AOSC 621AOSC 621Phase Function & RTE Solution

Solution of the RTE with scattering

Solving equation that includes scattering is difficult, because it is difficult to evaluate the integral in the scattering source term

The problem is that

Sscatt

p• I is peaked (huge peak at sun)• P is peaked (diffraction peak)

Peakiness is problem because while smooth functions can be integrated using a coarse angular resolution (e.g., using f L d l ial t s) ak f ti s d a hi h s l ti ( t s) a d this ak s th

dI

ds

dI

d I 1 a B a

1

4P ,

4 I d

2

few Legendre polynomial terms), peaky functions need a higher resolution (more terms), and this makes the calculations much more lengthy.

Sample ice crystal phase functionsp y p f

22° and 46° halos

3

Methods for solving the RTE with scattering

Approximate methods:

2 stream approximation 2-stream approximation Eddington approximationIntegral form of radiative transfer equationSingle scattering approximationSingle scattering approximation

Accurate methods:

Successive orders of scatteringPrinciple of reciprocityAdding-doublingDiscrete ordinatesSpherical harmonics

M t t th d

4

Most accurate method:Monte Carlo

Azimuthal Dependence

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

2N1

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

l 0

2N1

(cos)l 0

where Plis the lth Legendre Polynomial

Legendre PolynomialLegendre Polynomial

Legendre Polynomialsg y

• The Legendre polynomials comprise a natural g p y pbasis set of orthogonal polynomials over the domain (0 180)

• Legendre polynomials are orthogonal1

luPuduP lkkl

12

1)()(21 1

1

klklforlk for 0but , 1 where

Legendre expansion & Henyey-Greenstein Function

Goal: describe phase function (P) using few parameters so that it can be handled easily in equation of radiative transfer

P cos 2l1 2N 1

l Pl cos

Pl is lth order Legendre polynomial(function for any x between –1 & 1)

l is case specific Legendre coefficient, given by

l0

Pl x 1

2l l!dl

dxlx2 1 l

P0 x 1 P1 x x P2 x 1

23x2 1 P3 x 1

25x3 3x

1

1

d l 1

2P cos

1

Pl cos d cos 0 1

gdPdP cos21coscoscos

21 1801

1

Simple approximation that uses only three terms to get: Henyey-Greenstein phase function:

22 01

P 1 g2

1 g2 2gcos0 1 1 g 2 g2

Henyey and Greenstein (1941) devised an h h h l d d

8

expression which mimics the angular dependence of light scattering by small particles

Mie-Debye Phase FunctionMie Debye Phase Function

Azimuthal Dependence

• The first moment of the phase function is l d t d b th b l commonly denoted by the symbol g=

• This represents the degree of assymetry of h l i d i ll d hthe angular scattering and is called the

assymetry factor. Special values of g are• When g=0 - isotropic scattering• When g=-1 - complete backscatteringg p g• When g=+1- complete forward scattering

Azimuthal Dependence

• We can now expand the phase function

),;','()(cos uupp

where)'(cos)'();''(12

muupuupN

m

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

120

muupuup

Nm

)()'()12()2(),'( 0 uuluup ml

ml

mllm

m

Azimuthal Dependence

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

I( u ) Im2N1

( u)cosm(0 )I(,u,) Im 0

(,u)cosm(0 )

W i di i f • We can now write a radiative transfer equation for each component

Azimuthal Dependence

)(),,( IudI mm

),,(),,(

12a

uId

u m

)',',(),;''.(''4 10

uIuupduda mm

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

0

/0

0

BaeuX

m

m

),,()2(4

),( where

),, , ,()()(

000

0

upFauX mm

Sm

m

4

Examples of Phase Functions

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

pRAY (cos) 3

4(1 cos2)

4),;","( uupRAY

)"(cos)1)("1("143 22222 uuuu

)"cos()1()"1("24

2/122/12 uuuu

Rayleigh Phase Function

• The azimuthally averaged phase function is

pRAY (u',u) 1

2d ' pRAY

2

(u', ';u,)2 0

31 '2 2

1(1 '2 )(1 2)

41 u'2 u2

2(1 u'2 )(1 u2)

• In terms of Legendre polynomials

( ' ) 11

P ( )P ( ')pRAY (u',u) 12

P2(u)P2(u')

Rayleigh Phase Function

• The assymetry factor is therefore

1

1 )'(),'('1 uPuupdug RAYl

11

1

1

)(),(2

uPuupdug RAYl

1

1

0),'(''21 uupudu RAY

1

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 i l t ti i h ll iimplementation is challenging

• Scaling transformations

Scaling Transformationsg

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

• Approaches a delta functionpp• Can treat the forward peak as an unscattered

beam, and add it to the solar flux term.,

• Then the remainder of the phase function is d d i L d P l i lexpanded in Legendre Polynomials.

):''(ˆ)(cosˆ uupp

)(cosˆ)12()1()cos1(2

),:,()(cos12

NNN

Plff

uupp

)(cos)12()1()cos1(20

ll

l Plff

• This is known as the approximation

• There are simpler approximations• There are simpler approximations

The Isotropic Approximation

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

)1()'(2),'(ˆ fuufuup ISO

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

The Isotropic Approximation1)(dI

1

)',(),'('2

),(),( uIuupduauId

udIu

1

)',('2

)1(),(),( uIdufauafIuI 12

dI ˆ)ˆ(

or1

uIduauId

udIu

)',ˆ('2ˆ

),ˆ(ˆ

),ˆ( 1

1

f )1(where

1

afafadafd

1

)1(ˆ and )1(ˆ

The Isotropic Approximationf ispeakscatteringfor ardtheofstrengththef

iprelationsh by thegiven ispeak scattering forwardtheofstrength the

fuupudu ISO ),'(''21 1

1 fp ISO

),(2 1

1

The -Two-Term Approximation• A better approximation results by

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

)'(ˆ uup

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

),(1

uPuPlfuuf

uup

ll

l

l

TTA

)()()()()(0

ff lll

l

The -Two-Term Approximation• Substituting into the azimuthally averaged

radiative transfer equation:q

),ˆ(ˆ

),ˆ(

uId

udIu

11

)'ˆ()(')(ˆ)12(

uIuPduuPladl

10

h

),()()()12(2

uIuPduuPl lll

l

dˆˆ

where ffgfl

2and11

fffg

ffg l

l