METO 621

Lesson 11

Azimuthal Dependence• In slab geometry, the flux and mean intensity depend only on and . If we need to solve for the intensity or source function, then we need to solve for , and . However it is possible to reduce the latter problem to two variables by introducing a mathematical transformation.

p(,) (2l 1)( )Pl

l0

2N 1

(cos)

where Plis the lth Legendre Polynomial

Azimuthal Dependence

• The first moment of the phase function is commonly denoted by the symbol g=

• This represents the degree of assymetry of the angular scattering and is called the assymetry factor. Special values of g are

• When g=0 - isotropic scattering

• When g=-1 - complete backscattering

• When g=+1- complete forward scattering

Legendre Polynomials

• The Legendre polynomials comprise a natural basis set of orthogonal polynomials over the domain

• The first five Legendre polynomials are

• P0(u)=1 P1(u)=u P2(u)=1/2.(3u2-1)

• P3(u)=1/2.(5u3-3u) P4(u)=1/8.(35u4-30u2+3)

• Legendre polynomials are orthogonal

(0 180)

klklfor

luPuduP

lk

lkkl

for 0but , 1 where

12

1)()(

2

1 1

1

Azimuthal Dependence• We can now expand the phase function

l

m

mmlll

N

ll

muuuPuP

luupp

1/

12

0

)'(cos)()'(2)()'(x

)12(),;','()(cos

• Inverting the order of summation we get

)()'()12()2(),'(

where)'(cos),'(),;','(

12

0

12

0

uuluup

muupuup

ml

ml

N

mllm

m

N

m

m

Azimuthal Dependence

• This expansion of the phase function is essentially a Fourier cosine series, and hence we should be able to expand the intensity in a similar fashion.

I(,u,) Im

m0

2N 1

(,u)cosm(0 )

• We can now write a radiative transfer equation for each component

Azimuthal Dependence

),,()2(4

),( where

1)-,2N0,1,2,....=(m )()1(

),(

)',',(),;''.(,''4

),,(),,(

000

0

/0

1

1

2

0

0

upFa

uX

Ba

euX

uIuupduda

uId

udIu

mm

Sm

m

m

mm

mm

Examples of Phase Functions•Rayleigh Phase Function. If we assume that the molecule is isotropic, and the incident radiation is unpolarized then the normalised phase function is:

pRAY (cos) 3

4(1 cos2 )

)"cos()1()"1("2

)"(cos)1)("1("14

3

),;","(

2/122/12

22222

uuuu

uuuu

uupRAY

Rayleigh Phase Function

• The azimuthally averaged phase function is

pRAY (u',u) 1

2d ' pRAY

0

2

(u', ';u,)

3

41 u'2 u2

1

2(1 u'2 )(1 u2)

• In terms of Legendre polynomials

pRAY (u',u) 11

2P2(u)P2(u')

Rayleigh Phase Function

• The assymetry factor is therefore

1

1

1

1

1

0),'(''2

1

)'(),'('2

1

uupudu

uPuupdug

RAY

RAYl

Mie-Debye Phase Function

Mie-Debye Phase Function

• Scattering of solar radiation by large particles is characterized by forward scattering with a diffraction peak in the forward direction

• Mie-Debye theory - mathematical formulation is complete. Numerical implementation is challenging

• Scaling transformations

Scaling Transformations

Scaling Transformations

• The examples shown of the phase function versus the scattering angle all show a strong forward peak. If we were to plot the phase function versus the cosine of the scattering angle - the unit actually used in radiative transfer- then the forward peak becomes more pronounced.

• Approaches a delta function• Can treat the forward peak as an unscattered beam, and add it to the solar flux term.

Scaling Transformations

• Then the remainder of the phase function is expanded in Legendre Polynomials.

)(cosˆ)12()1()cos1(2

),:','(ˆ)(cosˆ12

0

l

N

ll

NN

Plff

uupp

• This is known as the approximation

• There are simpler approximations

The Isotropic Approximation

• The crudest form is to assume that, outside of the forward peak, the remainder of the phase function is a constant, Basically this assumes isotropic scattering outside of the peak. The azimuthally averaged phase function becomes

)1()'(2),'(ˆ fuufuup ISO

• When this phase function is substituted into the azimuthally averaged radiative transfer equation we get:

The Isotropic Approximation

1

1

1

1

)',('2

)1(),(),(

)',(),'('2

),(),(

uIdufa

uafIuI

uIuupdua

uId

udIu

af

afadafd

uIdua

uId

udIu

1

)1(ˆ and )1(ˆ

where

)',ˆ('2

ˆ),ˆ(

ˆ),ˆ(

or1

1

The Isotropic Approximation

fuupudu

f

ISO

),'(''2

1

iprelationsh by thegiven

ispeak scattering forward theofstrength the

1

1

1

The -Two-Term Approximation

• A better approximation results by representing the remainder of the phase function by two terms (setting N=1 in the full expansion). We now get:

)()'()12()1()'(2

),'(ˆ1

0

uPuPlfuuf

uup

ll

l

ll

TTA

The -Two-Term Approximation• Substituting into the azimuthally averaged radiative transfer equation:

2

1

1

1

0

and 11

ˆˆ

where

)',ˆ()(')(ˆ)12(2

),ˆ(ˆ

),ˆ(

ff

fg

f

fg

uIuPduuPla

uId

udIu

ll

ll

l

ll