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already frozen droplet in this manner will freeze in distinct modes that depended on
temperature. In the temperature range characteristic of ice multiplication, droplets
freeze such that the outer shell of the droplet freezes first. The expanding ice builds
internal pressure on the liquid in the interior of the drop until the shell ruptures at its
weakest point ejecting liquid, which freezes creating a protuberance or a fragment.
This process, called the Hallett–Mossop process after its discoverers, is believed to
be important in many cloud systems.
The final mechanism that has been studied extensively is droplet fragmentation
by shattering or splinter ejection during freezing. Droplet shattering during freezing
appears to be limited to droplets >50 mm diameter. Larger drops are more likely to
fragment than smaller drops. Even when droplets do not fragment, the nature of the
freezing process can cause them to eject an ice splinter as the outer shell ruptures.
Because of the large droplet criteria, this process tends to be favored in maritime
clouds with larger drop distributions, and much less favored in continental clouds,
particularly those with large updrafts, such as cumulus.
Despite the fact that these ice multiplication mechanisms are well understood,
they still cannot account for the observed high ice particle concentrations in many
clouds. One attractive hypothesis that fits the observations is that rapid nucleation of
ice occurs at cloud boundaries, particularly cloud top. It has been shown observa-
tionally and theoretically that a narrow layer of supercooled water frequently exists at
the top of cold clouds, even at temperat ures <20
C. This liquid layer is believed to
exist because shear, radiation, and entrainment enhance vertical velocities near cloud
top, and because ice crystals are naturally very small near cloud top. Because of
these factors, the condensate supply rate frequently exceeds the bulk ice particle
growth rate and liquid is produced. Broad droplet spectra are often present in these
regions. Entrainment of dry air leads to evaporation of droplets near cloud top.
Contact nucleation may be favored in these conditions. Also, evaporation of drops
can lead to giant aerosol particles that, in deep cumulus, may carry significant
charge. These particles may be more effective as ice nuclei. At present, the impor-
tance of each of these ice-forming mechanisms in different clouds is speculative at
best. Clearly, more research is required before we understand all the possibilities and
processes associated with the formation of the ice phase in clouds.
LIST OF SYMBOLS
A constant in IN concentration equation
a ice crystal axis along the basal plane
C constant in CCN, IN concentration equations, capacitance factor
C
v
specific heat of air at constant volume
c ice crystal axis along prism plane
D diffusion coefficient
E
col
collision efficiency
E
coal
coalescence efficiency
E collection efficiency
296 MICROPHYSICAL PROCESSES IN THE ATMOSPHERE
e
0
r
vapor pressure over a solution droplet
e
r
vapor pressure over a spherically curved water surface of radius r
e
r,i
vapor pressure over a spherically curved ice surface of radius r
e
s, 1
saturation vapor pressure over a plane water surface
e
si, 1
saturation vapor pressure over a plane ice surface
F Helmholz free energy
DF
i
free energy change at ice–liquid interface
DF
0
free energy change associated with molecular realignment
f
v
ventilation coefficient
h Planck’s constant
i van’t Hoff factor
J nucleation rate
K thermal conductivity, collision kernel
k Boltzmann constant
k constants in CCN, IN concentration equations
L
v
latent heat of vaporization
L
s
latent heat of sublimation
n(r) number of aerosol particles in the size range r, r þ d log r
m mass of a drop
m
s
mass of solute
M(r) total mass of aerosol particles with radii greater than r
M
w
molecular weight of water
M
s
molecular weight of the solute
N number concentration
N
a
Avagadro’s number
N
c
number of water molecules in contact wi th ice surface
N
CCN
number of CCN active at a specified supersaturation
N
IN
number of ice nuclei
N(r) total number of aerosol particles with radii greater than r
N
Re
Reynolds number
n
0
number of molecules of water
n
s
number of molecules of solute
p pressure
R large drop radius
R
universal gas constant
R
d
gas constant for dry air
R
v
gas constant for water vapor
RH
w
relative humidity with respect to water
r particle radius
r
eq
equilibrium radius of a droplet
S(r) total surface area of aerosol particles with radii greater than r
S
w,v
saturation ratio over a water surface
S
s,v
saturation ratio over a solution droplet
S
i,v
saturation ratio over an ice surface
s
w
supersaturation with respect to water
5 GROWTH OF ICE PARTICLES IN ATMOSPHERE 297
s
i
supersaturation with respect to ice
T temperature
T
s
surface temperature of a hailstone
t time
V(r) total volume of aeroso l particles with radii greater than r
V
T
terminal velocity
V
T,H
terminal velocity of a hailstone
x
0
collision cross section
a accomodation coefficient
b condensation coefficient, constant in IN concentration equation
n kinematic viscosity
r aerosol particle density
r
w
density of water
s
w,v
surface tension at a water–vapor interface
s
w,i
surface tension at a water–ice interface
s
i,v
surface tension at an ice–vapor surface
r
i
density of ice
REFERENCES
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Beard, K. V., and H. T. Ochs III (1984). Collection and coalescence efficiencies for accretion,
J. Geophys. Res. 89, 7165–7169.
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I–IV, J. Atmos. Sci. 31, 1814–1831, 2118–2135.
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different parts of the world, J. Rech. Atmos. 4, 41–58.
Chuang, C. C., and K. V. Beard (1990). A numerical model for the equilibrium shape of
raindrops, J. Atmos. Sci. 47, 1374–1389.
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Junge, C. E. (1963). Large scale distribution of condensation nuclei in the troposphere, J. Rech.
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Magono, C., and C. W. Lee (1966). Meteorological classification of natural snow crystals,
J. Fac. Sci. 7(2), 321–362.
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drops, J. Atmos. Sci. 52, 1924–1933.
Pruppacher, H. R., and J. D. Klett (1997). Microphysics and Clouds and Precipitation, 2nd ed.,
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REFERENCES 299
CHAPTER 19
RADIATION IN THE ATMOSPHERE:
FOUNDATIONS
ROBERT PINCUS AND STEVEN A. ACKERMANN
1 OVERVIEW
‘‘What is the weather like?’’ Meteorologists answer this question in thousands of
locations every day when we make observations to determine the state of the atmos-
phere. The answer is summarized in a few carefully chosen quantities, typically wind
speed and direction, relative humidity, and temperature. In the language of physics
these roughly correspond to momentum, mass (of water), and energy. Weather
forecasting is the science and craft of predicting how these interrelated quantities
will change with time.
Energy is transferred through the atmosphere via five processes: convection,
advection, conduction, phase change, and radiation. The first two processes involve
the movement of mass from one place to another; conduction occurs when two
bodies at different temperatures are in contact; phase changes release latent heat.
Radiation is fundamentally different because it allows energy to be transferred
between two locations without any intervening material.
Why Study Radiation?
Because radiation can transport energy even without a medi um, it is the only way in
which Earth interacts with the rest of the universe. It is radiation, in fact, that
determines Earth’s climate, since in the long term the planet must shed as much
energy as it absorbs. Even within the Ear th–atmosphere system radiation can be a
powerful player in determining the local energy budget. Radiation is why it is usually
Handbook of Weather, Climate, and Water: Dynamics, Climate, Physical Meteorology, Weather Systems,
and Measurements, Edited by Thomas D. Potter and Bradley R. Colman.
ISBN 0-471-21490-6 # 2003 John Wiley & Sons, Inc.
301
warmer during the day, when the sun shines, than at night, and why the surface air
temperature is higher on cloudy nights than clear ones.
Radiation is also the basis for remote sensing, the ability to measure the state of
the atmosphere from a remote location (usually the ground or outer space). Remote
sensing includes everything from simple cloud imagery, through radar estimates of
precipitation, to ground-based sounding. Understanding the capabilities and limits of
remote-sensing measurements requires learning something about radiation.
Earth’s atmosphere is primarily made of gases, but it also contains liquids and
solids in the form of aerosols and clouds. Radiation interacts in fundamentally
different ways with gases and condensed materials. Gases, as we will see, interact
with radiation in just a few ways but have complicated spectral structure. Clouds and
aerosols, on the other hand, affect the radiation in each spectral region in about the
same way but make the mathematics much more complicated.
Nature of Radiation
In the atmospheric sciences, the word radiation means ‘‘electromagnetic radiation,’’
which is different than the particle radiation emitted from radioactive decay. Visible
light is one kind of electromagnetic radiation, as are gamma- and x-rays, ultraviolet,
infrared, and microwave radiation, and radio waves.
When radiation is measured using very sensitive instruments at extremely low
light levels, it is observed that the energy does not arrive continuously but rather in
small, finite amounts. These and other observations described in physics texts,
including the photoelectric and Compton effects, suggest that radiation can be
thought of as a collection of photons, tiny but discrete packets of energy traveling
at the speed of light. This is the particle view of radiation.
We can also describe radiation as an electromagnetic phenomenon. The inter-
actions among electric and magnetic fields, matter, charges, and currents are
described by Maxwell’s equations. The only nonzero solution to these equations
in empty space is a traveling wave. Constants in Maxwell’s equations predict the
wave velocity, which is exactly the speed of light. Lig ht behaves as a wave in many
circumstances, too, diffracting when passed through a slit and reflecting from
discontinuities in the medium. These observations are the motivation to describe
radiation in purely electrom agnetic terms.
How do we reconcile these two views? It is tempting to say that light can be both
wave and particle, but this is not quite accurate; rather, there are circumstances in
which light behaves like a wave and others in which it behaves like a particle. In this
and the following chapter we will primarily use the wave model, which is usually
more useful in the context of meteorology.
Radiation is also the single aspect of atmospheric science in which quantum
mechanics plays a role. This theory, developed in the first few decades of the
twentieth century, is based on the idea that the world is not continuous at very
small scales, but is divided up (‘‘quantized’’) into discrete elements: An electron’s
angular momentum, for example, can take on only certain values. As we will see, a
302 RADIATION IN THE ATMOSPHERE: FOUNDATIONS
complete description of radiation requires us to invoke ideas from quantum
mechanics several times.
2 FOUNDATIONS
Geometry
The directional aspects of radiative transfer are most naturally expressed in spherical
coordinates, illustrated in Figure 1. Location is specified by radius r from the origin,
zenith angle y, and azimuthal angle j, with differential increments dr, rdy, and
r sin y dj. The amount of radiation depends on direction, which in three dimensions
is specified with solid angle O, measured in steradians (str). The differential element
dO is the product of the polar and azimuthal angle differentials. There are 4p
steradians in a sphere:
ð
2p
0
ð
p
0
sin y dy dj ¼
ð
O
dO ¼ 4p ð1Þ
Polar angle is often replaced with its cosine m ¼ cos y, dm ¼ sin y dy, so that
dO ¼ dm dj. Polar angle is measured relative to a beam directed upwards, so
y ¼p and m ¼1 for a beam pointing straight down, and m > 0 for radiation travel-
ing upwards.
Figure 1 Geometry in polar coordinates. Radius r is measured from the origin, zenith angle
y from the vertical, and azimuthal angle j from the south.
2 FOUNDATIONS 303
Describing Electromagnetic Waves
Electromagnetic waves can be characterized in terms of their velocity c, the
frequency of oscil lation n, and the wavelength l. These quantities are related:
c ¼ ln ð2Þ
Wavelength is the distance between successive maxima of field strength and has
dimensions of length, while frequency has dimensions of inverse time, so velocity is
measured in distance per time. In a vacuum the speed of light c ¼3 10
8
m=s. This
value can change in other materials depending on the index of refraction m, which
varies with frequency. In air m 1soc is nearly unchanged; in water at visible
wavelengths m 1.33, so within cloud drops the speed of light is diminished by
about 25%. The inverse of wavelength is the wavenumber k ¼1=l. In atmospheric
applications wavelength is commonly measured in microns (1 mm ¼10
6
m), nano-
meters (1 nm ¼10
9
m), or angstroms (1 A
˚
¼10
10
m), with frequency in megahertz
(1 MHz ¼10
6
s
1
¼0
6
Hz) or gigahertz (1 GHz ¼10
9
Hz) and wavenumber
expressed in inverse centimeters.
Name Spectral Region
X-ray l < 10 nm
Ultraviolet (UV) 10 < l < 400 nm
Visible 0.4 < l < 0.7 mm
Near-infrared (near-IR) 0.7 < l < 3.5 mm
Middle-IR 3.5 < l < 30 mm
Far-IR 30 < l < 100 mm
Microwave 1 mm < l < 1m
The plane in which the electric field oscillates determines the polarization of the
radiation. In the atmosphere, though, this plane is rarely constant (i.e., radiation in
the atmosphere is usually unpolarized) so we’ll ignore this aspect.
Describing Radiation
If we stand in a field on a clear day and look around the sky, we will notice that lots
of light comes from the sun and little from the other directions. It is brighter in the
open field than in the shadow of a tree, and darker at night than at noon. The amount
of radiation, then, depends on space, time, direction, and wavelength. Because light
travels so fast, we usually ignore the time dependence.
The fundamental measure of radiation is the amount of energy traveling in a
given direction at a certain wavelength. This measure is called spectral intensity I
l
and has dimensions of power per unit area per solid angle per spectral interval, or
units of W=m
2
str mm. Spectral inte nsity is assumed to be monochromatic, or
consisting of exactly one wavelength, and depends on position.
304 RADIATION IN THE ATMOSPHERE: FOUNDATIONS
If we want to know the total amount of energy traveling in a given direction (say,
the amount of energy entering a camera lens or a satellite detector), we must inte-
grate I
l
across some portion of the spectrum to compute intensity (or broadband
intensity) I:
I ¼
ð
I
l
dl ð3Þ
The limits of integration in (3) depend on the application. If the film in our
camera is sensitive only to visible light, for example, the integration is over the
visible por tion of the spectrum, while if the lens is behind a colored filter we include
only those wavelengths the filter passes . Intensity has units of W=m
2
str; we use the
term broadband intensity when the integration is over a large part of the spectrum
(e.g., all the infrared or all the visible). Unless the radiation interacts with the
medium neither I
l
nor I change with distance.
Imagine next a sheet of back plastic placed on the ground in the sunlight. How
much energy does the sheet absorb? If we start with just one wavelength, we see that
the sheet absorbs energy at a rate F
l
F
l
¼
ð
1
1
ð
2p
0
I
l
jmjdj dm ð4Þ
Weighting by m accounts for geometry: A ray encountering the surface at an angle
is spread over a wider area than a beam coming straight at the surface. F
l
is called
the spectral flux and has units of W=m
2
mm. Flux is traditionally divided into
upward- and downward-going components:
F
#
l
¼
ð
0
1
ð
2p
0
I
l
jmjdj dm
F
"
l
¼
ð
1
0
ð
2p
0
I
l
jmjdj dm
ð5Þ
The black sheet absorbs at all wavelengths, so the total amount of energy
absorbed F is computed by integrating over both solid angle and spectral interval:
F
#
¼
ð
0
1
ð
2p
0
ð
I
l
jmjdl dj dm ¼
ð
0
1
ð
2p
0
Ijmjdl dj dm ¼
ð
F
#
l
dl ð6Þ
Flux (or broadband flux) has units of W=m
2
. The terms radiance and irradiance are
also used in textbooks and the technical literature; these cor respond to our terms
intensity and flux.
Why are so many different quantities used to describe radiation? Because the
two main applications of radiation, remote sensing and energy budget computa-
tions, require fundamentally different kinds of information. Remote-sensing instru-
2 FOUNDATIONS 305