Salby M.L. Fundamentals of Atmospheric Physics

290

9

Aerosol and Clouds

The real and imaginary parts of m refer to the phase speed and absorption

of electromagnetic radiation in a medium relative to those in a vacuum. An

ensemble of scatterers with number density n has polarizability

3 (m2-~)

Ce -- 4--~ n m2+ "

(9.24)

At wavelengths of visible radiation, absorption by air molecules is small enough

for

m i

to be ignored, while

m r

is close to unity. Then (9.26) reduces to

,,~ mr-1

a- 2"n'n ' (9.25)

so the scattering cross section for an individual molecule is given by

327r3(mr- 1) 2

&s - 3n2A 4 9 (9.26)

Scattering then yields an optical depth for the atmosphere of

f0 ~

'r = &s(A) n(z)dz.

(9.27)

Rayleigh scattering explains why the sky appears blue. The

,~-4

depen-

dence of &s implies that shorter wavelengths are scattered by air molecules

much more effectively than longer wavelengths. Blue light (A ~ 0.42 /xm)

is scattered five times more than red light (A ~ 0.65 rim), so it arrives at

the earth's surface as diffuse radiation emanating from all directions. When

the sun is near the horizon, sunlight passes obliquely through the atmosphere

and therefore along an extended path. Shorter wavelengths are then scattered

out of the incident beam, leaving longer (red) wavelengths to illuminate the

sky. On average, about 40% of the SW flux is scattered out of the incident

beam in the near-UV region, whereas less than 1% is removed in the near-IR

region. Overall, about 10% of SW radiation incident on the atmosphere is

scattered by molecules, half of which is returned to space and contributes to

the earth's albedo (Fig. 1.27). Most of that scattering takes place in the low-

est 10 km, where

n(z)

is large. For this reason, increasing elevation witnesses

a darkening of the sky and an intensification and whitening of direct sunlight.

Rayleigh scattering influences ozone photochemistry and stratospheric heat-

ing, for example, by enhancing the SW flux available to the Chappuis bands

of 0 3 (Sec. 8.3.1).

Owing to its relationship to dipole radiation, Rayleigh scattering has cer-

tain polarization properties. In the forward and backward directions (O = 0,

180~ scattered radiation is unpolarized--like incident sunlight. However,

scattered sunlight becomes increasingly polarized at intermediate angles and

is fully polarized at directions orthogonal to the path of incident radiation

(O = 90~ It is for this reason that the sky assumes a deeper blue when

viewed through a polarizer at directions orthogonal to incident sunlight.

9.4 Radiative Transfer in Aerosol and Cloud 291

MIE SCATIqERING

Scattering from spherical particles of arbitrary dimension was first treated

by Mie (1908). Mie scattering applies to the interaction of radiation with aerosol

and cloud droplets. Figure 9.25 shows the refractive indices for water and ice,

as functions of wavelength. Although m r is of order unity, it varies with A,

which makes diffraction and reflection by spheres of condensate wavelength-

dependent. On the other hand, mi, which is proportional to the absorption

coefficient/3aa,

A

mi

=/~aa

4rr' (9.28)

is small in the visible. However, it increases sharply at A > 1 /zm for both

condensed phases, which makes all but tenuous clouds optically thick in the

IR.

Application of Mie scattering to atmospheric aerosol relies on particles be-

ing sufficiently separated for their interactions with the radiation field to be

treated independently. Mie theory solves Maxwell's equations for the electro-

magnetic field about a dielectric sphere of radius a in terms of an expansion

in spherical harmonics and Bessel functions (see Liou, 1980). Properties of

the scattered wave field emerge in terms of the dimensionless size parameter

a

x = 2rr-. (9.29)

A

1.7

I I I I I I I ! I

[

!

i

9 -Water /

ii

t . . ,ce

-I

i I /~"

I

/

"._ /

~ , I\s / /J

11

I,

,'

I

"

]

!1

I mr I

0.9 I I I I I t t t t t

0 2 4 6 8 10 12 14 16 18 20

WAVELENGTH (prn)

100 I - I I I I I

itl t I

.I ,,~ ,'- ~..;'~-~

ol ~

Water

I-- 10 l

.... Ice -

Z

LU

Z j,!J

O 10 1 l

O10

C)

rr 10 r

<

z

10"i -~

10 .8

:

10 .9

0 2 4 6 8 10 12 14 16 18 20

WAVELENGTH (/am)

Figure 9.25 Index of refraction for water (solid line) and ice (dashed line) as functions of

wavelength. (a) Real component and (b) imaginary component. Adapted from Liou

(1990).

292

9

Aerosol and Clouds

The dimensionless

scattering

and

extinction efficiencies

Qs = ,rfa 2

(9.30)

Qe = 7Ta2

represent the fractional area of the incident beam removed through interaction

with the sphere by scattering and

in toto.

The scattering efficiency is shown

in Fig. 9.26 for a sphere with

m r

:- 1.33 and for several values of

m i.

For

conservative scattering

(m i

= 0), Q= increases quartically for x < < 1 (which

corresponds to Rayleigh scattering) and attains a maximum near x ~ 27r (i.e.,

A ~ a). The scattering efficiency then passes through a series of oscillations

that result from interference between light transmitted through and diffracted

about the sphere.

In the limit of large radius, the oscillations in scattering efficiency damp

out to give

Qs

"~ 2 = const x ~ ~ (9.31)

for conservative scattering. Hence, a sphere much larger than the wavelength

of incident radiation scatters twice the energy it intercepts. Scattered radiation

then includes contributions from energy that is diffracted about the sphere

as well as energy that is redirected by reflection inside the sphere. This limit-

ing behavior applies to the interaction of visible radiation with cloud droplets

w , , Ij~t~ ,~ I ~ i

i I I ill

/ 88

- mi=O Y~t

---

_

~ I .,-, I/AV

I

I I t [ IIIII i i l I i illl

5 IO 50 IO0

x - 2~

Figure

9.26 Scattering efficiency from a dielectric sphere having real component of refractive

index mr = 1.33 for several imaginary components mi, as functions of the dimensionless size

parameter x = 27r(a/A). After Hansen and Travis (1974). Reprinted by permission of Kluwer

Academic Publishers.

9.4

Radiative Transfer in Aerosol and Cloud

293

(compare Fig. 9.23). The weak dependence on wavelength and the fact that

droplet size spectra span several oscillations in Q~ explain why clouds appear

white.

For nonconservative scattering

(m i > 0), Q~

exhibits similar behavior, but

the oscillations due to interference are damped out with increasing absorption

(since radiation emerges from the sphere progressively weaker). By

mi = 1,

only the maximum near x = 2~r remains. Increased attenuation inside the

sphere also reduces the limiting value of

Qs

for x --+ c~, by absorbing energy

that would otherwise contribute to the scattered wave field. Far from the

sphere, radiation exhibits strong forward scattering (e.g., Fig. 8.8b), but is

complicated for an individual sphere by rapid fluctuations with 19 that result

from interference.

Under the conditions of independent scattering, properties of a population

of spheres follow by collecting contributions from the individual particles. The

scattering and extinction coefficients are then given by

f

~s =

] &sdn(a),

,1

(9.32)

g

13e = /

[cdn(a),

J

where

dn(a) - (dn/da)da

describes the droplet size spectrum. The single

scattering albedo for the population is characterized by

o) = --. (9.33)

~e

Similarly, the optical depth of a cloud of thickness

Azc

can be estimated as

T c -- ~ eAZc,

(9.34)

where

~e

is representative of the cloud as a whole.

Figure 9.27 shows, for several wavelengths, the phase function (solid lines)

corresponding to a droplet size spectrum representative of cumulus (Fig. 9.23).

Compared to Rayleigh scattering (dashed line), Mie scattering possesses much

stronger directionality. All A shown exhibit strong forward scattering, especially

shorter wavelengths, which are sharply peaked about 19 = 0 and for which

absorption is small (Fig. 9.25b). A weaker maximum appears at backward

scattering for wavelengths in the near-IR and visible regions.

The mean extinction cross section for the population: (Ore) -

fle/n

(Fig.

9.28a), is nearly constant across the visible, where cloud droplets are large

(9.31). Weak absorption at those wavelengths leads to a single scattering

albedo (Fig. 9.28b) of near unity, so ~r e ~ ~r s and scattering is likewise wave-

length independent. Individual components of sunlight are then scattered with

equal efficiency, so the cloud appears white. The extinction cross section at-

tains a maximum near 5/xm, at a wavelength comparable to the mean radius

of cloud droplets (Fig. 9.23), and falls off at longer wavelength. Because oJ --- 1

294

9 Aerosol and Clouds

DIRECTIONALITY OF SCATTERING

106 ' ' ' ' ' I ' ' ' , , I , , , , ,

104 ~~kk,,, \\

z ~- 0.5pro

,~176 / oecue

10"1~

10 2

0 60 120 180

(9 (Degs)

SCATTERING ANGLE

Figure 9.27

Phase function for scatter-

ing from the cumulus droplet population

in Fig. 9.23,

for several wavelengths

(solid

lines),.

Phase function for Rayleigh scatter-

ing

(dashed line)

superposed. Source:

Liou

(1990).

for A < 10/zm, most of the extinction results from scattering, which dominates

over absorption at wavelengths of SW radiation. Exceptional are narrow bands

at A ~ 3 and 6/zm, where absorption spectra of water and ice are peaked

(Fig. 9.25b). At A > 10/zm, to decreases due to diminishing Qs and increas-

ing absorption. Similar optical properties follow from a droplet size spectrum

representative of stratus.

Extinction by cloud droplets greatly increases the SW opacity of the atmo-

sphere. For cumulus 1 km thick and a droplet number density of 300 cm -3,

(9.34) gives an optical depth at A = 0.5 /xm of ~'c = 50, so the cloud is op-

CLOUD OPTICAL PROPERTIES

1.5

~ o,

0.0

10 -1

, | , ,,,|| l , , ,,|| , | , ||||!

(a) Extinction

Cross Section

| , | | |l|,i | | | , ,,||i | | | , |,,

10 ~ 10 '1 102

WAVELENGTH (pm)

1.2

1.0

0.8

0.6

0.4

0.2

0.0

I I i I IIII i I I I I I IIII I I I i III

o-1

(b) Single

Scattering Albedo

I i i , lJi,i l I t , ,,,,l I I i ' '''

10 ~

101

102

WAVELENGTH (pro)

Figure 9.28 (a) Extinction

cross section normalized by

(tre)(0.5/xm) = 166/xm 2 and (b) single

scattering albedo for the cumulus droplet population

in Fig. 9.23.

Adapted from

Liou (1990).

9.4

Radiative Transfer in Aerosol and Cloud

295

tically thick to visible radiation. Organized cumulus of the same dimensions

can have

"i- c

> 100. Even shallow stratus have optical depths in the visible well

in excess of 10. In the IR,

"i" c

is smaller but still large enough to render all

but tenuous clouds optically thick to LW radiation. At 10 ~m, the foregoing

conditions for cumulus lead to an optical depth of about 20.

Scattering by nonspherical ice particles is complicated by their irregular

structure and anisotropy. Even though their rudimentary geometry is hexago-

nal, ice crystals assume a wide range of shapes (Fig. 9.14), which can produce

a myriad of optical effects associated with orientation, directionality, and po-

larization. The familiar halo of thin cirrus results from prismatic crystals with

a preferred orientation.

9.4.2 Radiative Transfer in a Cloudy Atmosphere

Influences of cloud and aerosol on atmospheric thermal structure can now

be evaluated with the radiative transfer equation. Consider a homogeneous

cloud layer that is illuminated by a solar flux

Fs

inclined at the zenith angle

0s = cos -1 ~s (Fig. 9.29). The source function for diffuse SW radiation (8.24)

then becomes

~~ f02~ f_l 4,'m'

J(~b,/,, ~') = 1I(4~ ',/,',

r)P(d),

IX; , )dd)'dl~'

+ ~w-----FsP(ck, tx; ,#',-tXs)e ~s,

(9.35)

q.'TT

I;*= O~

Multiple-Scattered

Single-Scattered

p = cos 0

Hs = COS 0 s

1;c

Figure

9.29 Transmission of diffuse SW radiation through a scattering layer.

296

9 Aerosol and Clouds

where the wavelength dependence is implicit and emission is ignored. The

first term on the right-hand side represents contributions to diffuse radiation

from multiple scattering of the diffuse intensity I, whereas the second term

represents the contribution from single scattering of direct solar radiation.

Transmission of diffuse SW radiation through a cloudy atmosphere is then

governed by the radiative transfer equation.

Scattering transforms (8.25) into an integrodifferential equation. It is then

treated by expanding the phase function P(cos| where 19 is the three-

dimensional scattering angle, in a series of spherical harmonics and the diffuse

intensity I in a like manner; see Liou (1980) for a formal treatment. Express-

ing P(cos 19) in terms of the spherical coordinates (4', 0) and integrating over

4' then yields the transfer equation governing the azimuthal-mean component

I(~, ~-) of the diffuse radiation field

dI

o, f I

/x~-~r - I-~- a I(/z', ~-)P(/x;

I,z')dtz' - --FsP(Ix;-la, s)e ~ ,

(9.36.1)

where

POx; ~') = ~

cjPj(lx)Pj(tx'),

(9.36.2)

J

Pj

is the Legendre polynomial of degree j, and the expansion coefficients

cj

follow from orthogonality properties of the

Pj(tz).

For single scattering, the

convolution integral is omitted. The diffuse intensity is then directly propor-

tional to the phase function P. Approximate solutions for multiple scattering

can be obtained by numerical techniques. A simple but enlightening solution

follows from the two-stream approximation.

Owing to the involvement of Legendre polynomials, the integral in (9.36)

can be evaluated efficiently with Gaussian quadrature at a finite number of

zenith angles/zj (see, e.g., Rektorys, 1969), each one describing a different

stream of radiation. The two-stream approximation then follows by taking just

two angles: j = +1, which correspond to the diffusivity factor/2 -1 = ~/3, that

is, to the upwelling and downwelling streams Ir = 1(+/2, ~-). Incorporating

the orthogonality properties of Legendre polynomials (Abramowitz and Ste-

gun, 1972) and the transformation (8.54) leads to a system of two ordinary

differential equations for the downwelling and upwelling fluxes

(-/2 r*)(9.37.1)

dF'l'd'r* = F'I" - w [fF "1" +

(1 -

f )F ~ ] - S +

exp ~ss '

where

dFr

)

= F t - o2 [fF t +

(1 -

f)F*] - S-

exp --~-* , (9.37.2)

d'r*

1 + g (9.37.3)

f= 2

9.4

Radiative Transfer in Aerosol and Cloud

297

and (1- f) represent the fractional energy that is forward and backward

scattered, respectively,

1f_1

g -- 2 1 P(cos | cos | 19) (9.37.4)

is an

asymmetry factor,

and

Fs

(1 +

3g[xl, Zs).

S i _ ~-~

(9.37.5)

The asymmetry factor g, which equals the first moment of the phase func-

tion, measures the difference in scattering between the forward and back-

ward half-spaces. For isotropic scattering, g - 0 and f - 1. The same holds

for Rayleigh scattering. However, the strong forward lobe of Mie scattering

1 with ~" 0.85 being representative for

(Figs. 8.8b and 9.27) leads to f > ~, g

water clouds (see, e.g., Liou, 1990).

Subtracting and adding (9.36), introducing the net and total-emitted fluxes

F = F 1" - F +, (9.38.1)

P = F 1" + F+, (9.38.2)

and differentiating with respect to r* results in the equivalent second-order

system

d2F y 2 ( ~ )

dr,2 F - Z exp

-~ss r* , (9.39.1)

where

d2F ( ~ )

dr,2

- a/'2/~ -- /~

exp -~ss r* , (9.39.2)

3, 2 - (1 - o9)(1 + w - 2mr)

(9.39.3)

and

/i-

z-' s+(1-o,)s

/Zs

2- -/2 S- (1+ w_ 2wf)~

/Zs

S=S+_S -

S=S++S -.

(9.39.4)

(9.39.5)

298

9

Aerosol and Clouds

The general solution of (9.39) can be expressed

F 1' - Ca+ exp(yr*) + Da_ exp(- yr*) + E+ exp (-/2 r*],

\

/Zs /

F* - Ca_ exp(yr*) + Da+ exp(- yr*) + E_ exp - -- r*

where

o~_1 - ---

1+/3

2

1--o9

~2

1 + w- 2a)f

E+ =

2

- ~s2

2

a = ~s2 z.

/~2 _ .s2r2 /~2 _ .s2r2

(9.40.1)

(9.40.2)

The integration constants C and D are determined by boundary conditions.

If r* is measured from the top of the cloud layer, which has an effective

optical depth r c, and if the underlying surface is black, the downwelling diffuse

flux vanishes at the top of the scattering layer and the upwelling diffuse flux

vanishes at the bottom

Then

F+(0) =0,

(9.41)

Ff(r *) =0.

E_~_

r

- E+~+

r ~)

C--

a 2 exp(yr*) - c~ 2 exp(-yr*) (9.42)

E+a_

exp(-~rc)-

E_a+

exp(yrc)

D~

~2+ exp(~c) - ~2_ exp(- ~c)

The solution can then be used to evaluate the cloud albedo and transmissivity

FI"(0)

d c - tzsFs, (9.43.1)

= F*(rc) + Ix~Fs exp(-~rc), (9.43.2)

i~Fs

where the direct solar contribution in (9.43.2) is negligible for cloud that is

optically thick. From these, the cloud absorptivity follows as 1 -dc - ~.

9.4 Radiative Transfer in Aerosol and Cloud

299

Figure 9.30 shows d C and ~ as functions of optical depth and solar zenith

angle for microphysical properties representative of cumulus (Fig. 9.28) and

for a wavelength of 0.5 /~m. Cloud albedo increases sharply at small optical

depth (Fig. 9.30a) and exceeds 50% for ~-c > 20, which is typical of water

clouds. Hence, even shallow stratus are highly reflective. Transmissivity de-

creases with increasing ~-c~nearly complementary to cloud albedo, so most

incident SW energy is either reflected or transmitted. The absorptivity is of

order 10% across much of the range of optical depth shown, with contributions

from droplets and surrounding vapor being comparable (Stephens, 1978a). Be-

cause the cloud layer is optically thick, most of the absorption occurs near its

top, where heating rates

~l/Cp

are 10 K day-aand greater. Cloud albedo also

increases with increasing solar zenith angle (Fig. 9.30b), which elongates the

1.0

0.8

0.6

0.4

0.2

(a) Os= 0

~ f . Albedo

, ~ ........ Transmissivity

%

-- .= ........

0.0

0. 20. 40. 60. 80. 100.

1.0

0.8

0.6

0.4

0.2

(b) %= 20

OPTICAL DEPTH

0.0 , i I I I I = I i i I I = I I I ,

0 10 20 30 40 50 60 70 80 90

SOLAR ZENITH ANGLE (Degs)

Figure 9.30 Albedo and transmissivity at 0.5/xm for a cloud layer with microphysical proper-

ties of cumulus in Fig. 9.23 and nearly conservative scattering (a) for overhead sun, as functions

of cloud optical depth, and (b) for a cloud optical depth of 20, as functions of solar zenith angle.

Salby M.L. Fundamentals of Atmospheric Physics

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