Salby M.L. Fundamentals of Atmospheric Physics

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280

Aerosol and Clouds

into which they advance has already been moistened and its stability reduced

by preceding parcels.

A developing cumulus is marked by well-defined protrusions along its

boundary, especially at its top. Those dissolve through evaporation as they

penetrate into unsaturated air and are replaced from the rear by new protru-

sions. The base of a developing cumulus is also well defined, being level or

even concave down--because warmer temperature in the thermal's core ele-

vates the LCL (5.24). Conversely, a decaying cumulus is rendered amorphous

by turbulent mixing, which leads to evaporation and quickly blurs detailed

features created during its development. The cloud's base may then become

concave up due to a reversal of vertical motion and the horizontal temperature

gradient. Cloud matter descending beneath a cloud's base is termed

virga,

and

referred to as precipitation only if it reaches the ground.

Thermal growth is inhibited by strong vertical stability. Figure 9.17 illus-

trates a thermal that encounters and partially penetrates through a layer of

strong stability. After penetrating the stable layer, negatively buoyant fluid

continues to advance and eventually diffuses via turbulent mixing with sur-

rounding fluid. Fluid deflected by the stable layer fans out to form an anvil.

Such behavior appears occasionally at intermediate levels in the troposphere

(Fig. 9.18) and invariably when a cumulonimbus encounters stable air at the

tropopause. Turbulent entrainment that accompanies penetrative overshoots

at the tropopause (Fig. 7.7) then mixes tropospheric and stratospheric air in-

side the resulting anvil.

The stratiform anvil of a mature cumulonimbus often evolves into small

cellular convection. LW emission at its top and absorption at its base

(Sec. 9.4.2) destabilize the anvil layer and introduce convective overturning,

Domes of sinking virga then define

mammatus

(meaning "breast-shaped")

Figure 9.17 Laboratory simulation of a thermal encountering a layer of strong static stability.

Upon reaching the stable layer, the negatively buoyant plume is deflected horizontally into an

anvil. Fluid penetrating the stable layer continues to advance and eventually diffuses through

turbulent mixing. After Scorer (1978).

9.3

Macroscopic Characteristics of Clouds

281

Figure

9.18 A developing cumulus that encounters and penetrates through a shallow layer

of strong static stability. Courtesy of Ronald L. Holle (Holle Photography).

clouds

(Fig. 9.19), the descent of which is promoted by evaporative cooling.

Spaced quasi-regularly, mammatus features resemble cells in Rayleigh-B6nard

convection (Fig. 9.20), which develops in a shallow fluid that is heated differ-

entially at its top and bottom. The horizontal dimension of the cells reflects

the depth of the layer involvedJ Stratiform cloud is often destabilized by

absorption of LW radiation from the surface and emission to space, which

can transform a continuous cloud layer into an array of cumulus cells hav-

ing a preferred dimension.

Stratocumulus

and

fair weather cumulus

develop

behind a cold front in this fashion (e.g., Fig. 9.21), when cold air that is ad-

vected over a warmer surface is heated at its base and cooled radiatively at its

top.

Clouds also form in the absence of buoyancy through forced lifting over

elevated terrain.

Orographic clouds

do not delineate a particular body of air

but, rather, air flows through them. The

lenticular

(meaning "lens-shaped")

cloud

forms inside an organized wave pattern (Fig. 9.22a) that develops when

the air stream is deflected over a mountain range. Each lenticular cloud in

Fig. 9.22a continually forms along its leading edge, where air is displaced

above its LCL, and dissolves along its trailing edge, where air returns below

5According to Rayleigh-B6nard theory, conversion of potential energy into kinetic energy be-

comes increasingly efficient with decreasing horizontal dimension of the cells. Energy conversion is

eventually offset by frictional dissipation (e.g., mixing of momentum across the walls of the cells),

which becomes important at horizontal dimensions comparable to the depth of the fluid. It is no

accident that cumulonimbus cells have dimensions comparable to the depth of the troposphere.

282

Aerosol and Clouds

Figure 9.19

Mammatus cloud

that develops from a cumulonimbus anvil. Supplied by the

National Center for Atmospheric Research, University Corporation for Atmospheric Research,

National Science Foundation.

its LCL. Also known as

mountain wave clouds,

lenticular clouds are often

found at great heights and, on occasion, in the stratosphere, where they are

termed

nacreous clouds.

Vertical motion accompanying wave clouds has been

harnessed by sailplanes to achieve record-breaking altitudes~as high as 50,000

feet. The vertical structure of lenticular clouds is controlled by environmental

conditions. When moisture is stratified in layers, lenticular clouds form in a

stack (Fig. 9.22b). Such stratification can result from towers of anomalous

moisture left upstream by prior convection, which are then sheared by the

circulation and folded into a layered structure.

The smooth form of lenticular clouds characterizes air flow that is turbu-

lence free or

laminar.

That steady motion is often replaced just downstream

by severe turbulence, which is made partly visible by

fractus

(meaning "frag-

mented")

clouds.

So-called

rotor clouds,

which are evident at levels beneath the

lenticular clouds on Figs. 9.22a and b, are "dynamic." Looping cloud matter in

Fig. 9.22c makes visible strong updrafts and downdrafts and severe turbulence,

which have been faulted for numerous aircraft disasters over mountainous ter-

rain.

Because they remain fixed, orographic clouds make cloud behavior unre-

liable as a tracer of air motion. Cumulus clouds more faithfully track the

9.3

Macroscopic Characteristics of Clouds

283

Figure

9.20 Laboratory simulation of Rayleigh-B6nard convection. A fluid layer heated at

its base evolves into a regular array of hexagonal cells with a preferred horizontal scale. Fluid

ascends in the center of each cell and descends along its edges. After Koschmieder and Pallas

(1974).

movement of individual bodies of air, but even they are complicated by non-

conservative behavior (e.g., condensation and evaporation), which limits their

usefulness for diagnosing air motion.

All of the foregoing clouds develop from upward motion, which reduces

the saturation mixing ratio of air through adiabatic cooling. Cloud can also

develop isobarically through heat rejection, which cools air below its dew point.

Conductive and radiative cooling lead to the formation of fog and marine

stratus over cold oceans.

The stratosphere is isolated from convective motions and therefore from the

source of water vapor at the earth's surface. Consequently, clouds rarely form

above the tropopause. Exceptional are nacreous clouds that develop in con-

nection with the mountain wave. Nacreous clouds are thought to be a subset

of more ubiquitous

polar stratospheric clouds

(PSCs). Because the stratosphere

284 9

Aerosol and Clouds

Figure 9.21 Visible image revealing cellular cloud structure behind a cold front. The cloud

pattern is organized into a quasi-regular array of cells with a preferred horizontal scale. The cold

front is part of a mature cyclone that advances across the British isles. Cloud cover ahead of the

cold front marks the warm sector of the cyclone, which is just occluding north of Ireland. After

Scorer (1986).

is very dry, PSCs form only under very cold conditions. Zonal-mean temper-

atures over the Antarctic fall below 190 K during Austral winter, more than

20 K colder than temperatures over the wintertime Arctic (Chapter 17). For

this reason, PSCs occur over the Antarctic with much greater frequency and

depth than over the Arctic.

Figure 9.22 (a) Schematic of the mountain wave that develops leeward of an elongated range.

Lenticular clouds

are fixed with respect to orography, while air flows through them. The positions

rotor clouds

are also fixed, but those fragmented features are highly unsteady, with turbulent

ascent (descent) along their leading (trailing) edges. Westward tilt with height (e.g., of wave

crests) corresponds to upward propagation of energy (Chapter 14), a feature that is often visible

in the structure of lenticular clouds. (b) Longitudinal view through the mountain wave. Smooth

lenticular clouds form in a stack, separated by clear regions, due to stratification of moisture.

These laminar features contrast sharply with turbulent rotor clouds below. (c) Transverse view

from the ground. Vorticity and severe turbulence are made visible by looping cloud matter that

comprises rotor clouds. Courtesy of V. Haynes and L. Feierabend (Soaring Society of Boulder).

9.3 Macroscopic Characteristics of Clouds 285

Even over Antarctica, PSCs are tenuous by comparison with tropospheric

clouds, almost all of which are optically thick. Some 3 to 5 km in depth, PSCs

form at heights of 15 to 25 km and are often layered. Their occurrence is

strongly correlated with temperature. Few clouds are sighted at temperatures

warmer than 195 K, whereas the likelihood of finding PSC at temperatures

below 185 K approaches 100% (WMO, 1988). Two distinct classes of PSCs are

observed. Type I PSC is comprised of frozen droplets of nitric acid trihydrate

(NAT), which has a low saturation vapor pressure. Submicron in scale, PSC

I particles form at temperatures of 195 to 190 K and have the appearance

of haze. Type II PSC may be nucleated by Type I particles or by background

aerosol, but contains much larger crystals of water ice. PSC II particles, which

are several tens of microns and larger, form at temperatures of 190 to 185 K

and have features similar to cirrus. Their larger sizes give type II particles

significant fall speeds, which makes them important in chemical considerations

(Chapter 17).

9.3.2 Microphysical Properties of Clouds

Microphysical properties vary with cloud type (Table 9.2). Stratus have number

densities n

"~ 300 cm -3

and droplet radii of a ,~ 4/zm. Liquid water content

is more variable, but of order 0.5 g

m -3.

Similar numbers apply for cumulus.

However, cumulonimbus are distinguished by much lower number density and

higher liquid water content, which imply significantly larger droplets. Com-

pared to water clouds, cirrus have much lower number densities that are ob-

Table 9.2

Microphysical Properties of Clouds*

Cloud type

n (cm -3)

(a) (/xm)

((Te) @ <Be> @

p~ (g m -3) 0.5 tzm (/zm 2) 0.5/xm

(km -1)

Stratus (St) 300. 3. 0.15 450. 135.

Stratocumulus (Sc) 250. 5. 0.3 120. 30.

Nimbostratus (Ns) 300. 4. 0.4 400. 120.

Cumulus (Cu) 300. 4. 0.5 200. 60.

Cumulonimbus (Cb) 75. 5. 2.5 500. 38.

Cirrus (Ci) 0.03 250. 0.025 t 1

• 10 4

0.5

Tropical cirrus (Cs) 0.10 800. 0.20* 4

x 10 4

0.4

PSC I < 1 0.5

10-6 t

10 0.005

PSC II <0.10 50. 0.05* 100 0.015

*Representative values of number density n, mean droplet radius (a), liquid water content Pt,

mean extinction cross section (be), and mean extinction coefficient (/3e) for different cloud types.

Typical values are subject to a wide range of variability, especially for cirrus. Sources: Carrier

et al.

(1967), WMO(1988), Dowling

et aL

(1990), Liou (1990), Knollenberg

et al.

(1993), and Turco

et al.

(1993).

t Ice water content.

286

Aerosol and Clouds

100

t- till

"~['~40

\,, -

0 10 20 30

DROPLET RADIUS

(/~'n)

10 -2 _

10 .3 _

10 .4 _

10 .5

10 .6

101

Tropical Cirrus

I i

102 103

CRYSTAL LENGTH (prn)

(b)

10 4

Figure 9.23 Cloud particle size spectra for (a) droplets inside water clouds and (b) crystals

inside ice clouds. Adapted from Liou (1990).

served over a wide range:

10 -7

to 10

cm -3

(Dowling

et al.,

1990; Knollenberg

et al.,

1993). Their ice water contents

are likewise smaller (e.g.,

0.1

g m-3). The characteristic dimension of cirrus ice crystals ranges from several

tens of microns to as large as 1 mm. PSCs have number densities similar to

cirrus and may form under analogous environmental conditions.

Figure 9.23 presents size spectra representative of several cloud types. Stra-

tus and cumulus have the simplest spectra, which possess a single maximum.

Spectra of nimbostratus and cumulonimbus exhibit secondary maxima at radii

larger than the mean: (a) -~ 5/xm. While comparatively unpopulated, the tails

of these size spectra (a > 20/xm) describe particles that grow most rapidly

through collision (Sec. 9.2.2) and consequently favor precipitation. Spectra of

cirrus can also possess a secondary maximum, but at much larger size (e.g., at

100 to 1000/zm). These larger particles have rapid fall speeds, which produce

the mares' tails characteristic of hook-shaped

cirrus uncinus

(Fig. 9.15).

9.3.3 Cloud Dissipation

Clouds dissolve through evaporation when unsaturated environmental air is

entrained into and mixed with saturated air. By diluting moisture inside a

thermal with cooler and drier environmental air, entrainment also siphons

off positive buoyancy, kinetic energy, and excess water vapor that would oth-

erwise be available to reinforce upward motion through latent heat release

(Sec. 7.4.2). Entrainment is the principal destruction mechanism for cumulus

cloud and must be offset for such cloud to grow.

9.4

Radiative Transfer in Aerosol and

Cloud

287

When particles grow so large that they can no longer be supported by the

updraft (e.g., a > 100/zm), clouds are dissipated by precipitation or, in the

case of cirrus, sedimentation. Evaporative cooling and downward drag exerted

on air by falling cloud particles oppose the updraft and the resupply of mois-

ture and positive buoyancy necessary to sustain the cloud against entrainment

(Problem 9.16). Strong LW cooling at the top of a cumulus tower, which pro-

motes subsidence, can likewise oppose cloud growth.

Entrainment is less effective in stratiform cloud because of its horizontal

extent and weaker vertical velocities. However, radiative heating at its base and

cooling at its top destabilize stratus, which stimulates entrainment of drier air

overhead and acts to dissolve such cloud. Radiative heating can also dissolve

cloud by elevating its temperature above the dew point, which breaks up fog

and marine stratus. This dissipation mechanism and formation mechanisms

associated with surface heating make diurnal variations an important feature

of the cloud field (Fig. 9.36).

9.4 Radiative Transfer in Aerosol and Cloud

The presence of liquid and solid particles greatly elongates the path trav-

eled by photons of SW radiation, which then undergo repeated reflection and

diffraction. Along with absorption inside particles, this sharply increases the

optical depth posed to incident radiation.

9.4.1 Scattering by Molecules and Particles

RAYLEIGH SCATFERING

The simplest treatment of scattering describes the interaction of sunlight

with molecules and is due to Rayleigh (1871).

Rayleigh scattering,

which ap-

plies to particles much smaller than the wavelength of radiation, considers a

molecule exposed to electromagnetic radiation as an oscillating dipole. The

strength of this oscillation is measured by the

dipole moment p,

which is related

to the electric field E 0 of incident radiation as

p - aE 0, (9.15)

where a is the

polarizability

of the scatterer: Established through interaction

with the wave's electromagnetic field, the dipole in turn radiates a scattered

wave with electric field E. At large distances r from the dipole, radiation

emitted at an angle 3' from p behaves as a plane wave whose amplitude satisfies

1 sin 7 a2P

E- c2 r Ot 2'

(9.16.1)

288

Aerosol and Clouds

where c is the speed of light (see, e.g., Jackson, 1975). Incorporating (9.15)

and plane wave radiation of the form

exp[ik(r- ct)] obtains

E = -Eok2a sin

exp[ik(r - ct)]. (9.16.2)

Consider sunlight at a scattering angle 19 from the path of incident radiation

(Fig. 9.24). Sunlight is unpolarized, so its electric field E 0 is distributed isotrop-

ically over directions orthogonal to the path of propagation. It can, therefore,

be represented in independent components

Eox and EOy of equal magnitude.

These incident electric fields induce dipole moments

Px and py, which, in turn,

radiate scattered waves. Radiation scattered at the angle 19 makes an angle

Yx = 7r/2 from Px and an angle

")/y --"

(7r/2)- 19 from

py.

Then (9.16.2) gives

Eoy

Incident

Path

Scattered

Path

Figure

9.24 Rayleigh scattering of SW radiation by air molecules. Unpolarized sunlight is

characterized by radiation with equal and independent electric fields

EOx

and

E0y ,

which induce

dipole moments

and

at the scatterer. Those in turn emit radiation at angles | from the path

of incident radiation with electric fields

and Ey that are determined by

and

(see text).

The orthogonal electric fields

Fox

and

Eoy

have been chosen to position the scattering angle O in

the plane formed by the path of incident radiation and E0y.

9.4 Radiative Transfer in Aerosol and Cloud

289

the corresponding electric fields of scattered radiation

E x -- -Eoxk2~ - exp[ik(r - ct)],

r (9.17)

Ey --

-EoykZcr cOs O exp[ik(r - ct)].

The intensity of radiation is proportional to the average of [El 2. Incident

sunlight of intensity I 0 is then scattered into the angle O with intensity

a 2 (1 -}- COS 2 O)

(9.18)

I(0, r) - Io k4 r-- T 2 "

In terms of the phase function (8.22) and wavelength A -

2rr/k,

(9.18) may

be cast into the canonical form

a 2 327r 4

I(| r) -- I0~- T 3A 4 P(| (9.19.1)

with

3 (1 n t- COS 2 O)

(9.19.2)

P(O) = ~

The scattered intensity (Fig. 8.8a) has maxima in the forward (O - 0 ~ and

backward (O - 180 ~ directions, with equal energy directed into each half-

space.

The scattered flux follows as in (9.19), but with the incident flux F 0 in

place of I 0. Integrating the scattered flux over a sphere of radius r obtains the

scattered power

F0cr 2 ~1287r5 (9.20)

3A 4 ,

which has dimensions of energy/time. Then the scattering cross section for an

individual molecule is

-- a 21287r5

3A 4 . (9.21)

Finally, the scattered intensity at distance r can be expressed

I(O, r) - I 0 6s P(O). (9.22)

47rr 2

Electromagnetic field theory relates the polarizability cr to the dimensionless

refractive index

m - m r - m i.

(9.23)