MET 60
Chapter 4:
Radiation & Radiative Transfer
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The layout of chapter 4 is:
• Basics of radiation
• Scattering, Absorption & Emission of radiation
• Radiative Transfer
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Basics of radiation
• Properties of radiation (pp. 113-117)
– wavelength, frequency etc.
– Intensity vs. flux
– Blackbody radiation
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Basics of radiation cont.
• Basic Radiation Laws (pp. 117-120)
– Wien’s Law & Stefan-Boltzman Law
– About the type and amount of radiation emitted
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Scattering, Absorption & Emission of radiation
• Emissivity, absorptivity, transmissivity, reflectivity (p.
120)
– All relate to things that can happen to radiation as it
passes through the atmosphere
• The Greenhouse Effect (p.121)
• The physics of scattering (pp. 122-125)
– Type & amount of scattering depends on number,
size & shape of particles in the air
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Scattering, Absorption & Emission of radiation contd.
• The physics of absorption
– Lots of details! (pp. 126-130).
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Radiative Transfer
Putting it all together to follow a beam of incident
radiation:
• From top of atmosphere to the surface
• And back up
• With interactions along the beam (scattering, absorption
etc.)
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Radiative Transfer contd.
With the result being:
• A vertical profile of heating rates due to radiation
• e.g., in the form of the values of
 T 

 z 
 t 
Remember that radiative heating drives the atmosphere!
– Vertical distribution (here)
– Horizontal distribution (climatology-related)
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Basics of Radiation
• The sun emits radiation (type? amount?)
• Earth intercepts it and also emits its own radiation (type?
amount?)
• Radiation is characterized by:
– Frequency () … measured in “per sec”
– Wavelength () … measured in micrometers (µm) or
microns
– Wavenumber (-1)
• Note: all EM radiation travels at the speed of light (c) with
c=
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The EM spectrum…
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Solar radiation consists mainly of:
• uv radiation (? – 0.38 µm)
• visible radiation (0.38 – 0.75 µm)
• IR radiation (0.75 - ? µm)
– Near-IR has  < 4 µm
– Far-IR has  > 4 µm
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Terms…
• Monochromatic intensity of radiation is the amount of
energy at wavelength  passing through a unit area (normal
to area) in unit time
I
• Adding over all wavelengths (all values of ), we get
radiance, or intensity: I
• I is also called radiance
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Terms…
• Monochromatic flux density (irradiance) is the rate of energy
transfer through a plane surface per unit area due to
radiation with wavelength 
F
• For, say, a horizontal surface in the atmosphere:
F   I  cos d
2
Integrate over ½ sphere
Monochromatic
radiance
Accounts for radiation
arriving in slanted direction
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Note…on confusion!
http://en.wikipedia.org/wiki/Irradiance
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Inverse Square Law…
• Flux density (F) obeys the inverse square law:
F  1/d2
where d = distance from source (sun!)
sun
earth
149 million km
mars
227.9 million km
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Blackbody Radiation…
A surface that absorbs ALL incident radiation is called a
blackbody
All radiation absorbed – none reflected etc.
• Hypothetical but useful concept
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Blackbody Radiation…
• Radiation emitted by a blackbody is given by:
c1
B (T ) 
c2 T
e
1
5


Planck Function

T
• c1 and c2 are constants
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Blackbody Radiation…
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Blackbody Radiation…
• Fig. 4.6 shows how emission varies with  for different
temperatures
• Choosing T values representative of the sun and of earth
gives Fig. 4.7 (upper)
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Wavelength of peak emission?
• Wien’s Displacement Law…
1
max 
T
• or
max 
2897
T
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Solar radiation …
Peaks in the visible
Concentrated in uv-vis-IR
Terrestrial radiation …
Peaks in the IR (15-20 m)
All in IR (far IR)
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Maximum intensity of emission?
• Stefan-Boltzmann Law…
F  T
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So the sun emits much more radiation than earth
since Tsun >> Tearth
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Example 4.6
Calculate blackbody temperature of earth (Te).
Assume: earth is in radiative equilibrium
 energy in = energy out
Assume: albedo = 0.3 (fraction reflected back to space)
Assume: solar constant = 1368 W/m2
= incoming irradiance/flux density @ top of atmosphere
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Incoming energy:
Given by solar constant
spread over area of Earth = area the beam intercepts
area = Re2
Thus incoming = 1368 x (1 – 0.3) x Re2
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Outgoing energy:
Given by
Fe = Te4
where we need to find Te
Now this is per unit area, so the total outgoing energy is
Fe = 4Re2Te4
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Equating:
4Re2Te4 = 1368 x (1 – 0.3) x Re2
 4Te4 = 1368 x (1 – 0.3)

Te = {1368 x 0.7 / 4}¼

Te = 255 K
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Non-blackbody radiation
A blackbody absorbs ALL radiation
A non-blackbody can also reflect and transmit radiation
Example – the atmosphere!
Actually, the gases that make up the atmosphere!
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Definitions:
I
emissivity   
B
actual radiation emitted / BB radiation
BB has  = 1
absorptivity
I  (absorbed )
 
I  (incident )
radiation absorbed / radiation incident
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reflectivity
I  (reflected )
R 
I  (incident )
radiation reflected / radiation incident
transmissivity
I  (transmitted )
T 
I  (incident )
radiation transmitted / radiation incident
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incident
reflection
absorption
absorption
transmission
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incident
reflection
scattering
absorption
absorption
transmission
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Kirchoff’s Law
   
emissivity = absorptivity (at Em)
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An example regarding the greenhouse effect…
1)
Pretend the atmosphere can be represented as a single
isothermal slab
The slab is transparent to solar radiation (all gets through!)
The slab is opaque to terrestrial radiation (none gets through!)
Everything is in Em.
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outgoing = F units for
balance
incoming = F units
F
“top”
F
2F
F units emitted downwards
F
z=0
Surface receives 2F units
Surface must emit 2F units for balance
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Now use the 2F units of radiation emitted by the surface to
compute Te via Stefan-Boltzman.
F = 1368 W/m2 modified by albedo
result:
Te = 303 K
 Greenhouse effect delivers 48 K of “warming” (single slab
model)
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2)
Pretend the atmosphere can be represented as two
isothermal slabs…or three etc. – see text
per p.122, Te = 335 K etc.
Note:
include more layers → steeper lapse rate in lower atmosphere
eventually …  > d … unstable atmosphere “predicted”
→ use a Radiative-convective model instead…”convective
adjustment”
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Physics of Scattering, Absorption & Emission
Need to understand physics of these processes to come
up with expressions for how much radiation is scattered
etc. from a beam.
Scattering
Consider a “tube” of incoming radiation – Fig. 4.10.
Radiation may be scattered by:
– gas molecules (tiny)
– aerosol particles (small – tiny).
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Physics of Scattering, Absorption & Emission
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Scattering contd.
Scattering amount depends on:
1)
2)
3)
Incident radiation intensity (I)
Amount of scattering gases/aerosols
Ability of these to scatter (size, shape etc.)
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Scattering contd.
For an incident intensity of I, an amount dI is lost by
scattering, with
dI   I K N ds.
N = number of particles (gas, aerosol) per unit volume.
 = c/s area of each particle
ds = path length (see diagram)
K = (scattering or absorption) efficiency factor (large “K”)
Note:
K(total extinction) = K(scattering) + K(absorption)
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Scattering contd.
For a gas, we write:
dI   I  rk ds.
r = is the mass of the absorbing gas per unit mass of air
 = air density
k = mass absorption coefficient (m2kg-1)(small “k”)
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Scattering contd.
For a column (“tube”) of air from height z to the top of the
atmosphere, we can integrate:

  rk dz.
z
This represents the amount of absorbing material in the
column down to height z.
Called the optical depth or optical thickness (  ).
Large   much extinction in the column.
Note that  is wavelength-dependent.
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Scattering is very complicated.
Scattering particles have a wide range of sizes and shapes
(and distributions).
Start by looking at a sphere of radius r.
???
How does this scatter?
Extinction is given by Eq. 4.16 – need to know K - provided
by theory (which we will not do!!)
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First…Fig. 4.11
y-axis: r = scattering radius (m)
x-axis:  = wavelength (m)
Plotted is:
x
2 r

Fig. 4.11 shows us the different regimes of scattering that
occur as a function of:
-
wavelength of radiation (solar vs. terrestrial)
size of scattering particle
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Results of theory…
With small particles (x << 1), we get Rayleigh Scattering
And theory gives:
K   4
Particles scatter radiation forward and backward equally!
Fig. 4.12a.
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As particle size increases, we get more forward scattering…Fig.
4.12 b,c.
For larger particles with x > 1, we get Mie Scattering.
In this case, values of K are oscillatory - Fig. 4.13.
Note: an index of refraction has entered
m = mr + imi
mr = (speed of light in vacuum) / (speed of light through particle)
mi = absorption (mi = 0  no absorption; mi = 1  complete absorption)
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Example 4.9…the sky is blue because…
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K (blue)  0.64  m 

  3.45
K (red )  0.47  m 
Blue light is scattered 3.45 times more efficiently than red
light!
ALSO…p.124 2nd column
…tells us that to understand satellite imaging and
retrievals, as well as weather radar etc., we need to
apply the ideas in this section.
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Absorption by non-gaseous particles
Not much information BUT read last sentence of p.126
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Absorption - and emission - by gas molecules
Energy arrives, is emitted and absorbed in discrete
amounts called photons
Having energy
E = h
And c = 

E = hc/
h = Planck’s constant
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Atomic energy states
An atom has electrons in orbit around the nucleus
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For the electron to jump into a higher orbit (higher energy
level), a discrete amount of energy must be absorbed
So only discrete orbits are allowed
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When this discrete amount of energy (E) is absorbed, a
spectrum of absorption versus wavelength shows a
spike.
absorption
wavelength (m)
Finite absorption at certain wavelengths.
Zero absorption otherwise (transparency).
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→ line spectrum
Each species → different line spectrum
All overlap & combine in the atmosphere
Adding molecules → additional complications
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Energy of a molecule, E is:
E = Eo + Ev + Er + Et
translational energy
Energy due to
electron orbits
in atoms
Energy due
to vibration
of molecule
Energy due to
rotation of molecule
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As a result, the spectrum is more complicated.
Adding E of energy (e.g., incident from the sun) can result
in changes to the rotational state of the molecule, ditto
vibrational, ditto electron states etc.
→ complex absorption spectrum (one for each species)
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Examples:
http://www2.ess.ucla.edu/~schauble/molecular_vibrations.htm
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Examples:
Atmospheric
Absorption spectra
for the main gases
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Examples:
http://en.wikipedia.org/wiki/Electromagnetic_spectroscopy
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