3 scattering of radiation by molecules and...

12.815, Atmospheric Radiation Dr. Robert A. McClatchey and Prof. Ronald Prinn

3. Scattering of Radiation by Molecules and Particles

a. Introduction

Here, we’ll deal with wave aspects of light rather than quantum aspects.

Consider components of electric field in 2 mutually perpendicular directions, parallel and perpendicular to the plane of propagation and propagating in the z direction:

tE = a e i(ω −kz) ω = circular frequency r r

i( t kz) E = a e ω − k = 2π (1)

The intensity is given by:

* 2 2I = E E* + E E = a + a (2) r r r

Let us first consider Single Scattering. We may consider a single particle or a small volume of particles such that scattering events will all be single scattering events.

Fig. 1

p = phase matrix p′′ = phase function

11 dV P I dV 0I = k P I0 or I = hsca sca4 R2 4 R2π π

k = scattering cross-section per unit volume sca

Nσscak = = 1 ∑ σ , i [ksca] = length −1

sca scadV dV i 1

[σsca] = length2

N = total no. of particles σ , i

sca Qsca,i = where G i = geometrical cross sec tion Gi

12.815, Atmospheric Radiation Lecture Dr. Robert A. McClatchey and Prof. Ronald Prinn Page 1 of 16

Figure by MIT OCW.

σsca ∑ σ , isca = Scattering efficiency = Qsca Qsca G ∑Gi

P is the phase matrix and provides the angular distribution and polarization of the scattered light. For our purpose here, lets consider the total intensity of the radiation whether polarized or not. Then the term, p11 , is the phase function or scattering diagram which defines the probability for scattering of unpolarized incident light in any direction. p11 is normalized such that:

∫p dΩ

= 1 where d Ω = element of solid angle (3)4π

Let us define <cos α> = ∫ cos α p dΩ

where α is the scattering angle (see Fig.4π

<cos α> = anisotropy parameter (or asymmetric parameter) and can vary between +1 and -1. <cos α> = 0 for isotropic scattering.

We define analogous terms for particle absorption and we have:

= k + kabs kext sca

σ = π r Qσext = σ sca + σ abs 2

Qext = Qsca + Qabs

(4) Then, we define:

σksca sca Qsca ω = single scattering albedo = = = kext σext Qext

For practical applications, ksca and ω can be taken as constants and is a function only of the scattering angle, α.

This special case is valid for:

(1) randomly oriented particles, each of which has a plane of symmetry (2) randomly oriented asymmetric particles, if half the particles are mirror

images of the others. (3) Rayleigh scattering and Mie scattering.

2 aπAnother important definition: χ = where a = particle radius.

=m n − ini where nr = real part of the index of refraction and ni = imaginary part. r

ni is responsible for absorption and nr is responsible for scattering.

For water, nr = 1.33 across the visible and near infrared. And ni will depend on the kinds of materials dissolved in the water drops. It will therefore be much more of a function of wavelength.

Returning to the analytical expression for the electric field as a beam of radiation passing through a single particle:

i( t kz)E(z, t) = E e ω − 0

The intensity of radiation varies as the square of the E-field. So, we have:

t2 2 i(ω −kz)I = E0 e k = 2π

and λ =λ0

⎡ − π2 z ⎤i ⎢ ( r −in ) + ωt⎥

⎦So, we have : E(z, t) = E0 e ⎣ λ0i

2 zni 2 z r ⎞− π ⎛ − πi ⎜⎜ + ωt ⎟⎟λ0 e ⎝ λ0 ⎠= E e0

⎛ − π π4 z i ⎞ ⎛ 4 z r ⎞

⎝ λ0 ⎠ e ⎝ λ0 ⎠⎜ ⎟⎜ ⎟ i ⎜⎜ − + ωt ⎟⎟

and I E02 e

2 aπand – using the definition of size parameter χ = where we will take z = 2a =

λdiameter of drop, we have:

− χ z2 4 ni i( 4 r + ω t)I = E0 e

absorption

− χ e

b. Mie scattering – still single scattering.

E at distance R in the

s 4 , i⎧ ⎫ exp( ikR + ikz) ⎪⎧S (α θ) S (α θ)⎫ ⎧E ⎫ far field⎪ ⎪ −

⎨ 1 , ⎪ ⎪ ⎪r r⎨ ⎬ = ⎬ ⎨ ⎬ (6)

s⎪ ⎪ ikR ⎩ 3 α θ 2 α θ ⎭ ⎪E i ⎪⎩ ⎭ ⎪S ( , ) S ( , )⎪ ⎩ ⎭E

If we consider isotropic, homogenous, spheres, we have:

⎧⎪S ( ) 0 ⎫⎪ 1 α S = ⎨

α⎬ I =

2 2 F 0 The S values are in general

⎪ 0 S ( )⎪ k R⎩ 2 ⎭ complex number and functions of scattering angle.

And we have the transformation matrix:

⎫⎧1 * * (S S − S S2 ) 0 0 ⎪⎪2 (S S + S S2 ) 1 * *

1 1 2 2 1 1 2

⎪ ⎪ ⎪ ⎪ ⎪1 * * (S S + S S2 ) 0 0 ⎪⎪ (S S − S S2 ) 1 * *

⎪⎪2 1 1 2 2 1 1 2 ⎪

⎪F = ⎨ ⎬ ⎪ 1 *⎪ 0 0 (S S + S S* ) i (S S* − S S* )

⎪⎪

⎪ 2 1 2 2 1 2 1 2 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ * * 1 * * ⎪ ⎪ 0 0 −

i (S S − S S1 ) (S S + S S1 )⎩ 2 1 2 2 2 1 2 2 ⎪⎭

which is proportional to the phase matrix: F = C P (8)

11 F dΩThe normalization condition on P leads to: C = ∫ 4π4π

and since σ = effective cross section, we have: sca

2 F11 IR d Ω dΩ k2σscaσ = ∫ and C = ∫ = sca 4π 4π4π I0 4π

So – we have:

F11 k2σ p 1sca * = = (S S* + S S ) defines phase function 2 1 1 2 24π

F21 k2σ p 1sca * = = (S S* − S S ) (9) 2 1 1 2 24π

21 43 * *F33 =

k2σsca p =

1 (S S* + S S ) F43 =

k2σsca p = −

i (S S* − S S )

2 1 2 2 1 2 1 2 2 14π 4π

For a single sphere, we have:

S1 = ∑ 2n + 1 ⎣anπ + b τ ⎤⎡ n n n ⎦

n 1 n(n + 1) =

S2 = ∑ 2n + 1

n 1 n(n + 1) ⎣⎡bnπ + τ ⎤ (10) n an n ⎦

π and τ are function only of α and relate to Legendre Polynomials. n n

na and nb are functions of 2 a

and r im n in= − and involve Spherical Bessel

functions. We also have:

* *sca n n n n2

2Q (2n 1)(a a b b )

= + +∑

ext e n n2n 1

2Q (2n 1) R (a b )

= + +∑ (11)

e n n 1 n n 1 e n n2n 1sca

4 n(n 2) 2n 1cos R (a a b b ) R (a b )

n 1 n(n 1)x Q

+ +< α > = + +

+ +∑

All above is for Single Scattering from a Single Sphere. In general, if optical thickness is not too large, single scattering can be applied to a distribution of particles assumed to be independent. We then have:

sca sca scar r

k (r)n(r)dr r Q (r)n(r)dr= σ = π∫ ∫

ext ext extr r

k (r)n(r)dr r Q (r)n(r)dr= σ = π∫ ∫ (12)

where n(r) = size distribution, describing the number of the particles having radii between r and r+dr over the range r1 to r2.

Figure by MIT OCW.

c. Geometric Optics:

When r >> λ, we can use ray theory of light due to Fresnel (see Van de Hulst, Ch. 3, Light Scattering by Small Particles). Terminology is:

0 – diffraction

1 – external reflection

2 – double refraction

3 – first rainbow

4 – second rainbow

θ = ∑P ( ), then 1 2π π

P ( ) sin θ θ φ , is for non-absorbing spheres and if P( ) θ θ d d ∫ ∫4π 0 0 =0

from Fresnel theory.

real = 1.33 = 2.00

always true in geometric optics 0 .500 .500

1 .033 .081 often sufficient to consider just these

2 .442 .364

3 .020 .043

4 .033 .008

5 .002 .004

For non-absorbing particles, diffraction = 1\2 of scattering light. Thus, the geometric optics limiting value of Qext = 2.0.

Figure by MIT OCW.

d. Rayleigh Scattering:

λ r λ and r where m = n + i irm

Radiation penetrated particle to be in homogenous quickly. ∴ particle own field is external electric field negligible in the process.

Particle can be considered

Ek2αp incident sin βe−ikn where β = angle between dipoleE = (13)sca R moment and direction of scatter

3 (1 + co 2 α) − 4

sin2 α 0 0 ⎫ ⎪ ⎪ ⎪−

4 3 sin2 α (1 + co 2 α) 0

⎪ ⎪⎪ 0 ⎪⎪α = ⎨

⎬ (14) ⎪ 0 0

p( ) cos α 0 ⎪

⎪ ⎪ ⎪⎪ 0 0 3

2 cos α⎪⎪ 0⎩ ⎭

p11 = 3 (1 + cos2 α) (15)

which gives angular distribution of intensity scattered by small particles – the Rayleigh scattering.

We also obtain:

2 2m − 1 ⎧m − 1⎫ = x Qabs = −4 x m ⎨ 2 ⎬ (16)Qsca

8 4 23 m + 2 ⎩m + 2⎭

note that Qabs > as x → 0Qsca

note 4th power of x or λ-4 dependence This result is the same as Mie scattering as limiting case as x → 0 (r<<λ).

12.815 Lecture Notes (Atmospheric Radiation)

Multiple Scattering

Refer back to Eq. 22 from the first set of Atmospheric Radiation lecture notes where we discussed Case III which arises due to the following two conditions:

>> νB (T) (1) Fν

I(θ’,φ’,τν)>>Bν(τν) (2)

The resulting equation of transfer is:

µ dIν θ φ, τ = I ( , ) θ φ, τν) (3) ( , ) θ φ, τ − J ( ,

τd ν ν ν ν ν

ω oθ φ τν) =

π ∫P( , , ', ') I ( ', ', 'd 'd ' 4 0, 0 νwhere J ( , , θ φ θ φ ν θ φ τν) sin θ θ φ +ω

e−τν / µ P( θ φ )F ν 4

and the formal solution is:

τ d 'µ µ φ φ ) e(τ −τ ') / µ τν

ν, , 0,I( τ µ µ φ φ) ↑ = ∫ J( τν ', , 0, , 0 ν ν µ > 00,

(4) τνs d '

νI( τ µ µ φ φ ↓ = − J( τ ', , 0, , 0 ( ' µ < 0ν, , 0, , 0) ∫ ν µ µ φ φ ) e − τ −τν ) / µ τν

τν µ

Due to the complexities of evaluating the integrals in Eq. 4, a number of techniques have been used to generate numerical results:

1. Discrete Ordinates 2. Doubling or Adding Method 3. Successive Orders of Scattering 4. Iteration of Formal Solution 5. Invariant Embedding 6. Method of X and Y Functions 7. Spherical Harmonics Method 8. Expansion in Eigenfunctions 9. Monte Carlo Method

We will focus some attention on the Discrete Ordinates Method and apply an available computer program to some exercises.

Radiative Transfer in a Scattering Atmosphere

1. Coordinate system in a “plane parallel” atmosphere

Here position defined by z (or τ) only. Recall that optical depth τ related to altitude z by dτ = -αdz where α is the extinction coefficient.

Notation

cos θ = µ ; θ normal

cos θo = µ o ; θo normal

cos θ‘ = µ‘ ; θ

= inclination to upward outward

‘ = inclination to upward outward

normal

Figure by MIT OCW.

From spherical geometry, the cosine of the scattering angle, α can be expressed in terms of the incoming and outgoing directions in the form:

' 2 1 2 2cos α = µµ + (1 − µ ) (1 − µ ' )

1 2 cos (φ − φ) (5)'

Let us now digress for a moment and examine the properties of Legendre polynomials (which come to play in a variety of ways in radiative transfer problems).

We may consider writing the phase function in terms of Legendre polynomials in the form:

P(cos α) = ∑ C P (cos α) (6) =0

Legendre polynomials have the following form, and orthogonal and recurrence properties:

µ = 1 P ( )1 d

n 2 nP ( ) n µ =

n(µ − 1) (n = 1,2.....) 0 2 n! d µ

3 2 1So, P ( ) 1 µ = µ P ( ) = µ − ..........2 µ

1 ⎧0 ≠ h⎪P ( )P ( ) d µ = ⎨ 2 (7) µ k µ∫

⎪ = h−1 ⎩2 + 1

+ 1 µ + P ( ) µP (h) = P ( ) µ 2 + 1 +1 2 + 1 +1

Using Eq. 5, the Phase Function defined above may be written as follows:

N 2 2 ′,P( , , ′ ′) = ∑ ⎢

(1 − µ′)12 cos ( θ − θ )⎤ (8)

µ θ µ φ C P ⎡µ µ + (1 − µ ) ⎥⎣ ⎦

From the orthogonality condition, the expansion coefficients are given by:

2 + 1 1

C = P ( ) P ( ) d µ = 0,1......N 2 −

where we note that the phase function is normalized to unity:

1 2π 1

µ µ φ ≡ 1 4π ∫ ∫ P( )d d

0 −1

There is an addition theorem for Legendre polynomials which allows us to write the Phase Function as follows:

m m µ mP( , , ′ ′) = ∑ ∑C P ( )P ( ) cos m( φ − φ ) (9)µ φ µ φ µ′ ′

=m 0 =0

where Pm denotes the Associated Legendre polynomials and:

( − m ! m )C (2 − δ0,m)C

( + m !, = m,......N 0 ≤ m ≤ n (10) )

δ ⎧1, m = 0

0, m = ⎨ ⎩0, otherwise

In view of the expansion of the phase function, the diffuse intensity may also be expanded in a cosine series in the form:

τ µ φ = ∑ Im τ µI( , , ) ( , ) cos m( φ − φ ) (11)0m 0 =

Substituting Eqs. 9 and 11 into Eq. 3, and using the orthogonality of the associated Legendre polynomials, the equation of transfer splits into (N+1) independent equations, and may be written as:

Nω m mµ dIm ( , )

= I ( , ) 4 =m

µτ µ m τ µ − (1 + δ 0,m) ∑ C P ( ) (12)

m µ′ m τ µ′) d µ′x ∫ P ( )I ( , −1

Nω m m µ m

4π =0

C P ( )P (−µ0)F e− ∑ 0

τ− µ

m = 0, 1, …… N

Let us rewrite these equations as follows:

τ µ m τ µ − Jm ( , ) (13)µ dIm ( , )

= I ( , ) τ µ dτ

with the source function given by:

N 1 m m mm τ µ µ P ( ) I ( ,

4J ( , ) = (1 + δ 0,m)

ω ∑ C P m ( ) ∫ µ′ τ µ′) d µ′

=m −1 (14) Nω m m µ m+ ∑ C P ( )P (−µ )F e 0

τ− µ

4π 0 =m

To proceed with the solution of Eq. 13, we first discretize the equation by replacing µ with µi (i= -n,…., n, with n= 1, 2,….) and the integral with a sum with the weights, aj

f( )d jµ µ = ∑ f( µ j) a (15)∫ −1 j=−n

The homogeneous solution for the set of first-order differential equations may be written:

n m m m m µ kj τI ( , τ µ ) = ∑ Lj ψ j ( ) e − (16)i i

j=−n

mwhere ψ ( ) and kj denote the eigenvectors and eigenvalues, respectively, j µ m i

mand Lj are coefficients to be determined from appropriate boundary

conditions. On substituting Eq. 16 into the homogeneous part of Eq. 13, the eigenvectors may be expressed by

nm m m m

ψ j ( ) = (1 + δ0,m) ω

∑ C P ( ) ∑ a P ( µq) ψ (µq) (17)µi m µi q m

j4(1 + µ j k ) =m q=−nj

The particular solution may be written in the form

m m µ −τ

µ0 (18)I ( , τ µ ) = Z ( )e p i i

From Eq. 13, we have

N m ω m mZ ( ) µ = ∑ C P ( ) (19)µi ⎛4 1 +

µ⎟µ

j ⎞ =m i

⎜⎝ 0 ⎠

⎛ n m m m x ⎜ ∑ a P ( µq)Z ( µ ) + P ( −µ )

π ⎠

⎞⎟q q 0

⎝ q=−n

m mEquations 17 and 19 are linear equations in Ψ j and z and may be solved

numerically. The complete solution for Eq. 13 is the sum of the general solution for the associated homogeneous system of the differential equations and the particular solution. Thus,

I ( , m m −kj τ m µ −τ

µ0m τ µ ) = ∑ Lj ψ j ( ) e m

+ Z ( ) e (20)i µi ij=−n

i = -n, ………. +n

mIn order to determine the unknown coefficients, L , aq, boundary conditions j

must be imposed.

In the discrete-ordinates method for radiative transfer, analytical solutions for the diffuse intensity are explicitly given for any optical depth. Thus the internal radiation field can be evaluated without additional computational effort. And furthermore, useful approximations can be developed from this method for flux calculations.

Advantages of Discrete Ordinate Method

a) In principle - numerical computations can be done for any order of approximation.

b) The internal radiation field is determined - not just the Reflection & Transmission.

c) Accurate results (to about 1%) are achievable with only a few streams (3-4) for most cases.

We will utilize the Discrete Ordinate computer program to do a few excercises.

Multiple Scattering Computational Techniques

1. Discrete Ordinates (We’ll discuss in detail in a few minutes.) 2. Doubling or Adding

Principle: If reflection and transmission is known for each of two layers, the reflection and transmission from the combined layer can be obtained by computing the successive reflections back and forth between the two layers. If the two layers are chosen to be identical, the results for a thick homogenous layer can be built up rapidly in a geometric (doubling) manner.

3. Successive Orders of Scattering Principle: Intensity is computed individually for photons scattered once, twice, three times, etc. with the total intensity obtained as the sum over all orders. If the intensity is expanded in a Fourier series, the high frequency terms arise from photons scattered a small number of times. Therefore, most Fourier terms can be obtained with some accuracy by computing a few orders of scattering.

4. Iteration of Formal Solution Direct solution of integral over source function by dividing atmosphere into layers with small optical thickness.

5. Invariant Imbedding Differential Equations are developed which give the change of reflection and transmission matrices when an optically thin layer is added to the atmosphere. It is a special case of the Doubling or Adding technique.

6. Method of X and Y Functions Involves the determination of integral equations for functions which depend upon only one angle and are directly related to Reflection and Transmission matrices. The integral equations need to be solved numerically. The integral equations are completely specified by a character function depending on the particular phase function. This method is due to Chandrasekhar.

7. Spherical Harmonic Method Intensity is immediately expanded into a finite number of spherical harmonics and then the Phase Function is expanded in Legendre polynomials similar to the Discrete Ordinate method.

8. Expansion in Eigenfunctions Standard technique for solving differential equations. Find homogenous solution and particular solution. Apply boundary condition. Direct application to complete RTE is ponderous. Discrete Ordinates technique depends on this approach for solving discretized set of equations.

9. Monte Carlo Method Scattering of an individual photon can be considered to be a stochastic process, with the Phase Function being the probability density function for scattering at a given angle. Photons are allowed to play a game of chance in a computer and by recording the history of a sufficient number of photons, the radiation field can in principle be determined to an arbitrary accuracy. The basic simplicity of this method allows great flexibility, and hence it can be applied to complicated problems which would be virtually insoluble by other methods.

Isotropic Scattering and Discrete Ordinates

Pertinent RTE:

ω µ dI( , , ) µ φ τ −

4π ∫∫P ( , , ′, ′, ′, ) d ′ dµ φ τ = I( , , )

µ φ µ φ′) I( µ φ τ µ φ′

τ− µ0 P ( µ φ )F (1) 4 0, 0

For isotropic scattering, we have:

, P ( , , ′ ′) = 1 and I ( , ) µ φ τµ φ µ φ µ τ = I( , , ) d φ (2)2π ∫

i.e. – Intensity is azimuthally independent.

− τ

µdI( , )

τ µ −ω 1

I( , ′ τ µ

= I( , ) ∫ τ µ′) d µ −ω

F e µ0 (3)dτ 2 −1 4π

Applying Gaussian Quadrature, and setting Ii = I (τ, µi), we have:

+nω − τ

µ dIi = I i − ∑ I aj −

ωF e µ0 (4) i d 2τ j=−n

i = −n,........., +n

Since this is linear differential equation, we need to seek the general solution (sometimes called the homogenous solution) and then the particular solution.

Homogenous solution:

kTry (guess) I i = i − τ where gi and k are constants. g e

dIi = Ii − ∑ I aj (5)i dτ 2 j j

ω∴ g (1 + µ k) = ∑a g i i 2 j

So, gi must be of the form L1 + µ k where L is a constant.

Substituting this back into Eq. 5, we get the characteristic equation for eigenvalue k

+n aj n ajω ∑ = ω ∑ 2

= 1 2 j=−n (1 + µ j k) j 1 (1 − µ2 k )=

Note difference in summation

This Eq. has 2n roots, ±kα α = 1……..n which when ω = 1 includes 2Kα values of zero.

General Solution is:

I i = ∑ L ± α e∓kατ

i = −n,......, +n (7) α=1 1 ± µki α

Particular Solution:

ω = F he

τ− µ0 i = −n,......, +nTry: I i π i4

+n−µWe have: i hi

1 ∑ a hj − 1

µ= hi −

j=−nj

+n⎛ ior h 1 + µµ

⎞ =ω ∑ a hj + 1 (8) i ⎜

⎝ 0 ⎠⎟ 2 j=−n

γhi must be of the form =

µ+ (9)

i1 µ0

with γ = ⎜1 − ω ∑ aj ⎟

⎞−1

(10)⎜

j 1 [1 − µ j4µ2 ⎟⎝ = 0 ⎠

Adding the homogenous and particular solutions, we obtain:

+ωF γ e

I i = ∑L e− τ

τ− µ0

(11) j 1 1 + µi k j π ⎜

⎛ µ

i= 4 1 + µ ⎞⎝ 0 ⎠

i = -n, …….. +n

The Lj are determined from boundary conditions.

3 scattering of radiation by molecules and...

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