# Meteorology 342

```8. Atmospheric Radiation
8.1 Basic concept and definition
8.1 Basic concepts and definitions
* Energy propagated in the form of an advancing electric and
magnetic field disturbance.
* Travels in wave form at the speed of light (c).
* Wavelength (  ) is the physical distance between adjacent
maxima or minima in the electric or magnetic field. unit:  m
* Wave number ( k ) is the number of wavelengths in a unit
distance, i.e., k  1 /  unit: 1/cm
* Frequency ( ) is the number of successive maxima or
minima passing a fixed point in a unit of time,   c / 
unit: cycle-per-second (cps) or 1/s
* The energy associated with electromagnetic radiation is
contained in discrete packets called photons.
* The energy contained in a photon is proportional to
frequency, and is given by W  h
where h is Planck’s constant, h  6.6261034 J s
2. Electromagnetic spectrum:
* Classified into bands
in terms of frequency
(or wavelength)
* Most significant spectral
regions associated with
in atmosphere lie
between ultraviolet light
and microwaves.
3. Solid angle
• The ratio of the area of
the sphere intercepted by
the cone to the square of
–    / r2
• What is the area cut out
of a sphere by one
• What is the solid angle
representing all directions
at a point?
A meteorological satellite circles the earth at a height h above the earth’s
surface. Let the radius of the earth be ae and show that the solid angle
under which the earth is seen by the satellite sensor is
  2 [1  (2ae h  h2 )1/ 2 /(ae  h)]
– Energy/time (J/s or Watt)
– W/m2
– Defines the irradiance coming from a particular direction.
5. Absorption
• a = absorbed/incident = spectral absorptance
• r = reflected/incident = spectral reflectance
•  = transmitted/incident = spectral transmittance
• a + r +  = 1
6. Scattering
* A physical process by which a particle in the path of an
electromagnetic wave continuously abstracts energy from
the incident wave and reradiates that energy in all directions.
~ 104 m
~ 1m
* Gas molecules
Aerosols
Water droplets
Ice crystals
~ 10m
~ 100m
Large raindrops and hails
~ 1cm
* Small particle: scatter light, equally in forward and backward directions
Larger particle: scatter more in forward direction, more complex
* Size parameter:
The ratio of the particle circumference to the incident wavelength
x  2a / 
a
Rayleigh scattering:
x
Lorenz-Mie scattering:
&lt;&lt; 1
x 1
* Multiple scattering
Scattering more than once, important
process for the transfer of radiant
energy when aerosols and clouds
are involved.
* Extinction:
A result of scattering plus absorption, which
removes energy from a beam of light traversing
the medium and the light beam is attenuated.
Why does grass look green?
Why is the sky blue?
Planck’s Law
* Blackbody
– all incident radiation is completely absorbed.
– In all ’s, the maximum possible emission is realized.
* The amount of radiation emitted is uniquely determined
by temperature.
– Planck’s law.
* Relate the emitted intensity to the wavelength and the
temperature of emitting substance.
* Rayleigh-Jeans distribution:
Wien distribution:   0

Stefan-Boltzmann Law
• Total radiant intensity emitted by a blackbody.
• Integrate Planck’s law over all wavelengths.
• F =  T4
: Stefan-Boltzmann constant
Wein’s Displacement Law
• Wavelength of maximum intensity
• m = 2897/T,
units: m (m), T (K)
• Allows brightness temperature to be inferred from the radiation
emitted by an object
Kirchhoff’s Law
• A molecule which absorbs radiation of a
particular wavelength also is able to emit
• Rate of emission is a function of temperature
and wavelength
• Consider:
–
–
–
–
Two parallel plates of infinite extent (wall 1 and wall 2)
Both plates are at the same temperature.
Radiative transfer must be same in all directions.
Wall 1 has an absorptance a1 = 1
Absorptivity ( a ) and emissivity (   )
Kirchoff’s law:
a   
strong emitters  strong absorbers
weak emitters  weak absorbers
* Non-blackbodies absorb less radiation than do blackbodies.
* The ratio of absorbed monochromatic irradiance to blackbody
absorbed monochromatic irradiance is known as absorptivity.
* Non-blackbodies emit less radiation than do blackbodies.
* The ratio of emitted monochromatic irradiance to blackbody
monochromatic irradiance is known as emissivity.
* For a non-blackbody (or a gray body), a     1
the Stefan-Boltzmann law is F  T 4
Practice Problem
reaching the earth’s orbit is 1380 W/m2. Nearly
all the radiation is emitted from the outer-most
visible layer of the sun, which has a mean radius
of 7x108 m. Calculate the equivalent blackbody
temperature of this layer? The mean distance
between the earth and sun is 1.5 x1011 m.
Practice Problem
• Calculate the equivalent blackbody temperature
of the earth, assuming a planetary albedo of
0.30. Assume that the earth is in radiative
equilibrium, so that there is no net energy gain
– Planetary albedo  the fraction of the total incident
solar radiation that is reflected back into space
without absorption.
Meteorology 342
Homework (8)
1. a) Find the equivalent blackbody temperature of Mars. Assume the irradiance at the top of the earth’s
atmosphere is 1370 Wm 2, the distance from the earth to the sun is 1.5 1011 m , the distance from
Mars to the sun is 228106 km , and that Mars has an albedo of 0.17.
and the radiative temperature of the earth.
2. An infrared scanning radiometer aboard a meteorological satellite measures the outgoing radiation
emitted from the earth’s surface in the 10 m window region. Assuming that the effect of the atmosphere
between the satellite and the surface can be neglected, what would be the temperature of the surface
if the observed radiance at 10  m is 9.8Wm 2 m1sr 1 ?
o
3. A black land surface with temperature of 15 C emits radiation at all frequencies. What would be
1
the emitted radiance at 0.7  m , 1000 cm , and 31.4 GHz? Use appropriate Planck functions in
the calculations.
o
4. Assuming the average normal body temperature is 98 F , what would be the emittance of the body?
If it is not a blackbody but absorbs only 90% of the incoming radiation averaged over all wavelengths,
what would be the emittance in this case? Also, at which wavelength does the body emit the maximum
energy?
1. The equation of radiative transfer
* Reduction of the radiant intensity due to absorption
* Increase in the radiant intensity due to emission and multiple scattering
dI
 I  J 
k ds
k : mass extinction cross section
2. Beer-Bouguer-Lambert Law
The decrease in the radiant intensity traversing a homogeneous
extinction medium is in accord with the simple exponential function
whose argument is the product of the mass extinction cross section
and the path length.
I   I  (0)e 
s
    k  ds
Optical depth:
0
3. Schwarzschild’s equation
dI
  I   B (T )
k ds
* The increase in the radiant intensity arising from blackbody
emission of the material.
* Reduction of the radiant intensity due to absorption
1. An energy source: the Sun
* Composed mostly of hydrogen (75% by mass) and
helium (25%), with traces of iron, silicon, neon, and
carbon
* Solar interior
a. Core:
Temperatures of 15 million K,
Density as high as 160,000 kg/m 3
Thermonuclear reactions generate gamma- and x-rays
Sun produces radiation in its high density core
2. The earth’s orbit around the sun
* All of the planets of the solar system revolve in elliptic orbits around
the sun, which is one of the foci of the ellipses.
* All of the planets except Mercury and Pluto orbit nearly in the same
plane, known as the ecliptic.
* The eccentricity of the earth’s orbit is currently about 0.0167, so it is
very close to being a circle.
The average earth-sun distance is defined as an astronomical unit (AU)
and has a value of 149,597,870 km.
The earth is farthest from the sun (1.0167AU) at aphelion, which occurs
near July 4.
The earth is closest to the sun (0.9833AU) at perihelion, which occurs
near January 2.
3. Solar constant:
The total solar irradiance reaching the top of the earth’s atmosphere
on a surface perpendicular to the solar beam at the mean distance
between the sun and the earth.
Solar Spectrum
The recently proposed solar constant
2
is about 1366 Wm . It varies with the
irradiance from the sun and changes
with the sunspot cycle, but it doesn’t
vary by much.
The solar constant value is critical in the interpretation of
measured solar absorption and heating rates in the
atmosphere.
Two methods for determining the solar constant:
1) Satellite measurement with self-calibrating radiometers
Making use of Beer-Bouguer-Lambert law
4. Solar insolation
Insolation is defined as the solar irradiance on a horizontal area, is a
Insolation depends on the zenith angle and the variable distance of
the earth from the sun.
The insolation integrated with time over the hours of daylight will give
the energy per unit area per day received at a point on the earth.
Example: On a clear day, measurements of the direct solar flux density F at
the earth’s surface in a wavelength interval give the following values:
Zenith angle (degree)
40
50
60
70
F ( Wm 2 )
13.95 12.55 10.46 7.67
Find 1) the solar flux density at the top of the atmosphere, and
2) the transmissivity of the atmosphere for normal incidence in this
wavelength interval.
5. Atmospheric absorption and scattering of solar radiation
* Absorption
is coming in from an infinitely small solid angle. For such radiation we
dF  sec0k F dz
dF F 0

  exp(   )
dz
H
* Scattering
Rayleigh scattering:
For a scatter of a given size, the scattered intensity is
inversely proportional to the fourth power of wavelength.
Shorter wavelengths are scattered much more efficiently than
longer wavelengths.
I ~
1
4
6. Solar heating rates
The absorption of solar radiation results in the heating
of the atmosphere.
The heating rate is proportional to the amount of energy
absorbed.
1 dF( z )
 T 
  
c p dz
 t  s
1. Infrared spectrum
* As the sun emits radiation over
all wavelengths, the earth and
the atmosphere also emit
* Solar and infrared spectra are
separated into two spectral
ranges above and below ~5m,
and the overlap between them
is relatively small.
* The emitted Planck radiance from the earth and the atmosphere
is smaller than that of solar radiation, but the wavelength for the
intensity peak is longer.
* The equilibrium temperature of the earth-atmosphere system
Solar Spectrum
Infrared Spectrum
2. Absorption of infrared radiation in the atmosphere
* The greenhouse effect (the atmospheric effect)
Energy emitted from the earth is absorbed by atmospheric
gases such as carbon dioxide, water vapor, ozone).
* The atmospheric window
The atmosphere is relatively transparent from wavelength
8 m to 12.5 m , except for ozone which has an absorption
band in the 9.6 m region.
Atmospheric absorbers in the IR region:
* Ozone (consists of three oxygen atoms)
O3 Is a very strong absorber in the UV, has rotational and vibrational transitions,
and absorbs strongly in the IR centered on the wavelength near9.6. m .
* Carbon dioxide (a linear molecule)
CO2 Has no permanent electric dipole, but as it vibrates a dipole can be
generated. Thus, it has vibrational bands, which are very important
for the atmosphere. Major bands are centered at 15m and 4.3m .
* Water vapor (a bent molecule)
H 2O has strong absorption in many
bands throughout the IR region, and is
a strong absorber in the microwave
region.
* Nitrous oxide, methane, and
carbon monoxide
N 2O and CH 4 have two very strong
absorption bands in the IR.
CO has one strong band in the IR.
3. Emission and infrared radiative transfer equation
A pencil of infrared radiation traversing an absorbing and
emitting medium will be weakened by the absorption, but it
may be strengthened by thermal emission from the medium.
(or intensity) I .
dI 
 k  sec 0 ( I    B )
dz
dI 
 k  sec 0 ( B  I  )
dz
4. Infrared cooling rates
Because the atmosphere loses radiative energy to space
through thermal infrared emission, it is normally cooled
by this process.
 /2
F  2 
0
I   cos sin d

F  ( z )   F  d
0
F ( z)  F  ( z)  F  ( z)
1 dF( z )
 T 
  
c p dz
 t ir
Infrared radiation between cloud and surface
1. Radiation budget at the top of the atmosphere (TOA)
F  (1  r )Q  Fir
energy in the tropics
and subtropics
Losses in the polar
regions
2. Radiation budget at the surface
F (0)  Fs (0)(1 rs )  [ sTs4   s Fir (0)]
3. Radiative heating/cooling rates of the atmosphere
Atmospheric solar heating
is mainly produced by the
absorption of water vapor
in the troposphere and of
ozone in the stratosphere.
serves to cool the atmosphere,
amount of energy equivalent
to the solar input, maintaining
solar heating almost everywhere.
cooling in the earth’s
atmosphere is compensated for
by other forms of energy in the
atmosphere and by the
transport of heat from the
surface.
Equilibrium temperature vs Surface temperature
Q  S /4
Q(1  r )  Ta4  (1   )T 4  0
Q(1  r  A)  Ta4  T 4  0
Example: Assume that the atmosphere acts as a single
isothermal layer with a temperature T that transmits