1. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer 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). Ch 13: Radiative Transfer with Multiple Scattering. Primer: Saturn’s Moon Enceladus. How do we know if it’s water vapor, ice particles, or liquid water? Why do we use this wavelength range? Why not use visible or UV?

2. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Two Stream Approximation: Multiple Scattering in 1 dimension. h, ground level 0, top of atmosphere z z + dz arbitrary layer dz I ↓ (z) I ↑ (z) I ↓ (z+dz)I ↑ (z+dz)  ext dz =  abs dz +  sca dz. P ↓↑ = P ↑ ↓. P ↑ ↑ = P ↓ ↓.  ext dz = Probability a photon undergoes extinction in dz.  abs dz = Probability a photon is absorbed in dz.  sca dz = 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. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Conservation of energy in dz for downward intensity (or flux): (seeking relationships between the fluxes above and below dz). h, ground level 0, top of atmosphere z z + dz arbitrary layer dz I ↓ (z) I ↑ (z) I ↓ (z+dz)I ↑ (z+dz)  ext dz =  abs dz +  sca dz. P ↓↑ = P ↑ ↓. P ↑ ↑ = P ↓ ↓. 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) =  abs dz I ↓ (z) +  sca P ↓ ↑ dz I ↓ (z) + I ↓ (z+dz) absorption scattering transmission

4. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Conservation of energy in dz for upward intensity (or flux): (seeking relationships between the fluxes above and below dz). h, ground level 0, top of atmosphere z z + dz arbitrary layer dz I ↓ (z) I ↑ (z) I ↓ (z+dz)I ↑ (z+dz)  ext dz =  abs dz +  sca dz. P ↓↑ = P ↑ ↓. P ↑ ↑ = P ↓ ↓. 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) =  abs dz I ↑ (z+dz) +  sca P ↑ ↓ dz I ↑ (z+dz) + I ↑ (z) absorption scattering transmission

5. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Form Differential Equations from the Difference Equations Derived h, ground level 0, top of atmosphere z z + dz arbitrary layer dz I ↓ (z) I ↑ (z) I ↓ (z+dz)I ↑ (z+dz)  ext dz =  abs dz +  sca dz. P ↓↑ = P ↑ ↓. P ↑ ↑ = P ↓ ↓.

6. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Aside: Asymmetry Parameter of Scattering, g. -1<g<1 I0I0 Is()Is()  n r =1.33 =0.6328 D=20 um g=0.874

7. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Scattering Relationships: Example and the Asymmetry Parameter g particle incoming photons forward scattered photons back scattered photon Here P ↓↓ =3/4. P ↓↑ =1/4. P ↓↓ + P ↓↑ = 1. g≡ P ↓↓ - P ↓↑ in 1-D. g = P ↓↓ - (1- P ↓↓ ) Solving, P ↓↓ =(1+g)/2 = P ↑↑ P ↓↑ =(1-g)/2 = P ↓↑

8. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer 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. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Special Case: No Absorption, Single Scatter Albedo = 1. Reflection and Transmission Coefficients, R and T. h, ground level  0, top of atmosphere  I ↓ (0)≡I 0 R≡I ↑ (0)/ I 0 T≡I ↓ (  ) / I 0 ground is a totally absorbing surface, I ↑ (  ) ≡0.

10. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Solving for the Case where Single Scattering Albedo=1 (no absorption)

11. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer 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. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Features of the Multiple Scattering Solution Continued... “Photons are lost to the downward stream only if they are scattered in the opposite direction” R1 T

13. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Direct and Diffuse Transmitted Radiation cloud H I0I0 ItIt IrIr LWP = Cloud Water Mass / Area Q ext = Cloud droplet extinction efficiency CCN = # cloud condensation nuclei Cloud optical depth n r =1.33 =0.6328 D=20 um g=0.874  figure 1   Diffuse = Total - Direct

14. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer 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. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Summary of Multiple Scattering Equations: 1 D model.

16. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer R and T:

17. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Reminder from Chapter 7 Presentation

18. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Optical Depth from k ext : Liquid Water Path Liquid Water Path z bot z top Somewhere there has to be an integral over z!

19. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer 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 0.95. 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. T dir = exp(-  ext h)= 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. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Cloud Liquid Water Path, Effective Radius, And Cloud Albedo grams / m 2 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. Does this make sense? Why? How do things change when the single scattering albedo is not equal to 1, and absorption happens?

21. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Cloud above a Reflecting Ground I ↓ (0)≡1 R T ground has reflectance, (or albedo) = A g. TA g TA g R TA g RA g T2AgT2Ag T 2 A g RA g TA g RA g R TA g RA g RA g T 2 A g (RA g ) 2 T 2 A g (RA g ) n T(A g R) n TA g (A g R) n R total = R+T 2 A g + T 2 A g RA g + T 2 A g (RA g ) 2 +... + T 2 A g (RA g ) n +... R total = R+T 2 A g / (1- A g R) T total = T+TA g R+ T(A g R) 2 +... + T(A g R) n +... T total = T / (1- A g R)

22. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Features of a Cloud above a Reflecting Ground I ↓ (0)≡I 0 R T ground has reflectance, (or albedo) = A g. TA g TA g R TA g RA g T2AgT2Ag T 2 A g RA g TA g RA g R TA g RA g RA g T 2 A g (RA g ) 2 T 2 A g (RA g ) n T(A g R) n TA g (A g R) n General Relationship: R total = R+T 2 A g /(1- A g R) T total = T / (1- A g R) A g =0 R total = R, T total = T A g =1 R total = R +T 2 /(1- R) T total = T/(1-R) A g =1, R+T=1 (conservative case) R total = 1 T total = 1 Cloud Absorption, A A total = (In - Out)/I 0

23. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Weirdest thing (study the A g = 1 case, R + T = 1 conservative case) I ↓ (0)≡I 0 R T ground has reflectance, (or albedo) = A g =1 in this case. T TR T2T2 T2RT2R TRR T 2 (R) 2 T 2 (R) n T(R) n A g =1 R total = R +T 2 /(1- R) T total = T/(1-R) A g =1, R+T=1 (conservative case) R total = 1 T total = 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 = R total 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.

24. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Conservative Case: Example of T total

25. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Energy Conservation for the Conservative Case I ↓ (0)≡I 0 R T ground has reflectance, (or albedo) = A g. TA g TA g R TA g RA g T2AgT2Ag T 2 A g RA g TA g RA g R TA g RA g RA g T 2 A g (RA g ) 2 T 2 A g (RA g ) n T(A g R) n TA g (A g R) n General Relationship: R total = R+T 2 A g /(1- A g R) T total = T / (1- A g R) Energy Conservation: In to system = Out of system 1= R total + A gnd = 1 (can show with algebra) Total Absorption by Ground, A gnd A gnd = (In gnd - Out gnd )/I 0 A gnd = T total (1- A g ) Can show with algebra, or by inspection

26. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Additional Relationships and Limits for the General Case: deep multiple scattering with some absorption

27. Pat Arnott, ATMS 749 Atmospheric Radiation Transfer An Example