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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?. Ch 13: Radiative Transfer with Multiple Scattering. Primer: Saturn’s Moon Enceladus.

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Ch 13: Radiative Transfer with Multiple Scattering. Primer: Saturn’s Moon Enceladus.

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  1. 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? Ch 13: Radiative Transfer with Multiple Scattering.Primer: Saturn’s Moon Enceladus. 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 arbitrary layer dz extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓. 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 arbitrary layer dz extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓. 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 arbitrary layer dz extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓. 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 arbitrary layer dz extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓. 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. 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 R “Photons are lost to the downward stream only if they are scattered in the opposite direction”

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

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

  15. R and T:

  16. Reproduce the figure on the next slide using the simple model with absorption for the values of  ≠1. Calculate the cloud albedo as a function of effective radius and liquid water path for single scattering albedo equal to 0.999, 0.98, and 0.95. For each case, assume that the absorption is caused by black carbon aerosol embedded in the cloud. Calculate the absorption coefficient necessary to give each 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). Homework Problem (see website for the other problem). 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).

  17. 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.

  18. Cloud above a Reflecting Ground I↓(0)≡1 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. Rtotal= R+T2Ag+ T2AgRAg+ T2Ag(RAg)2+ ... + T2Ag(RAg)n + ... Rtotal= R+T2Ag / (1- AgR) Ttotal= T+TAgR+ T(AgR)2+ ... + T(AgR)n + ... Ttotal = T / (1- AgR)

  19. Features of a Cloud above a Reflecting Ground Cloud Absorption, A Atotal= (In - Out)/I0 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) Ag=1, R+T=1 (conservative case) Rtotal= 1 Ttotal = 1

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

  21. An Example

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