Salby M.L. Fundamentals of Atmospheric Physics
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200 8
Atmospheric
Radiation
sorbed at high levels in connection with photodissociation and photoionization
of 02 and 03. As a result, very little of the UV incident on the top of the
atmosphere reaches the tropopause. By contrast, the bulk of the spectrum in
the visible and near-IR regions arrives at the tropopause unattenuated, with
the exception of narrow absorption bands of water vapor and carbon dioxide.
However, in passing from 11 km to the ground, the remaining SW spectrum
is substantially absorbed in the IR by water vapor and carbon dioxide, ab-
solute concentrations of which are large in the troposphere. Consequently,
the spectrum of SW radiation reaching the surface is concentrated at visible
wavelengths, for which the atmosphere is mostly transparent.
In contrast to solar radiation, LW radiation emitted by the earth's surface
is almost completely absorbedmmost notably by H20 in a wide band centered
at 6.3/zm and another in the far infrared, by CO2 in a band centered at 15/zm
near the peak of the LW emission spectrum, and by a variety of trace gases
including 03,
CH4,
and N20. Consequently, most of the LW energy emitted
at the ground is captured in the overlying air. That energy must be reemitted,
half upward and half back downward. The upward component then undergoes
repeated absorption and reemission in adjacent layers until it is eventually
rejected to space. This sequence of radiative exchanges traps LW radiation
and, through mechanisms developed later in this chapter, elevates the surface
temperature of the earth.
A comparison of Figs. 8.1b and 8.1c indicates that most of the LW energy
emitted by the surface is absorbed in the troposphere. Only in the atmospheric
window at wavelengths of 8 to 12/zm is absorption weak enough for much of
the LW radiation emitted by the surface to pass freely through the atmosphere.
The 9.6-/xm band of ozone, which is positioned inside this window, is the only
strong absorber at those wavelengths. Most of that absorption takes place in
the stratosphere, where ozone concentrations are large.
8.1.1 Spectra of Observed SW and LW Radiation
The most important property of the SW spectrum is the solar constant, which
represents the flux of radiant energy integrated over wavelength reaching the
top of the atmosphere at the mean earth-sun distance. The solar constant
is weakly variable, having a value of about 1370 W m -2 according to recent
satellite measurements. When distributed over the globe, this SW flux leads to
the daily insolation shown in Fig. 1.28, which follows primarily from the length
of day and solar inclination for particular latitudes and times of year.
Spectra of observed SW and LW radiation are shaped by the absorption
characteristics described above. At the top of the atmosphere (Fig. 8.2), the
solar spectrum resembles the emission spectrum of a blackbody at about 6000
K. That temperature is characteristic of the sun's photosphere, where most of
the radiation is emitted. Wavelengths shorter than 300 nm deviate from this
simple picture, instead reflecting temperatures of 4500 to 5000 K, which are
8.1
Shortwave and Longwave Radiation
201
SOLAR RADIATION
E
c-
Ckl
i
E
LIJ
O
Z
< 1
a
<
rr
rr
TOP OF THE ATMOSPHERE
SEA LEVEL
BLACKBODY EMISSION (5900 K)
!o
~o
,, ~o
/,, ~o,co~
H~O,C%
O0 ...........................................................................................................................................................................................................................
400 800 1200 1600 2000 2400 2800 3200
WAVELENGTH (nm)
Figure 8.2 Spectrum of SW radiation at the top of the atmosphere (solid) and at the earth's
surface (shaded), compared against the emission spectrum of a blackbody at 6000 K (dashed).
Individual absorbing species indicated. Adapted from Coulson (1975).
found somewhat higher near the base of the sun's
chromosphere.
At the earth's
surface, the wings of the solar spectrum have been mostly stripped off, in the
UV by ozone and molecular oxygen and in the IR by absorption bands of
water vapor and carbon dioxide, leaving wavelengths of 0.3 to 2.4/zm.
Wavelengths shorter than 300 nm, which constitute a small fraction of the
solar energy flux, are absorbed fairly high in the earth's atmosphere through
photodissociation and photoionization. As shown in Fig. 8.3, these energetic
components penetrate to altitudes in inverse relation to their wavelengths.
Wavelengths shorter than 200 nm do not penetrate appreciably below 50 kin,
whereas A < 150 nm tend to be confined to the thermosphere. Exceptional
is the discrete
Lyman-a emission
at 121 nm, which corresponds to transitions
from the gravest excited state of hydrogen. The Lyman-a line happens to co-
incide with an optical window in the banded absorption structure at these
wavelengths that allows it to reach the upper mesosphere and lower thermo-
sphere, where it is involved in photoionization and photodissociation.
The wavelength of SW radiation is also inversely related to the altitude in
the solar atmosphere where it is emitted. As indicated in Fig. 8.4, A of 300 nm
and longer originate in the photosphere, whereas A < 200 nm are emitted by
the chromosphere, and very energetic components with A < 50 nm originate
higher yet in the corona. Solar variability increases with altitude in the solar
atmosphere. Consequently, temporal variations in the SW spectrum are most
202 8
Atmospheric
Radiation
200
I I I I
I -- NzO r,.
i 9
02 ~-
.4-02. .D.
150 +
2 "' /
<
100
UJ ~ZuJ 50 F
LYMANOr.
13.
1 1 1 1
0 50 100 150 200
03 ..
25,0 300
WAVELENGTH (nm)
Figure
8.3 Penetration altitude, where SW flux at wavelength A is attentuated by a factor of
e -1. Absorbing species and thresholds for photoionization indicated. After Herzberg (1965).
10-1 /~ Solar I~ \ ........
10"2-- J X flare 3B Lyc4~ X
10 -4
sun~/ / ~ ,,~,,,,, /,,iv ~ \
Z 10_t /~Quiet / X -
a
< ]- /
m \
rr 10
10j Corona ~I~ ~ 1'tosphere X _
10"
10 1 1 10 10 2 10 3 10 4 10 5 10 6
WAVELENGTH (nm)
Figure
8.4 Spectrum of solar irradiance (flux) as a function of wavelength, under normal and
disturbed conditions. Visible band and emission levels in the solar atmosphere indicated. Adapted
from Smith and Gottlieb (1974)with permission of Kluwer Academic Publishers.
8.2 Description of Radiative Transfer 203
apparent in the far-UV region, which makes only a small contribution to the
solar constant.
Two phenomena dominate solar variability. The 27-day rotation cycle of
the sun leads to variations of several percent in the UV and fall to under
1% for wavelengths longer than 250 nm (WMO, 1986). Sunspot activity also
evolves through an l 1-year cycle that modulates solar emissions. Between its
extremes,
solar-max
and
solar-min,
activity changes noticeably in the outer
reaches of the sun's atmosphere. Owing to the inverse relationship between
solar altitude and the emitted wavelength, SW variability decreases sharply
with increasing A (Fig. 8.4), which, in turn, strongly limits the altitudes in the
earth's atmosphere affected by such variations. At h = 160 nm, the peak-to-
peak variation in flux between solar-max and solar-min is about 10% and it
drops to less than !% by 300 nm. In terms of the solar constant and most of
the energy reaching the earth's surface, the variation between solar-max and
solar-min is of order 0.1% (Willson
et aL,
1986).
The spectrum of LW radiation emitted to space (Fig. 8.5) also
resembles that of a blackbody. However, it corresponds to a temperature of
288 K only in the atmospheric window: wavenumbers h -1 of 800 to 1200
cm -~, where radiation emitted by the surface passes to space relatively
freely. Other wavenumbers are absorbed and reemitted in the overlying
atmosphere by water vapor and clouds, which emit radiation at colder
temperatures.
The strong absorption bands of
CO 2
and
0 3
sharply reduce emission to
space at 15 and 9.6/xm, respectively. Each corresponds to a blackbody tem-
perature distinctly colder than that within the atmospheric window, where
outgoing radiation emanates from the earth's surface. The 15-/xm emission of
CO2, which is positioned near the center of the LW spectrum, corresponds to
a blackbody temperature of about 220 K and an altitude in the upper tropo-
sphere. The 9.6-/xm emission of ozone corresponds to a temperature of about
270 K and an altitude in the upper stratosphere. Along with water vapor, these
minor constituents trap LW radiation emitted by the earth's surface and ele-
vate its temperature over what it would be in the absence of an atmosphere
(e.g., over the effective blackbody temperature of 255 K). Known as the
green-
house effect,
this phenomenon follows from the atmosphere's transparency to
visible radiation and opacity to infrared radiation, which are controlled by
optically active species.
8.2 Description of Radiative Transfer
8.2.1 Radiometric Quantities
Understanding how radiation influences atmospheric properties requires a
quantitative description of radiative transfer, which is complicated by its three-
(I_WO/I_JSz_LUO i_OgS ~Je ) 33NVICIVEI
WAVELENGTH
(prn)
240
200
I60
I20
80
40
0
0
200
400
600
000
1000
I200
1400
1600
1800
2000
WAVE
NUMBER
(cm-l)
Figure
8.5
Spectrum of outgoing LW radiation over
(215"
W,
15"
N)
observed by Nimbus-4
IRIS,
as a function of wavenumber
A-'.
Blackbody spectra for different temperatures and individual absorbing species indicated. Adapted from Liou
(1980).
8.2
Description of Radiative Transfer
205
= I~b.
/
/
/
/
/
/
/
Figure
8.6 A pencil of radiation that occupies the increment of solid angle dI~ in the direction
.0 and traverses a surface with unit normal n. The monochromatic
intensity
or
radiance
passing
through the pencil: I A = Ix.O, describes the rate at which energy inside the pencil crosses the
surface per unit area, steradian, and wavelength. Integrating the component normal to the surface:
Ix(.O. n) = I x cos0, over the half-space of 2~r steradians in the positive n direction yields the
monochromatic forward flux or
irradiance F +.
dimensional nature and its dependence on wavelength and directionality. As
illustrated in Fig. 8.6, radiant energy traversing a surface with unit normal
n can have contributions from all directions, which are characterized by the
unit vector g~. Such multidimensional radiation is termed
diffuse,
whereas, if
it arrives along a single direction (like incoming solar), it is termed
parallel-
beam radiation.
The monochromatic
intensity
or
radiance I A - lag]
repre-
sents the rate that energy of wavelengths A to A + dA flows per unit area
through an increment of solid angle df~ in the direction g]. Having dimen-
sions of power/(wavelength, area. steradian),
IA(x, g])
is a function of posi-
tion x and direction g~ and characterizes a pencil of radiation that traverses
the surface at a
zenith angle 0
from its normal. Then the rate energy flows
in the direction g~ per unit cross-sectional area and wavelength interval is
Idi).
Integrating the component of intensity normal to the surface" I~.
n -
IA(O.n )
= I~
cos 0, over the half-space of solid angle in the positive n direction
206 s
Atmospheric
Radiation
defines the monochromatic flux or
irradiance
crossing the surface
F+ = f2~ Ix " ndg~+
= .~-~~ I x cos
Odf~ +
f02~r f0 ~r/2
= Ia(~b, 0)cos 0sin
OdOdch,
(8.1)
which has dimensions of power/(wavelength 9 area). The
total flux
then follows
by integrating over the electromagnetic spectrum
F + = F+dA,
(8.2)
which has the dimensions of energy flux: power/area. It follows that the
monochromatic flux F + represents the spectral density at wavelength A of
the total flux F +.
The forward energy flux in the n direction is described by F +. The backward
energy flux F~- may be obtained by integrating I in the negative n direction over
the opposite half-space. Both are nonnegative quantities. The monochromatic
net flux
in the n direction is then the difference between the forward and
backward components:
Fx = F + - F~-. (8.3)
By applying (8.1) and (8.3) to the three coordinate directions, we obtain the
net flux vector Fx. In the special case of
isotopic
radiation, for which the
intensity is independent of direction: Ix # Ix(J2), (8.1) reduces to
F~- = 7rI A. (8.4)
Since the forward and backward components are then equal, the net flux in
any direction vanishes.
Electromagnetic radiation can interact with matter through three basic
mechanisms: absorption, scattering, and emission. A pencil of radiation oc-
cupying the solid angle dO is attenuated in proportion to the density and
absorbing characteristics of the medium through which it passes. Energy can
also be scattered out of the solid angle, which likewise attenuates energy
flowing through dO. Together, absorption and scattering constitute the net ex-
tinction
of energy passing through the pencil of radiation. Conversely, energy
emitted into the solid angle or scattered into dO from other directions intensi-
fies the flow of energy through the pencil of radiation. These interactions are
governed by a series of laws that describe the transfer of radiation through
matter.
8.2 Description of Radiative Transfer
207
8.2.2 Absorption
In the absence of scattering, absorption of energy from a pencil of radiation
is expressed by the following principle.
LAMBERT'S LAW
The fractional energy absorbed from a pencil of radiation is proportional
to the mass traversed by the radiation. For an incremental distance
ds,
this
principle may be written
dlA
-- -PO'aAds ,
(8.5.1)
Ix
where the constant of proportionality trax is the specific
absorption cross sec-
tion
(also referred to as the
mass absorption coefficient),
which has units of
area/mass. The cross section symbolizes the area of the pencil lost by passing
through an increment of mass
pdAds
or, equivalently, the effective absorbing
area of that massmeach for the wavelength A. The foregoing principle can
also be expressed in terms of the particle number density n (e.g., of molecules
or aerosol) through which the radiation passes
dlA
= -n&a,~ds,
(8.5.2)
Ix
where the absorption cross section graA is referenced to an individual particle
and has dimensions of area. Thus, &aA symbolizes the effective absorbing area
posed by a particle in the path of the radiation. In either representation,
Lambert's law for attenuation of energy is linear in the intensity I a.
The
absorption coefficient
[3 aA -- PO'aA
: nO'ax,
(8.6.1)
which has dimensions of inverse length, measures the characteristic distance
over which energy is attenuated. The density in (8.5) corresponds to the mass
of the absorber, which, in a mixture, may follow from several constituent gases.
Collecting contributions from different constituents then gives
flax -- Z ripO'ahi
(8.6.2)
i
= ~ rirt~raX i,
(8.6.3)
i
where
r i
denotes the mass mixing ratio of the ith absorbing species.
By integrating Lambert's law along the path of radiation, we obtain
I~(s) = I~(O) exp (- foSptra~ds') .
(8.7)
208 s Atmospheric
Radiation
In the absence of scattering and emission, the intensity inside a pencil of
radiation decreases exponentially with the
optical path length
U(S) -- pO'axds'
-
flaxds',
(8.8)
which is the dimensionless distance traversed by radiation weighted according
to the density and absorption cross section of the medium. Because Lambert's
law involves no directionality, it applies to flux as well as intensity.
The monochromatic
transmissivity,
which describes the fraction of incident
radiation remaining in the pencil at a given distance, is then
I (s)
(s) = I (O)
= e -u(s).
(8.9)
Transmissivity decreases exponentially with optical path length through an
absorbing medium. Conversely, the monochromatic
absorptivity a A
represents
the fraction of incident radiation that has been absorbed from the pencil
during the same traversal. Since
+ ax- 1 (8.10)
in the absence of scattering and emission, the absorptivity follows as
ax(s) - 1 - e -u(s).
(S.ll)
Absorptivity exponentially approaches unity with path length through an ab-
sorbing medium, which, in that limit, is said to be "optically thick."
8.2.3 Emission
To maintain thermal equilibrium, a substance that absorbs radiant energy must
also emit it. Like absorption, the emission of energy into a pencil of radiation is
proportional to the mass involved. The basis for describing thermal emission
is the theory of blackbody radiation, which was developed by Planck and
contemporaries near the turn of the century.
Blackbody radiation is an idealization that corresponds to the energy emit-
ted by an isolated cavity in thermal equilibrium. Such radiation is characterized
by the following properties:
9 The radiation is uniquely determined by the temperature of the
emitter.
9 For a given temperature, the radiant energy emitted is the maxi-
mum possible at all wavelengths.
9 The radiation is isotropic.
8.2 Description of Radiative Transfer
209
In addition to being a perfect emitter, a blackbody is also a perfect absorber:
Radiation incident on a blackbody is completely absorbed at all wavelengths.
PLANCK'S LAW
To explain blackbody radiation, Planck postulated that the energy E of
molecules is quantized and can undergo only discrete transitions that satisfy
AE = An. hv, (8.12)
where n is integer, h is Planck's constant (Appendix C), and u = c/A is the
frequency of electromagnetic radiation emitted or absorbed to accomplish the
energy transition AE. Hence, the radiation emitted or absorbed by individual
molecules is quantized in photons that carry energy in integral multiples of
hu. On this basis, Planck derived the following relationship between the spec-
trum of intensity B A emitted by a population of molecules and their absolute
temperature:
2hc 2
BA(T)- AS(eK-~-cT -- 1)' (8.13)
where K is the Boltzmann constant. Plotted in Fig. 8.7, the Planck or blackbody
spectrum (8.13) possesses a single maximum at wavelength
A m
and increases
with temperature. At wavelengths shorter than Am, B a decreases sharply,
whereas a comparatively wide band is involved at longer wavelengths. Because
blackbody radiation is isotropic, the form of the Planck spectrum applies to
flux (8.4) as well as to intensity.
WIEN'S DISPLACEMENT LAW
The wavelength of maximum intensity follows from (8.13) as
2897
/~m=
T [/zm], (8.14)
which decreases with increasing temperature of the emitter. This feature of the
emission spectrum allows the brightness temperature of a body to be inferred
from radiation emitted by it. For instance, SW radiation incident on the top
of the atmosphere is concentrated in the visible and peaks near a wavelength
of 0.480/xm (Fig. 8.2), which corresponds to blue light. Wien's displacement
law then implies an effective solar temperature of about 6000 K. On the other
hand, the 288 K mean surface temperature of earth corresponds to maximum
emission at a wavelength of about 10/zm (Fig. 8.1). Similar reasoning can
be applied to individual wavelengths via (8.13). For instance, features in the
observed spectrum of outgoing LW radiation (Fig. 8.5) at 15 and 9.6/zm can
be identified with emission by CO2 in the upper troposphere and by 0 3 in the
upper stratosphere. Each is significantly colder than the surface temperature,
which is manifested in the outgoing spectrum only in the atmospheric window
at 8 to 12/xm.