Salby M.L. Fundamentals of Atmospheric Physics

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270

9 Aerosol and Clouds

EQUILIBRIUM SUPERSATURATION

0.5

0.4

0.3

0.2

0,1 "

0.0

-0.1

-0.2

at(2)

~\ (1) (1) Pure Water

i \\ (2) NaCI 1016g -

/,~ \\ (3) NaCI 1015g

,..,[ \'\ (4) NaCI 10"4g

tt ~k~N\\

(5) NaCl 10"2g i

0.1 .10 1.0 10.

a (pm)

DROPLET RADIUS

Figure 9.9 Critical supersaturation necessary to sustain a droplet of radius a and containing

specified amounts of solute (solid lines). Critical supersaturation for a pure-water droplet (dashed

line) is superposed. For a given amount of solute, the corresponding

K6hler curve

describes

different forms of equilibrium, depending on whether a is smaller or larger than the threshold

radius at the maximum of the curve,

a t.

For a <

at,

a droplet is in stable equilibrium: Perturbations

in a are opposed by condensation and evaporation, which restore the droplet to its original state.

For a >

at,

a droplet is in unstable equilibrium: Perturbations in a are reinforced, so the droplet

either evaporates toward

a t

or grows through condensation away from its original state.

becomes negative (i.e.,

< 100%). Hence, nuclei that are more soluble

activate sooner and are more favorable to droplet growth.

The equilibrium described by a given K6hler curve differs according to

whether the droplet radius is larger or smaller than a at the maximum of the

curve, which is hereafter referred to as the threshold radius

a t.

A droplet with

a < a t

is in stable equilibrium: Adding a vapor molecule drives the droplet

along its K6hler curve to higher Ec. The actual supersaturation E then lies

beneath the equilibrium value, so the droplet evaporates back to its original

size. Removing a water molecule has the reverse effect. In either case, a

perturbed droplet is restored to its original state. Droplets with a <

a t

cannot

grow through condensation and comprise

haze.

Conversely, a droplet with

a > a t

is in unstable equilibrium: Adding a vapor molecule drives that droplet

along its K6hler curve to lower Ec. The actual supersaturation then exceeds

the equilibrium value, so the droplet continues to grow through condensation.

Removing a water molecule has the reverse effect. In either case, a perturbed

droplet evolves away from its initial state, analogous to the behavior of a

pure-water droplet (Fig. 9.8). Droplets with a >

a t

can enlarge into cloud

drops.

Only a small fraction of tropospheric aerosol actually serves as CCN be-

cause many particles are too small, not wettable, or insoluble. Continental

9.2 Microphysics of Clouds

271

aerosol is the dominant source of CCN because of its greater number den-

sity and high concentration of soluble sulfate, which is found in many cloud

droplets. By contrast, sea salt is not common in cloud droplets~even over

maritime regions. Instead, CCN in maritime clouds are comprised chiefly of

sulfate particles, which, through anthropogenic production, have nearly dou-

bled globally (Houghton

et al.,

1990). Sulfate aerosol also forms through con-

densation of dimethylsulphide (DMS), which is emitted by phytoplankton at

the ocean surface.

Cumulus clouds have droplet number densities of order 102 cm -3 over mar-

itime regions and

10 3

cm -3 over continental regions, as is reflected in observed

droplet size spectra (Fig. 9.10). The greater number density of continental

cloud droplets mirrors the greater number density of continental aerosol. It

also implies a smaller mean droplet size because the density of condensate,

liquid water content Pt

(gm-3), does not differ appreciably between these

cloud categories. In addition to being more numerous and smaller, droplets

inside continental cumulus are more homogeneous in size than droplets in-

side maritime cumulus. The wider size spectrum of maritime cloud droplets

.~.

40 -(a) Mariti

.9.o 30

z 20

rr 10

ILl

rrl

.......

Z C,,,

0 10 20

RADIUS

(pro)

!--

200

Ill

IJJ

100

(b) Continental (Cu)

0 !

30 0 10 20 30

RADIUS (/An)

Figure 9.10 Droplet size spectra for cumulus cloud in (a) maritime environment and (b) con-

tinental environment. Note the compressed scale in (b). Area under each curve reflects the overall

number density n, which is an order of magnitude greater in continental cumulus than in maritime

cumulus. Source: Pruppacher (1981).

272

9 Aerosol and Clouds

>.-

~1..-

0 VISUAL EXTENT OF CLOUD

= 1000 rn t

Figure 9.11 Variation of liquid water content Pl and vertical motion across a cumulus com-

plex. After Warner (1969).

follows from the fact that they form chiefly on soluble sulfates that activate

over a wide range of sizes.

Liquid water content varies strongly over dimensions of a cumulus complex

(Fig. 9.11), but remains highly correlated with upward motion. Liquid water

content

increases yertically to a maximum in the upper half of a cloud

and then decreases sharply to zero at the cloud top. It is noteworthy that

observed in nonprecipitating clouds seldom approaches adiabatic values

p~d,

which describe air parcels rising in isolation from environmental air. In fact,

pl/p'] d

< 0.5 and vanishes at the cloud top, which reflects entrainment of

drier environmental air through the walls of the cloud system (Sec. 7.4.2).

For similar reasons, the greatest supersaturations are found in the core of a

convective cloud, where moisture is relatively undiluted.

Before precipitation can occur, cloud droplets must enlarge by several or-

ders of magnitude. After passing over the peak of its K6hler curve, a droplet

can grow through condensation. The droplet's mass m then increases at the

rate vapor diffuses across its surface

am dp,

(9.10)

d t

= 41rrZu

dr '

where r denotes radial distance, v is the diffusion coefficient for water vapor,

and p~ is its density. By integrating from the droplet radius r = a to infinity

9.2 Microphysics of Clouds

273

and expressing the droplet's mass in terms of the density of liquid water

Pw,

we obtain

da v

= ~[po(ee)- pv(a)],

dt apw

which, with the gas law, can be written

da [ pv(c~) ] e(cxz) - e(a)

a~- -u . (9.11)

Pw e(c~)

For droplets larger than 1 /xm, the curvature and solute effects (Figs. 9.8

and 9.9) are small enough to take

e(a)~ ew.

Then (9.11) can be expressed

da~

[pv(oe) ] (9.121)

a-d- )- - u e,

which, if the right-hand side is approximately constant, gives

1[ ]/1

a(t)-

2u p~(c~) e . (9.12.2)

Droplet growth through condensation (Fig. 9.12) is rapid at small radius,

but slows as a droplet enlarges. This implies that a droplet population will

become increasingly homogeneous in size as it matures. While explaining the

preliminary stage of droplet growth, condensation is too slow at large a to

account for observed droplets in precipitating clouds.

9.2.2 Droplet Growth by Collision

Clouds that lie entirely beneath the freezing level are called

warm clouds.

Droplets inside warm clouds also grow through coalescence, when they collide

with other droplets. Large droplets have faster fall speeds than small droplets.

A large

collector drop

then sweeps through a volume of smaller droplets.

The collection efficiency

which is a dimensionless quantity that reflects the

effective collecting area posed to other droplets, is small for collector radii less

than 20/xm and increases sharply with collector size. For this reason, the tail

of the droplet spectrum at large a is instrumental in forming precipitation. 3

The rate at which a collector drop grows through collision with a homoge-

neous population of smaller droplets can be expressed as

-- "rra2EPlAW,

(9.13)

3The collection efficiency also increases with the radius of the droplet population encountered

by the collector (small droplets being swept aside without impact) up to a maximum. Beyond that

droplet radius, coalescence is not favored (see, e.g., Wallace and Hobbs, 1977).

274

Aerosol and Clouds

I I I I

am= 15pm

\ ;" I

\ ~,'

,' ~

Condensation|

. \

,'~'

--- Collision /

~ ,' ,, ~^ , -

}

I/ ii

I I

0.05

," ~ -

"" ,"

_.~. 0 10"4

0 0 5

0 20 40 60 80 1 O0

DROPLET RADIUS

(pm)

Figure 9.12 Droplet growth rate

da/dt

due to condensation (solid lines) and collision (dashed

lines) inside a cloud with liquid water content of Pt = 1.0 g m -3, as functions of droplet radius

a. Growth rate due to condensation shown for different supersaturations ~ = [ pv(oo)p,o .! ] 9 (see text).

Growth rate due to collision shown for a droplet population whose size spectrum has a maximum

at radius

am.

Adapted from Matveev (1967).

where a refers to the collector and Aw is the difference in fall speed between

the collector and the population of smaller droplets. Rearranging as in the

derivation of (9.12) yields

da Ew Pt

= ~ ~, (9.14)

dt 4 Pw

where Aw = w has been presumed for the collector. Because w and E both in-

crease with the radius of the collector, (9.14) predicts growth that accelerates

with the size of the droplet. Therefore, growth due to collision eventually dom-

inates that due to condensation. According to Fig. 9.12, droplets smaller than

20/zm grow primarily through condensation (solid lines), because fall speeds

are small enough for such droplets to move in unison with the surrounding

air and make collisions infrequent. However, collision (dashed lines) becomes

dominant for droplets larger than 20/xm. Since it favors large drops, collision

tends to broaden the size spectrum from the homogeneous population favored

by condensation. It thus provides a mechanism for a small fraction of the drops

to grow much faster than the rest of the population, with a > 100/zm domi-

nating precipitation reaching the ground. Marine cumulus, which have larger

9.2

Microphysics of Clouds

275

droplet sizes due to comparatively fewer CCN, are more likely to precipitate

than continental cumulus of similar size. Collision also tends to limit the size

of cloud droplets to a couple of millimeters, larger drops fragmenting during

impact.

9.2.3 Growth of Ice Particles

Clouds extending above the freezing level are called

cold clouds.

Cold clouds

are seldom composed entirely of ice. In fact, nearly 50% of clouds warmer

than -10~ and virtually all clouds warmer than -4~ contain no ice at

all (Fig. 9.13). Droplets in such clouds remain in the metastable state of

supercooled water (Problem 3.2). However, 90% of clouds colder than -20~

contain some ice.

The reason cold clouds are not composed entirely of ice, or

glaciated,

that ice can form only under special circumstances. Homogeneous nucleation

of ice is not favored at temperatures above -36~ The free energy of an

ice embryo formed when water molecules collect inside a droplet increases

for particles smaller than a critical dimension--analogous to homogeneous

nucleation of droplets (Fig 9.7). Therefore, such particles tend to disperse.

Because the critical dimension is greater than 20/zm for temperatures above

-36~ homogeneous nucleation of ice occurs only in very high clouds (see

Fig. 7.7).

Heterogeneous nucleation occurs when water molecules collect onto a spe-

cial type of particle known as a

freezing nucleus,

which has a molecular struc-

ture similar to that of ice. Because it assumes the size of the freezing nucleus,

an ice embryo formed in this manner can grow at temperatures much warmer

than that for homogeneous nucleation. Maritime clouds, whose droplet size

spectrum is broad (Fig. 9.10), glaciate more readily than continental clouds.

Crustal aerosol composed of clay is a prevalent form of freezing nucleus. But

1.00[~..~

_~ .80

z~ .60

.40

.20

0 ---'1----4

0 -4 -8 -12 -16 -20 -24 -28 -32 -36 -40

TEMPERATURE (C)

Figure

9.13 Fractional occurrence of unglaciated cloud, as a function of temperature, ob-

served over Germany (solid line) and Minnesota (dashed line). Source: Pruppacher (1981).

276

9 Aerosol and Clouds

even at -20~ less than one out of every hundred million aerosol particles

serves as an ice nucleus. This leaves many cloud droplets supercooled, which

makes aircraft encountering such clouds prone to icing.

Once formed, ice particles can grow through condensation of vapor, or

de-

position.

Since the saturation vapor pressure with respect to ice is lower than

that with respect to water (4.40), a cloud that is nearly saturated with respect

to water may be supersaturated with respect to ice. 4 A particle's growth is

therefore accelerated if it freezes and, in a mixed population, ice particles will

grow faster than droplets. Ice particles also grow through collision with super-

cooled droplets, or

riming.

This produces a layered structure about the original

ice particle. Beyond a certain dimension, the original structure is no longer

discernible. The resulting particle is then referred to as

grauple,

which is irreg-

ularly shaped and often proliferated with cavities. Last, ice particles can grow

through coagulation, which is favored at temperatures above -5~ These

mechanisms lead to a wide array of shapes and sizes that include bullets, plates,

slender needles, and complex configurations of hexagonal crystals (Fig. 9.14).

I Nlc 0 Pla ~ CP2b

Elementary sheath Hexagonal plate Bullet with dendrites

N2b ~ Plb ~ Rla

Combination of sheaths Crystal with Rimed needle crystal

sectorlike branches

Pie ~ Rib

(~ Cla Crystal with Rimed columnar crystal

Pyramid broad branches

Solid bullet Ordinary dendritic crystal Rimed plate or sector

q~ P2a ~ R2a

Cld Stellar crystal Densely rimed

Hollow bullet ~ with plates at ends plate or sector

$ P2d ~ Rna

O Cle Dendritic crystal Hexagonal graupel

Solid column with sectorlike ends

$ P2g .... R4b

Clf Plate with "'"^

Hollow column dendritic extensions ~ Lump graupel

Clg ~ P4b ~ I1

Solid thick plate Dendritic crystal Ice particle

with 12 branches

CPIa

Column with plates

Figure 9.14 Ice crystals observed in cold clouds. Adapted from Pruppacher and Klett (1978).

Reprinted by permission of Kluwer Academic Publishers.

4 Saturation mixing ratios with respect to water and ice typically differ by 10% or less, but at

very cold temperatures representative of the upper troposphere the differences can be substantial

(see Problems 5.27 and 17.9).

9.3 Macroscopic Characteristics of Clouds

277

9.3 Macroscopic Characteristics of Clouds

9.3.1 Formation and Classification of Clouds

Most clouds develop through vertical motion, when moist air is lifted above

its lifting condensation level (LCL) and becomes supersaturated. Clouds fall

into three broad categories:

1. Stratiform

(meaning "layered")clouds develop from large-scale lift-

ing of a stable layer. Characteristic of sloping convection, this process is

exemplified by warm moist air that overrides cold heavier air along a warm

front. Vertical motion accompanying stratus cloud development is of order

1 cm s -1 and the characteristic lifetime of these clouds is of order 1 day.

2. Cumuliform

(meaning "piled")

clouds

develop from isolated air

plumes that ascend buoyantly. Associated with cellular convection, cu-

mulous clouds grow through positive buoyancy supplied via sensible heat

transfer from the surface and latent heat released to the air during con-

densation, both of which make these clouds dynamic. Updrafts are of

order 1 m s -1 in developing cumulus, but can be several tens of m s -1 in or-

ganized mature cells like

cumulus congestus

and

cumulonimbus.

The char-

acteristic lifetime of cumulus clouds ranges from a few minutes to hours.

3. Cirriform

(meaning "fibrous")

clouds

develop through either of the

preceding forms of lifting. Found at high altitudes, cirrus clouds are com-

posed chiefly of ice particles, the larger components of which descend in

fallstreaks or

mares' tails

that give these clouds their characteristic wispy

appearance (Fig. 9.15).

Additional terms like

nimbo

and

alto,

which denote precipitation bearing and

midlevel, respectively, are used to specify particular cloud types (WMO, 1969).

Cumulus clouds develop in association with a plume of rising air known as a

thermal

(Fig. 5.4), which can be treated as a succession of buoyant air parcels.

Laboratory simulations reveal the development of a thermal to be closely

related to its dissipation (Fig. 9.16a). A mass of buoyant fluid is transformed

into a toroidal vortex ring (analogous to a smoke ring), which is characterized

by a hole in its center and a complex array of protrusions along its advancing

surface. Buoyant fluid penetrates its surroundings by entraining environmental

fluid into its center (Fig. 9.16b). This process dilutes its buoyancy and causes

the thermal to expand at an angle of about 15 ~ from the vertical. The buoyant

mass is turned completely inside out after advancing about 1.5 diameters,

during which it has become thoroughly mixed with surrounding fluid and its

buoyancy has been destroyed. The direct involvement of vorticity is illustrated

in the trajectory of individual air parcels, which may complete several circuits

about the advancing vortex ring before falling out of its influence.

Expansion is less evident in cumulus cloud because mixing with drier environ-

mental air leads to evaporation along its periphery. Because observed liquid water

278 9

Aerosol and Clouds

Figure 9.15 Hook-shaped

cirrus uncinus.

After WMO (1969).

contents are well below adiabatic values, mixing occurs throughout a cumulus

cloud and some environmental air is communicated into its very core. Conse-

quently, a cumulus can grow to appreciable height only by expanding enough

to isolate saturated air in its interior from drier environmental air being mixed

across its walls. Rising motion inside the thermal is compensated by gradual

sinking motion outside the thermal, which is distributed over a broader area.

That subsidence is promoted by evaporative cooling when saturated air mixes

with unsaturated environmental air. By inhibiting ascent, subsiding motion or-

ganizes new parcels into the existing cloud, which, in turn, favors its growth.

Likewise, buoyant parcels are most likely to produce cloud growth if the region

Velocity vectors

~ Contours

upward

component

velocity

-Contours

outward component

velocity

Stagnation pOlntS

+Sinks

Figure

9.16

(a) Laboratory simulation of buoyant fluid inside a thermal. After advancing 1 to

diameters, a mass of buoyant fluid has been

turned inside out. Environmental fluid entrained into its core produces a hole in the center of the buoyant mass.

(b)

Motion and streamlines

about the moving body

fluid. Mixing with surrounding fluid causes the advancing mass to expand at an angle from the vertical

about

15".

Adapted from Scorer

(1978).