1. Ch 13: Radiative Transfer with Multiple Scattering
Ch 13: Radiative Transfer with Multiple Scattering. Primer: Saturn’s Moon Enceladus.
Why do we use this wavelength range? Why not use visible or UV?
How do we know if it’s water vapor, ice particles, or liquid water?
Figure 13: Heat map (within white box) of the thermally active field of fractures in saturn’s moon Enceladus, measured at wavelengths between 12 and 16 micrometres, superimposed on a visual-light image. One of the four fractures (right) was only partially imaged. (wikipedia).

2. Two Stream Approximation: Multiple Scattering in 1 dimension.
0, top of atmosphere
I↓(z)
I↑(z) z
extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓.
arbitrary layer dz
z + dz
I↓(z+dz)
I↑(z+dz)
h, ground level
extdz = Probability a photon undergoes extinction in dz.
absdz = Probability a photon is absorbed in dz.
scadz = Probability a photon is scattering in dz.
P↓↑= P↑ ↓ = Probability a downward photon is scattered up, and vica versa.
P↑ ↑= P ↓ ↓ = Probability an upward photon is scattered up, and vica versa.
P↓↑+ P↑ ↑ = 1 ⇒ all of the choices for a scattered photon in 1 dimension.

3. Conservation of energy in dz for downward intensity (or flux): (seeking relationships between the fluxes above and below dz).
0, top of atmosphere
I↓(z)
I↑(z) z
extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓.
arbitrary layer dz
z + dz
I↓(z+dz)
I↑(z+dz)
h, ground level
Gain of downward flux by layer dz = Loss of downward flux by layer dz.
(No ↓ flux is generated in the layer by emission. Easy to do emission later.)
I↓(z)+ sca P↑ ↓ dz I↑(z+dz) = absdz I↓(z) + sca P↓ ↑ dz I↓(z) + I↓(z+dz)
absorption scattering transmission

4. Conservation of energy in dz for upward intensity (or flux): (seeking relationships between the fluxes above and below dz).
0, top of atmosphere
I↓(z)
I↑(z) z
extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓.
arbitrary layer dz
z + dz
I↓(z+dz)
I↑(z+dz)
h, ground level
Gain of upward flux by layer dz = Loss of upward flux by layer dz.
(No ↑ flux is generated in the layer by emission. Easy to do emission later.)
I↑(z+dz)+ sca P↓ ↑ dz I↓(z) = absdz I↑(z+dz) + sca P↑ ↓ dz I↑(z+dz) + I↑(z)
absorption scattering transmission

5. Form Differential Equations from the Difference Equations Derived
0, top of atmosphere
I↓(z)
I↑(z) z
extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓.
arbitrary layer dz
z + dz
I↓(z+dz)
I↑(z+dz)
h, ground level

6. Aside: Asymmetry Parameter of Scattering, g. -1<g<1
nr=1.33
=0.6328
D=20 um
g=0.874
Is()  I0

7. Scattering Relationships: Example and the Asymmetry Parameter g
Here P↓↓=3/4. P↓↑=1/4.
P↓↓+ P↓↑ = 1.
incoming
photons
back scattered
photon
g≡ P↓↓ - P↓↑ in 1-D.
g = P↓↓ - (1- P↓↓ )
Solving,
P↓↓=(1+g)/2 = P↑↑
P↓↑=(1-g)/2 = P↓↑
particle
forward scattered
photons

8. Relationships for Extinction, Scattering, Absorption, and the Single Scatter Albedo: Coupled de’s for the fluxes.
Fundamental equations
we use for everything.
Fluxes are coupled
by scattering.

9. ground is a totally absorbing surface,
Special Case: No Absorption, Single Scatter Albedo = 1. Reflection and Transmission Coefficients, R and T.
R≡I↑(0)/ I0
I↓(0)≡I0
0, top of atmosphere

T≡I↓() / I0
h, ground level
 
ground is a totally absorbing surface,
I↑( ) ≡0.

10. Solving for the Case where Single Scattering Albedo=1 (no absorption)

11. Features of the Solution for R and T with no Absorption
g and  are not uniquely determined by R and T measurements,only the product 1-g) is uniquely determined.
g=1, forward scattering only, then R=0, T=1.
g=-1, R≠1 because of multiple scattering, R=    
However, dilute milk will be colored blue (Rayleigh scattering)

12. Features of the Multiple Scattering Solution Continued ...1 T
“Photons are lost to the downward stream only if they are scattered in the opposite direction”

13. Direct and Diffuse Transmitted Radiation
Diffuse = Total - Direct
LWP = Cloud Water Mass / Area
Qext = Cloud droplet extinction efficiency
CCN = # cloud condensation nuclei
Cloud optical depth
cloud H I0 It Ir
nr=1.33
=0.6328
D=20 um
g=0.874
figure 1   

14. Can show that the downwelling diffuse dadiation in the single scattering limit is matches expectations from direct integration of the radiative transfer equation in the single scattering approximation (done in class for Rayleigh scattering).

15. Summary of Multiple Scattering Equations: 1 D model.

16. R and T:

17. Reminder from Chapter 7 Presentation

18. Optical Depth from kext: Liquid Water Path
ztop
Liquid Water Path
zbot
Somewhere there has to be an integral over z!

19. Homework Problem: Aerosol Indirect Effect
Reproduce the figure on the next slide using the simple model with absorption for the values of  = 1 and  ≠1. Calculate the cloud albedo as a function of effective radius and liquid water path for single scattering albedo equal to 1.0, and In the second problem, assume that the absorption is caused by black carbon aerosol embedded in the cloud. Calculate the absorption coefficient necessary to give the value of the single scattering albedo as a function of the liquid water path. Comment on the likelihood of observing these absorption coefficients. Finally, comment on how aerosol light absorption impacts the aerosol indirect effect (i.e. the increased cloud albedo because of smaller more numerous droplets).
Note: the mean free path of photons between scattering events is = 1 / sca.
Tdir = exp(- exth)= exp(-) = probability that photons pass through the general medium without interaction with the scatterers and absorbers. (Ballistic, unscattered photons useful for imaging in scattering medium with fast lasers that can gate out scattered photons that arrive later due to their larger path length).

20. Cloud Liquid Water Path, Effective Radius, And Cloud Albedo
Does this make sense? Why?
How do things change when the single scattering albedo is not equal to 1, and absorption happens?
grams / m2
Global Survey of the Relationships of Cloud Albedo and Liquid Water Path with Droplet Size Using ISCCP.Preview By: Qingyuan Han; Rossow, William B.; Chou, Joyce; Welch, Ronald M.. Journal of Climate, 7/1/98, Vol. 11 Issue 7, p1516.

21. Cloud above a Reflecting Ground
T2Ag
T2AgRAg
T2Ag(RAg)2
T2Ag(RAg)n
TAg
TAgRAg
TAgRAgRAg
T(AgR)n
TAg (AgR)n T
TAgR
TAgRAgR
ground has reflectance, (or albedo) = Ag.
Rtotal= R+T2Ag+ T2AgRAg+ T2Ag(RAg) T2Ag(RAg)n + ...
Rtotal= R+T2Ag / (1- AgR)
Ttotal= T+TAgR+ T(AgR) T(AgR)n + ...
Ttotal = T / (1- AgR)

22. Features of a Cloud above a Reflecting Ground
I↓(0)≡I0 R
T2Ag
T2AgRAg
T2Ag(RAg)2
T2Ag(RAg)n
TAg
TAgRAg
TAgRAgRAg
T(AgR)n
TAg (AgR)n T
TAgR
TAgRAgR
ground has reflectance, (or albedo) = Ag.
Ag=0
Rtotal= R, Ttotal = T
General Relationship:
Rtotal= R+T2Ag /(1- AgR)
Ttotal = T / (1- AgR)
Ag=1
Rtotal= R +T2 /(1- R)
Ttotal = T/(1-R)
Cloud Absorption, A
Atotal= (In - Out)/I0
Ag=1, R+T=1 (conservative case)
Rtotal= 1
Ttotal = 1

23. Weirdest thing (study the Ag = 1 case, R + T = 1 conservative case)
I↓(0)≡I0 R T2
T2R
T2(R)2
T2(R)n T TR
TRR
T(R)n T TR
TRR
T(R)n
ground has reflectance, (or albedo) = Ag=1 in this case.
Ag=1
Rtotal= R +T2 /(1- R)
Ttotal = T/(1-R)
Let R=0.99. With 100 photons incident, 99 immediately reflect upwards and are lost.
One photon passes through and reflects 99 times between the ground and cloud before being lost by transmission to the upward direction.
This is the basis of an optical buildup cavity and integrating spheres.
Ag=1, R+T=1 (conservative case)
Rtotal= 1
Ttotal = 1
Case shown: R ≈ 1. Radiation is ‘trapped’ between the cloud and ground. A very small amount is reflected besides the first reflection. Energy is conserved because In = Out, or 1 = Rtotal

24. Conservative Case: Example of Ttotal

25. Energy Conservation for the Conservative Case
I↓(0)≡I0 R
T2Ag
T2AgRAg
T2Ag(RAg)2
T2Ag(RAg)n
TAg
TAgRAg
TAgRAgRAg
T(AgR)n
TAg (AgR)n T
TAgR
TAgRAgR
ground has reflectance, (or albedo) = Ag.
Energy Conservation:
In to system = Out of system
= Rtotal + Agnd = 1 (can show with algebra)
General Relationship:
Rtotal= R+T2Ag /(1- AgR)
Ttotal = T / (1- AgR)
Total Absorption by Ground, Agnd
Agnd= (Ingnd - Outgnd)/I0
Agnd = Ttotal (1- Ag)
Can show with algebra, or by inspection

26. Additional Relationships and Limits for the General Case:
deep multiple scattering with some absorption

27. An Example