Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Figure 13: Heat map (within white box) of the...

$: Pat Arnott, ATMS 749 Atmospheric Radiation Transfer Figure 13: Heat map (within white box) of the thermally active field of fractures in saturn’s moon.$
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?


Two Stream Approximation: Multiple Scattering in 1 dimension.

h, ground level

0, top of atmosphere

z

z + dzarbitrary layer dz

I↓(z) I↑(z)

I↓(z+dz) I↑(z+dz)

extdz = absdz + scadz. P↓↑= P↑ ↓. P↑ ↑= P↓ ↓.

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.⇒


Conservation of energy in dz for downward intensity (or flux):(seeking relationships between the fluxes above and below dz).

h, ground level


z


I↓(z) I↑(z)



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


Conservation of energy in dz for upward intensity (or flux):(seeking relationships between the fluxes above and below dz).

h, ground level


z


I↓(z) I↑(z)



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


Form Differential Equations from the Difference Equations Derived

h, ground level


z


I↓(z) I↑(z)




Aside: Asymmetry Parameter of Scattering, g. -1<g<1

I0

Is()

nr=1.33=0.6328D=20 umg=0.874


Scattering Relationships: Example and the Asymmetry Parameter g

particle

incomingphotons

forward scatteredphotons

back scatteredphoton

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↓↑


Relationships for Extinction, Scattering, Absorption, and the Single Scatter Albedo: Coupled de’s for the fluxes.

Fundamental equationswe use for everything.

Fluxes are coupledby scattering.


Special Case: No Absorption, Single Scatter Albedo = 1.Reflection and Transmission Coefficients, R and T.

h, ground level


I↓(0)≡I0R≡I↑(0)/ I0

T≡I↓() / I0

ground is a totally absorbing surface,I↑() ≡0.


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


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)


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


Direct and Diffuse Transmitted Radiation

cloud

H

I0

It

Ir

LWP = Cloud Water Mass / AreaQext = Cloud droplet extinction efficiencyCCN = # cloud condensation nuclei

Cloud optical depth

nr=1.33=0.6328D=20 umg=0.874

figure 1

Diffuse = Total - Direct


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).


Summary of Multiple Scattering Equations: 1 D model.


R and T:


Reminder from Chapter 7 Presentation


Optical Depth from kext: Liquid Water Path

Liquid Water Path

zbot

ztop

Somewhere there has to be an integral over z!


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.

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).


Cloud Liquid Water Path, Effective Radius, And Cloud Albedo

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.

Does this make sense? Why?

How do things change when the single scattering albedo is not equal to 1, and absorption happens?


Cloud above a Reflecting Ground

I↓(0)≡1 R

T

ground has reflectance, (or albedo) = Ag.

TAg

TAgR

TAgRAg

T2Ag T2AgRAg

TAgRAgR

TAgRAgRAg

T2Ag(RAg)2 T2Ag(RAg)n

T(AgR)nTAg (AgR)n

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)


Features of a Cloud above a Reflecting Ground

I↓(0)≡I0 R

T


TAg

TAgR

TAgRAg

T2Ag T2AgRAg

TAgRAgR

TAgRAgRAg


T(AgR)nTAg (AgR)n

General Relationship:Rtotal= R+T2Ag /(1- AgR)


Ag=0Rtotal= R, Ttotal = T

Ag=1Rtotal= R +T2 /(1- R)Ttotal = T/(1-R)

Ag=1, R+T=1 (conservative case)Rtotal= 1Ttotal = 1

Cloud Absorption, AAtotal= (In - Out)/I0


Weirdest thing (study the Ag = 1 case, R + T = 1 conservative case)

I↓(0)≡I0 R

T

ground has reflectance, (or albedo) = Ag=1 in this case.

T

TR

TR

T2 T2R

TRR

TRR

T2(R)2 T2(R)n

T(R)n

T(R)n

Ag=1Rtotal= R +T2 /(1- R)Ttotal = T/(1-R)

Ag=1, R+T=1 (conservative case)Rtotal= 1Ttotal = 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

Let R=0.99. With 100 photons incident, 99immediately reflect upwards and are lost.One photon passes through and reflects 99 timesbetween 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.


Conservative Case: Example of Ttotal


Energy Conservation for the Conservative Case

I↓(0)≡I0 R

T


TAg

TAgR

TAgRAg

T2Ag T2AgRAg

TAgRAgR

TAgRAgRAg


T(AgR)nTAg (AgR)n

General Relationship:Rtotal= R+T2Ag /(1- AgR)


Energy Conservation:In to system = Out of system1= Rtotal + Agnd = 1

(can show with algebra)Total Absorption by Ground, Agnd

Agnd= (Ingnd - Outgnd)/I0

Agnd = Ttotal (1- Ag)

Can show with algebra, or by inspection


Additional Relationships and Limits for the General Case:

deep multiple scattering with some absorption


An Example

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