inherent limitations in
the vertical resolution that is achievable with remote
temperature soundings.
The appeal of the radiative transfer approach for
retrieving temperatures from radiances is that it
requires no prior data on the relationship between
the radiances measured by the instrument and in situ
temperature measurements. It is therefore the only
method applicable to atmospheric soundings on
planets lacking in situ measurements. An approximate solution to the radiative transfer equation can
be obtained by representing the atmosphere in terms
of n isothermal layers with its own temperature Tn.
Rewriting (4.58) as
Ii WsBi(Ts) Wi,n Bi(Tn)
(4.60)
n
where Ws is the weighting function for radiation
emitted from the Earth’s surface, Wi,n is the effective
weighting function for the atmospheric emission in
the nth atmospheric layer, and Bi(Tn) is the radiance
in the ith channel emitted by a blackbody at temperature Tn. The resulting set of equations, one for each
channel, in which the Ii are measured and the Wi,n
are estimated from (4.59) can be solved by a nonlinear iterative scheme27 to obtain the temperatures
Tn that yield the appropriate blackbody emissions.
Because of measurement errors, a constraint needs
to be applied in the retrieval process.
The statistical approach is less computationally
intensive than the radiative transfer approach. Other
advantages are that it takes into account the systematic biases and random error characteristics of the
radiance measurements and it offers the option of
combining radiance measurements with in situ temperature measurements and other pertinent observational data. The radiances in the channels measured
by a satellite-borne instrument are used to estimate
See K. N. Liou, An Introduction to Atmospheric Radiation, Academic Press, p. 391 (2002).
144
Radiative Transfer
the departures of the temperatures from some prescribed “reference sounding” defined at a prescribed
set of levels using standard linear regression techniques. The reference sounding incorporates in situ
data capable of resolving features such as the
tropopause that tend to be smoothed out in the
retrievals based on solutions of the radiative transfer
equations. Statistically based temperature and moisture retrievals are carried out routinely as an integral
part of the multivariate data assimilation protocols,
as explained in the supplement to Chapter 8 on the
book Web site.
Absorbed Solar Radiation
0
40
80
120
160
200
240
280
320
360
410
W m–2
Outgoing Longwave Radiation
4.6 Radiation Balance at the Top
of the Atmosphere
This chapter concludes with a brief survey of some of
the global fields relating to the energy balance at the
top of the atmosphere, defined on the basis of the
analysis of 1 year of satellite observations. The map
projection used in this section is area preserving, so
that the maps presented here can be interpreted as
depicting local contributions to global-mean quantities.
The top image in Fig. 4.34 shows the annual mean net
(downward) shortwave radiation, taking into account
the geographical variations in solar declination angle
and local albedo. Values are 300 W m2 in the tropics, where the sun is nearly directly overhead at midday
throughout the year. Within the tropics the highest
values are observed over cloud-free regions of the
oceans, where annual-mean local albedos range as low
as 0.1, and the lowest values are observed over the
deserts where albedos are 0.2 and locally range as
high as 0.35. Net incoming solar radiation drops below
100 W m2 in the polar regions where winters are dark
and the continuous summer daylight is offset by the
high solar zenith angles, widespread cloudiness, and the
high albedo of ice-covered surfaces.
The corresponding distribution of outgoing longwave radiation (OLR) at the top of the atmosphere,
shown in the bottom image in Fig. 4.34, exhibits a
gentler equator-to-pole gradient and more regional
variability within the tropics. As shown in Exercise
4.56, the observed equator-to-pole contrast in surface
air temperature is sufficient to produce a 2:1 difference in outgoing OLR between the equator and the
polar regions, but this is partially offset by the fact
that cloud tops and the top of the moist layer are
higher in the tropics than over high latitudes. The
regions of conspicuously low OLR over Indonesia
110
130
150
170
190
210
230
250
270
290
310
330
W m–2
Fig. 4.34 Global distributions of the annual-mean radiation
at the top of the atmosphere. [Based on data from the NASA
Earth Radiation Budget Experiment. Courtesy of Dennis L.
Hartmann.]
and parts of the tropical continents reflect the prevalence of deep convective clouds with high, cold tops:
the intertropical convergence zone is also evident as
a local OLR minimum, but it is not as pronounced
because the cloud tops are not as high as those associated with convection over the continents and the
extreme western Pacific and Indonesia. Areas with
the highest annual mean OLR are the deserts and
the equatorial dry zones over the tropical Pacific,
where the atmosphere is relatively dry and cloud
free, allowing more of the radiation emitted by the
Earth’s surface to escape unimpeded.
The net downward radiation at the top of the
atmosphere (i.e., the imbalance between net solar
and outgoing longwave radiation at the top of the
atmosphere) obtained by taking the difference
between the two panels of Fig. 4.34 is shown in
Fig. 4.35. The surplus of incoming solar radiation
over outgoing longwave radiation in low latitudes
and the deficit in high latitudes has important impli-
Exercises 145
(f) The colors of the stars are related to their
temperatures, whereas the colors of the
planets are not.
Net Radiation
(g) The color temperature of the sun is slightly
different from its equivalent blackbody
temperature.
(h) The equivalent blackbody temperature of
Venus is lower than that of Earth even
though Venus is closer to the sun.
–180 –150 –120 –90
–60
–30
0
W m–2
30
60
90
120
150
Fig. 4.35 Global distribution of the net imbalance between
the annual-mean net incoming solar radiation and the outgoing longwave radiation. Positive values indicate a downward
flux. [Based on data from the NASA Earth Radiation Budget
Experiment. Courtesy of Dennis L. Hartmann.]
cations for the global energy balance, as discussed in
Section 10.1. It is notable that over some of the
world’s hottest desert regions, the outgoing longwave
radiation exceeds absorbed solar radiation.
The exclusive focus on radiative fluxes in this section is justified by the fact that radiative transfer is
the only process capable of exchanging energy
between the Earth and the rest of the universe. The
energy balance at the Earth’s surface is more complicated because conduction of latent and sensible heat
across the Earth’s surface also plays important roles,
as discussed in Chapter 9.
Exercises
4.11 Explain or interpret the following in terms of
the principles discussed in this chapter.
(a) Species of plants that require a relatively
cool, moist environment tend to grow on
poleward-facing slopes.
(i) The equivalent blackbody temperature of
the Earth is lower than the global-mean
surface temperature by 34 °C.
(j) Frost may form on the ground when
temperatures just above the ground are
above freezing.
(k) At night if air temperatures are uniform,
frost on highways is likely to form first on
bridges and hilltops.
(l) Aerosols in the atmosphere (or on
windows) are more clearly visible when
viewed looking toward a light source than
away from it.
(m) Clouds behave as blackbodies in the infrared
region of the spectrum, but are relatively
transparent in the microwave region.
(n) Sunlit objects appear reddish around
sunrise and sunset on days when the air is
relatively free of aerosols.
(o) A given mass of liquid water would produce
more scattering if it were distributed among
a large number of small cloud droplets than
among a smaller number of larger droplets
(assume that both sizes of droplets fall
within the geometric optics regime).
(p) The upper layers of the atmosphere in
Fig. 4.36 appear blue, whereas the lower
layers appear red.
(b) On a clear day a snow surface is much
brighter when the sun is nearly overhead
than when it is just above the horizon.
(q) Smoke particles with a radius of 0.5 mm
appear bluish when viewed against a dark
background but reddish when viewed
against a light background (e.g., the sky).
(c) Light-colored clothing is often worn in hot
climates.
(r) The disk of the full moon appears uniformly
bright all the way out to the edge.
(d) The stars twinkle, whereas the planets do not.
(s) The absorption coefficient of a gas is a
function of temperature and pressure.
(e) The radiation emitted by the sun is isotropic,
yet solar radiation incident on the Earth’s
atmosphere may be regarded as parallel
beam.
(t) The absorptivity of greenhouse gases in the
troposphere is enhanced by the presence of
high concentrations of N2 and O2.