RADIATION BASICS
• Significance of radiation in the atmosphere
• Electromagnetic spectrum
• Absorption/emission
• Scattering
• Line broadening
SIGNIFICANCE OF RADIATION IN THE ATMOSPHERE
1)
CLIMATE
The Sun is the principal source of energy for the Earth. The global energy balance between solar
and terrestrial radiation determines the global climate. This can be expressed as:
Te  F (1   )
4
(27.1)
where  is Boltzmann’s constant, Te is the effective blackbody temperature of the Earth
(~276 K), F is the solar constant averaged over the atmosphere (~345 W/m2), and  is the
average albedo of the Earth (~0.3 or 30%).
Note: The effective blackbody temperature (also called the radiation temperature) is about 31 K colder than the global
mean surface temperature, which is about 278 K. This difference is called the natural greenhouse effect.
Eq. 27.1 is deceptively simple. We could, for example, use it to try to answer the question
“what happens to Te if F decreases?” The naïve answer is that Te also decreases. But this may
not be so, because the albedo is likely a non-linear function of Te (due to clouds, ice caps and
vegetation variations). Hence the conclusion is no longer so obvious. It is even harder to use
Eq. 27.1 to infer the effect on the surface temperature of a change in F, unless we know more
about the climate system and its feedbacks.
The sketch below shows the interaction of incident solar radiation (normalized to 100 units) with
the Earth and its atmosphere.
The following figure shows typical spectra of solar and terrestrial radiation, and some of the
principal absorbers.
2) WINDS
The distribution (latitudinally and seasonally) of radiation over the atmosphere determines the
temperature gradients (particularly the meridional temperature gradient), and these drive
atmospheric winds. If you have taken dynamic meteorology, you may recall the thermal wind
equation:

vg
g ˆ 

k   pT
z
fT
(27.2)
Eq. 27.2 states that the vertical wind shear is controlled by the horizontal temperature gradient.
Dynamics students will also know that if the vertical wind shear is sufficiently strong, zonal
flow can become baroclinically unstable, leading to the occurrence of mid-latitude cyclonic
systems, which are associated with our weather. The sketch below illustrates the fact that, on
average, there is a net radiation surplus near the Equator and a net radiation deficit at the
Poles. This energy imbalance leads to the creation of the meridional temperature gradient
alluded to above. The general circulation of the atmosphere and oceans results from these two
“fluids” attempting to erase this temperature gradient in order to come to thermal equilibrium.
Since the system is always forced by the Sun, the time average climatological state represents a
balance between radiative forcing, which acts to strengthen the gradient, and
atmosphere/ocean dynamics, which act to weaken the gradient.
Sandström’s theorem states that a closed, steady circulation can be maintained in an
atmosphere only if the heat source is maintained at a higher pressure than the heat sink. This
occurs in the Earth’s atmosphere, because the atmosphere is largely transparent to solar
radiation but is quite opaque to parts of the terrestrial radiation spectrum. Absorption of
solar radiation by water vapour, however, is emerging as an important component of the
overall radiation balance. Integrated over solar wavelengths, modern amounts of water vapour in
the atmosphere absorb approximately 19% of incoming solar radiation.
3) REMOTE SENSING
Satellites now permit vertical sounding of the atmosphere for temperature and the concentration
of various atmospheric components. This can provide much greater horizontal spatial resolution
than with the current rawinsonde system. However, at the moment, the vertical resolution is not
as good as with rawinsondes, and clouds can interfere with the remote sensing necessary to
obtain vertical profiles of temperature. The following figures show some typical “remotely
sensed” spectra. After this series of lectures on radiation, you should be able to interpret them
in terms of the physical processes that are occuring, and affecting the outgoing radiation.
Note: these plots can be animated if you go to the University of Oregon’s site below. These
static images are for December only.
http://geography.uoregon.edu/envchange/clim_animations/
Note: these plots can be animated if you go to the University of Oregon’s site below. These
static images are for December only.
4) METEOROLOGICAL OPTICS
The scattering of visible radiation in the atmosphere, by liquid water droplets and ice crystals,
leads to a number of aesthetically pleasing, and sometimes theoretically challenging,
phenomena—rainbows, double rainbows, haloes, glories, sunsets, etc. Sadly, we will not have
time to cover them in this course.
ELECTROMAGNETIC SPECTRUM
Electromagnetic radiation is a transverse wave that propagates at c=2.997925x108 m/s in a
vacuum. In a dielectric medium, c=c(). This is known as dispersion. All waves satisfy the
relation c=  where  is their wavelength and  is their frequency. The reciprocal of
wavelength is called the wave number. Spectroscopists seem to prefer it to wavelength. The
electromagnetic spectrum can be divided into seven broad categories. The table below gives
the upper limit of the wavelengths associated with each category.
Electromagnetic waves
Maximum wavelength (m)
-rays
10-4
X-rays
10-2
ultraviolet
0.39
visible
0.76
infrared
103
microwave
106
radio

INTERACTION OF MATTER AND RADIATION
Consider an isolated molecule. The kinetic and electrostatic potential energy of the electrons
are quantized. So are the rotational and vibrational kinetic energies of the atoms. The
translational kinetic energy, however, is not quantized. The energy state of the molecule (total
quantized energy) is the sum of the electronic, rotational, and vibrational energies. There are
several ways in which changes in the energy state of a molecule may occur.
EMISSION
Emission occurs when a collision leads to an excited energy state, followed by a radiational
transition to a lower state. In this case, thermal (internal) energy is converted into radiant
energy. The emitted energy is in the form of a photon of frequency  given by E=h,
where E is the energy difference between the two states (quantized) and h is Planck’s
constant=6.626x10-34 Js.
Since only certain energies are permitted, the frequency takes on discrete values, which are
characteristic of a particular molecule. This gives rise to a line spectrum Emission lines
associated with electronic orbital transitions are usually in the X-ray, ultraviolet, and visible
portions of the spectrum. Pure vibrational transitions are usually in the infrared (e.g. water
molecules in food vibrating in your microwave), while pure rotational transitions are
frequently in the microwave region of the spectrum. Some molecules (e.g. CO2, H2O, O3) can
undergo simultaneous rotation and vibration collisions, giving rise to line clusters known as
rotation-vibration bands.
ABSORPTION BANDS
A molecule may absorb a photon of a frequency appropriate to raise it to a quantized higher
energy state. Following this absorption the molecule may return to a lower energy state during a
collision. The quantized nature of the energy state transitions of an atom or molecule means that
absorption and emission spectra are quite similar for a given substance.
The net result is that some of the photon’s energy is converted into thermal (translational)
energy of the molecule. Remembering our kinetic theory of gases, this implies that we could
increase the temperature of a gas by shining radiation on it at those wavelengths where the
gas absorbs strongly. This is the case for ozone in our stratosphere (with global climatic
implications).
SCATTERING
A molecule absorbs a photon, following which transition to the original state occurs by
emission of an identical photon, but in a different direction.
PHOTOCHEMICAL DISSOCIATION
A molecule absorbs a photon, which causes it to break down into its atomic components (i.e. it
breaks the chemical bond). An example in the upper atmosphere is the photo-dissociation of
molecular oxygen (O2) into atomic oxygen (O). This process is a precursor to the formation of
ozone. This process involves a continuum of photon energies above a particular threshold,
and these photons come from the Sun.
O2  h  O  O
where h is the energy of the photon and must be higher than that associated with wavelengths
of 0.24 m (i.e. <0.24 m).
PHOTOIONIZATION
A molecule absorbs a photon, which leads to the removal of one or more electrons. This usually
requires <0.1 m, and, as for photochemical dissociation, the photon need not have a discrete
frequency.
MOLECULAR INTERACTION EFFECTS
1)
NATURAL BROADENING
So far we have considered an isolated molecule (except for “instantaneous collisions” that inject
or remove energy). Such a molecule exhibits a line spectrum, whose line widths are determined
solely by the Heisenberg uncertainty principle Et~h/2, where E is the uncertainty in the
transition energy and t is the lifetime of the excited state. This is known as natural
broadening, but in the atmosphere it is too narrow to be of any consequence.
Note: natural broadening can also be explained “classically” in terms of the damping of a harmonic oscillator during
emission (see Liou, p. 16).
2) DOPPLER AND COLLISION BROADENING
Within a gas, natural broadening of absorption lines is greatly enhanced by Doppler
broadening, due to the random translational velocities of the molecules with respect to the
observer, and by collision broadening, which results from the interaction of the electrostatic
force fields of the molecules during collisions, and the resulting small changes in their
quantized energy states. Below about 30 km in the atmosphere, collision broadening is the
dominant mechanism, and it always dominates in the wings of spectral lines.
3) LIQUIDS AND SOLIDS
Here the interaction between molecules is strong because of their proximity, with the result
that the line spectrum is “smeared out” into an essentially continuous spectrum.
RADIATION BASICS II
• Radiance
• Irradiance
• Solar constant
• Heating rate
DEFINITIONS AND CONCEPTS
We will begin by defining several terms used to describe aspects of atmospheric radiative
transfer. Unfortunately, the terminology and symbology in this field have not been completely
standardized, so you will encounter other terms and symbols. I have indicated some of these
in parenthesis below.
Monochromatic (spectral, spectral density) implies radiation of a particular frequency or
wavelength. The term is also used to indicate the value of some quantity per unit interval of
frequency or wavelength.
Integrated implies that the quantity in question has been integrated over a finite wavelength or
frequency interval.
Solid angle is a “three-dimensional” angle. In two-dimensions, the radian measure of an angle
is the ratio of the circumference subtended by the angle to the radius. The equivalent definition
of solid angle is the ratio of the area subtended by the solid angle (think of the angle at the apex
of a cone, for example) to the square of the radius. Solid angles are measured in steradians (sr).
the steradian is a non-dimensional quantity. The solid angle, , can be related to the polar
angle (zenith angle), , and the azimuth angle, , in polar coordinates:
d  sin dd
(28.1)
Radiant flux (radiant power, luminosity), P, is the total radiant energy, Q, per unit time emitted
by or incident upon a surface. Its units are Watts.
dQ
P
(W )
dt
(28.2)
As an example, our Sun has a radiant flux of 3.9x1026 W.
Irradiance (flux density, emittance), F, is the radiant energy per unit time passing through unit
area. It usually refers to the radiant energy arriving from 2 steradians; that is, from the
hemisphere on one side of the surface. In general, the irradiance depends on the orientation of
the surface, which can be specified by the unit vector normal to the surface. Hence the
irradiance can be thought of as a vector whose direction is opposite to that of the surface
normal. Its units are Watts per square metre.
F
dP
dA
(28.3)
Radiant intensity, J, is the radiant flux per unit solid angle from a point source. Its units are
Watts per steradian.
J
dP
d
(28.4)
Radiance (intensity, specific intensity, brightness, luminance), L, is the radiant energy per
unit time and per unit solid angle, passing through unit area normal to the beam and arriving
from a particular direction, or travelling along a particular direction (although this definition
breaks down for a collimated beam). Its units are Wm-2 sr-1 .
Note: In British books, the symbol used for radiance is often I.
d 2P
L ( x, y , z , t ,  ,  ,  ) 
dA cos d
The schematic below illustrates the definition of L and Lambert’s cosine law.
(28.5)
Lambert's cosine law states that the intensity of radiation along a direction which has angle
with the normal to the surface is: L=Lncos where Ln is the intensity of radiation in the
normal direction.
The relation between F and L is simply:
F   L cos d
(28.6)
2
In general, there will be radiant energy travelling in both directions through a particular
surface (e.g. a slab of atmosphere). In order to distinguish between the irradiance in the two
directions, some convention is established (e.g. up and down if the surface is horizontal) and
the two irradiance values are denoted as F+, F- or F, F. The difference between the two
irradiances is called Fnet. As a general rule, one may say that irradiance is measured by a
non-imaging device (e.g. a pyranometer), while radiance is measured by an imaging device
(e.g. a telescope).
Isotropic radiation is a radiation field in which LL(,). In this case F=L. An example of
isotropic radiation that we will encounter next is blackbody radiation.
Diffuse radiation is radiation emitted from a source with a finite solid angle; for example, sky
radiation.
Parallel beam radiation (collimated beam) is radiation from a point source. For our purposes,
the Sun is sufficiently distant that it can be considered to be a point source, and the radiation
from the Sun is essentially parallel beam radiation. For parallel beam radiation, the relation
between F and L becomes:
F  L cos 
where  is the solid angle of the source. For the Sun,
(28.7)
L  2 107 Wm 2 sr 1
The solar constant is the solar irradiance at the annual mean distance of the Earth, measured
on a surface above the atmosphere that is perpendicular to the solar beam.
F  L 
The solar constant is approximately 1353 Wm-2 .
(28.8)
RADIANT ENERGY DENSITY
We will derive an equation for the radiant energy density u (Jm-3 ), in a medium through which
electromagnetic energy is flowing.
After a time dt, the energy which has crossed the area dA in a particular direction will be
located in a cylinder of length cdt, where c is the speed of light in the medium. Dividing the
energy by the volume of the cylinder leads to:
du 
LdtddA cos  Ld

dA cos cdt
c
(28.9)
Integrating over all possible directions from which the radiation may come:
u
1
Ld

c 4
(28.10)
For isotropic radiation, Eq. 28.10 becomes:
u
4L 4 F

c
c
(28.11)
RADIANT HEATING RATE
The radiant heating rate h (Wm-3 ) is simply proportional to the divergence of the irradiance.
Note: this is the analogue of the continuity equation for conservation of mass.
This can be incorporated into the first law of thermodynamics to give:
 
dT
h    F  c p 
dt
(28.12)
where:

F  Fxiˆ  Fy ˆj  Fz kˆ
(28.13)
Fx   L cos  x d
(28.14)
and
4
and x is the angle between the beam and the x-axis (that is, Fx is the net irradiance passing
through the y-z plane; some thought should convince you that it is the net irradiance that
determines the heating rate—hence the integration over 4 steradians rather than 2 steradians).
The other components of the irradiance vector can be obtained by symmetry. Substituting
Eq. 28.14 and the other components into Eq. 28.13, and thence into Eq. 28.12, and using the
fact that cosx =dx/ds, where s is the distance along the beam, leads to:
dL
d
ds
4
h 
(28.15)
Thus, in order to determine the heating rate, we need to know how the radiance varies along
the beam. This will be the subject of Beer’s Law coming up.
RADIATION IN THE ATMOSPHERE
We present here some definitions (consistent with WMO standards) of various quantities
relevant to atmospheric radiation.
Short wave radiation (solar radiation) is taken to be radiation with <2.5 m.
Long wave radiation (terrestrial radiation) is taken to be radiation with >2.5 m.
Direct solar radiation is the radiation from only those directions defined by the Sun’s disk,
falling on a surface normal to the beam.
The vertical component of direct solar radiation (downward direct solar) is the direct solar
radiation received by a horizontal surface.
Diffuse solar radiation (sky radiation) is the downward solar radiation received on a horizontal
surface from 2 steradians (i.e., the upper hemisphere), omitting the solar disk.
Note: At solar elevation angles greater than about 70 o, the ratio of diffuse to direct solar radiation is about 10%.
Global solar radiation is the downward direct and diffuse solar radiation received by a
horizontal surface from 2 steradians.
Atmospheric radiation is the radiation emitted by the atmosphere (both up and down) and the
radiation emitted upward by the Earth’s surface.
Net radiation is the net of the upward and downward solar radiation, or the net of the upward
and downward terrestrial radiation.
BLACKBODY RADIATION
• Planck function
• Stefan-Boltzmann Law
• Wien’s Law
• Kirchoff’s Law
A blackbody is one which absorbs all incident radiation. A blackbody also emits the maximum
possible irradiance at all wavelengths, for a given temperature. It is an idealized concept, but
nevertheless one that can be approximated quite well in the laboratory by a radiant cavity.
The Sun’s emission spectrum is very much like that of a blackbody. Paradoxically, snow emits
and absorbs infrared radiation like a blackbody.
PLANCK FUNCTION
The Planck function describes the blackbody spectrum, that is the blackbody monochromatic
radiance, LB (also denoted by B in some books), as a function of temperature and
wavelength:
2hc 2
1
LB  5
(29.1)
 exp( hc / kT )  1
where h is Planck’s constant (6.625x10-34 Js), and k is Boltzmann’s constant (1.38x10-23 JK-1 ).
The blackbody radiance can also be expressed as a function of frequency using the fact that
LBd=LBd. Since blackbody radiation is isotropic, FB= LB.
STEFAN-BOLTZMANN LAW
Although the Stefan-Boltmann law was derived before Planck’s law, it gives an expression for
the integrated irradiance over all wavelengths (the area under the Planck curve, in fact):

2 4 k 4 4
4
FB   FB d 
T


T
2 3
15
c
h
0
(29.2)
where  is the Stefan-Boltzmann constant, 5.67x10-8 Wm-2 K-4 .
WIEN’S LAW
Wien’s law gives the wavelength of the peak of the Planck function. It can be found by setting
FB
0

(Note that the peak in FB does not occur at the same wavelength.)
This leads to:
p 
2898
T
 ( m),
with
T (K )
(29.3)
KIRCHOFF’S LAW
Absorptivity, a, is defined as the ratio of absorbed radiant energy to incident radiant energy.
That is:
a 
Fabs
Finc
(29.4)
Emissivity, , is defined as the ratio of emitted radiant energy to the radiant energy
emitted by a blackbody at the same temperature and wavelength:
 
F
FB
(29.5)
Kirchoff’s law states that under the conditions of local thermodynamic equilibrium (which
requires that the molecular collision frequency should be much greater than the frequency
of emission or absorption by gaseous molecules, which holds below about 60 km in
the Earth’s atmosphere):
   a
at
same T
(29.6)
Kirchoff’s law is sometimes colloquially phrased as “a good absorber of radiant energy is a
good emitter of radiant energy.” This is essentially true, but keep in mind the caveats that the
wavelength and temperature must be the same for absorption and emission, and local
thermodynamic equilibrium must also hold.
A grey body is also an idealization in which the emissivity is independent of wavelength. It is
a way of sidestepping the need to take into account the line spectrum of real gases by
essentially assuming that their emission is a constant fraction of the emission of a blackbody.
We conclude by defining two additional concepts that will be useful in the upcoming sections.
Reflectivity (albedo) is the ratio of reflected to incident irradiance:
r 
Fr
Finc
(29.7)
Transmissivity (also called transmittance) is the ratio of transmitted to incident irradiance:
t 
Ft
Finc
(29.8)