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 d is called the spectral net flux

dAdtd

EdF

3

• We define two positive energy flows in two separate hemispheres

dAdtd

EdFv

3

123

.. HzmWdAdtd

EdF

Radiative Flux / Irradiance

• Net energy flow in the +ve direction is EdEdEd 333

• The net flux is also written in a similar manner FFF

• Summing over all frequencies we obtain the net flux or net irradiance

2

0

.mWFdF

Spectral Intensity

• Defined as

ddAdtd

EdI

cos

4

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

dddAdtIEd cos4

• The rate at which energy flows into each hemisphere is obtained by integrating the separate energy flows

EdEdEdEd 4343 ,

Flux and Intensity

• We can now define expressions for the half-range flux

IdwdAdtd

EdF

IdwdAdtd

EdF

cos

cos

3

3

• Where the spectral net flux is given by

12

4

.. cos HzmWIdFFF

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

ddr

drdr

r

Ad sin.

.sin..22

Average intensity and energy density

• Averaging the directionally dependent intensity over all directions gives the Spectrally Average Intensity

12

4

..).(4

1)( HzmWrdwIrI

• The energy density is given the symbol U , where

dc

I

cdtddA

ddAdtdI

dVd

EddU

cos

cos4

Energy density

• The energy density in the vicinity of a collection of incoherent beams traveling in all directions is given by

U 4 d U

1

cd I

4

4cI (J.m 3.Hz 1)

The total energy density is the sum over all frequencies

U dU0

(J.m 2)

Isotropic distribution• Assume that the spectral density is independent of

direction.• We have previously defined the flux in either

hemisphere as

I

I

dId

dId

IdF

2/

0

2

2

sin2

)(sinsin

sincos

cos

Isotropic distribution

• Note that

c

IU

FFF

4

0

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.

F I d d sin cos

0

/ 2

0

2

I d d sin cos

/ 2

0

2

I I

U I

cd d sin

0

/ 2

0

2

I

cd d sin

/ 2

0

2

2cI I

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

)ˆˆ()ˆ( 0 SS FI

)cos(cos)()ˆˆ( 000

0

0

2/

2/

2

0

0

cos

)cos(coscossin)(

F

ddFF

where