Lesson 3
METO 621
Basic state variables and the Radiative
Transfer Equation
• In this course we are mostly concerned with the
flow of radiative energy through the atmosphere.
• We will not cover polarization effects in any
detail. This is known as the scalar
approximation, in contrast with the vector
description.
• Of central importance is the scalar intensity.
• Its specification as a function of position,
direction, and frequency conveys all of the
desired information about the radiation field
(except for polarization).
Geometrical optics
• The basic assumptions of the radiative transfer
theory are the same as those for geometrical optics.
• The concept of a sharply defined pencil os radiation
was first defined in geometrical optics.
• A radiation pencil is obtained when light emitted
from a point source passes through a small hole in
an opaque screen.
• We can view this pencil by allowing it to fall on a
second opaque screen.
• An examination of this spot of light shows that its
edges are not geometrically sharp, but consist of a
series of bright and dark bands.
• These bands are called diffraction fringes.
Picture of diffraction fringes
Geometrical optics
• The size of the region over which these bands occur
is of the order of the wavelength of the light.
• If the diameter of the pencil is much larger that the
wavelength of the light then the diffraction effects
are small and can be ignored.
• The propagation can then be described in purely
geometrical terms.
• These rays are not necessarily straight. In general
they are curves whose direction is determined by
the gradient of the refractive index of the medium.
• In this course we will assume that the index of
refraction is constant and ignore it.
Geometrical optics
• For our purposes it is convenient to replace the
concept of a pencil ray with the concept of
incoherent (non-interfering) beams of radiation
• A beam is defined in analogy with a plane wave. It
carries energy in a specific direction and has infinite
extension in the transverse direction.
• When a beam of sunlight is scattered by the Earth’s
atmosphere it is split into an infinite number of
incoherent beams propagating in different directions.
• It is also convenient to define an angular beam as
an incoherent sum of beams propagating in various
directions inside a small cone of solid angle dw ,
centered around a direction W.
Flow of radiative energy
Radiative Flux / Irradiance
• Net energy of radiative flow (power) per unit
area within a small spectral interval dn is called
the spectral net flux
3

d E
2
1
Fn 
W .m .Hz
dAdtdn

• We define two positive energy flows in two
separate hemispheres
F n
d 3E 

dAdtdn
3 
d
E

Fv 
dAdtdn
Radiative Flux / Irradiance
• Net energy flow in the +ve direction is

d E  d E d E
3
3
3

• The net flux is also written in a similar manner


Fn  Fn  Fn
• Summing over all frequencies we obtain the net
flux or net irradiance

F   dn Fn
0
W .m 
2
Spectral Intensity
• Defined as
4
d E
In 
cos dAdtddn
The spectral intensity is the energy per unit area,
per unit solid angle, per unit frequency and per unit
time. cos.dA is the projection of the surface
element in the direction of the beam.
Intensity is a scalar quantity and is always positive.
Flux and intensity
• We can rewrite the equation for Iν
d E  In cos  dAdt d dn
4
• The rate at which energy flows into each
hemisphere is obtained by integrating the
separate energy flows
d E  d E
3

4


, d E  d E
3

4


Flux and Intensity
• We can now define expressions for the halfrange flux
3 
d
E

Fn 
  dw cos In
dAdtdn 
3

d E
F 
   dw cos In
dAdtdn


n
• Where the spectral net flux is given by
Fn  Fn  Fn   d cos In
4
W .m
2
.Hz 1

Polar diagram
Polar Coordinate system
• Each point on the surface of a sphere can
be represented by three coordinates
• The distance of the point from the origin, r
• The angle in the xy plane, θ, known as the
polar angle.
• The angle in the xz plane, Φ, known as the
azimuthal angle
Polar coordinate system
• The area bounded by dθ and dΦ has
dimensions of r.dθ and r.sin θ. dΦ and the solid
angle associated with this area is
A r.d .r sin  .d
d  2 

d

.
sin

d

2
r
r
Average intensity and energy density
• Averaging the directionally dependent intensity
over all directions gives the Spectrally Average
Intensity
1
In (r ) 
4
 dwIn (r.) W .m
2
.Hz
1

4
• The energy density is given the symbol Un ,
where
d E In cosdAdtdnd In
dUn 

 d
dVdn
dA cos cdtdn
c
4
Energy density
• The energy density in the vicinity of a
collection of incoherent beams traveling in all
directions is given by
Un 

4
1
4
d Un   d In 
In
c 4
c
(J.m3 .Hz 1 )
The total energy density is the sum over all frequencies
U

 dnUn
0
(J.m2 )
Isotropic distribution
• Assume that the spectral density is independent of
direction.
• We have previously defined the flux in either
hemisphere as
Fn   d cos  In
  d cos  In sin  d
  d In sin  d (sin  )
 /2
 sin  
 In 2 

 2 0
  In
2
Isotropic distribution
• Note that


Fn  Fn  Fn  0
4 In
Un 
c
Hemispherically Isotropic Distribution
• This situation is the essence of the two stream
approximation - to be discussed later.
• Let I+ denote the value of the constant intensity in the
positive hemisphere and I- that in the negative
hemisphere.
• For a slab medium we have replaced an angular
distribution of intensity with two average values, one for
each hemisphere.
• Although a first sight this may seem inaccurate, for
some radiative transfer calculations it is surprisingly
accurate.
2
 /2
0
0
2
 /2
Fn  In  d



0
 /2
 d sin  cos  In  d  d sin  cos

  In  In 
In
Un 
c
2
In
 d  d sin   c
0
0

2  
In  In 

c
2

0
 /2
 d  d sin 
Collimated distribution
• Approximation used for the intensity of an
incoming solar beam (the finite size of the sun
is ignored). We write the solar intensity as
S
ˆ
ˆ 
ˆ )
I ()  Fn  (
0
S
n
where
ˆ 
ˆ )   (   ) (cos  cos )
 (
0
0
0

n
Fn  F
2
 /2
 d  (  )  sin  cosd  (cos  cos )
0
0
 Fn cos 0
0
 /2