Potter T.D., Colman B.R. (co-chief editors). The handbook of weather, climate, and water: dynamics, climate physical meteorology, weather systems, and measurements

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CCN due to human activities in the United States has been estimated at about 14%

of that from natural sources. Cloud and fog droplets sampled over land frequently

contain residue of combustion products and organic material, often of biogenic

origin. Forest ﬁres and sugar cane burns, for example, produce large concentrations

of CCN. Studies of subequatorial and Saharan air over West Africa suggest that CCN

are produced from bush ﬁre smoke, bacterial decomposition of plants and associated

sulfur-containing gases, and emission of droplets rich in soluble substances by

plants.

The surface of a pure water droplet consists of a layer of water molecules, each

with the potential to evaporate and enter the humid air above the surface. Vapor

molecules within the humid air also collide with and enter the drop. When a non-

volatile solute is present in the droplet, solute molecules or ions occupy some sites

on the droplet surface. Thus, fewer water molecules are available to evaporate into

the air. However, vapor molecules can enter the solution at the same rate as before.

Equilibrium can only be established when the vapor pressure decreases over the

solution so that the rate that water molecules leave the solution equals the rate at

which they enter the solution from the air. Raoult’s law,

þ n

 m



 m



þ im

¼ 1 þ

 m



1

ð12Þ

which can be derived from principles of equilibrium thermodynamics, describes this

process formally. Raoult’s law states that the vapor pressure over a solution droplet

) is reduced from that over a pure water dropl et (e

) by an amount equal to the

mole fract ion of the solvent. In (12), n

is the number of molecules of water in the

droplet, n

the number of molecules of solute, m

is the mass of the solute, and M

the molecular weight of the solute. In nature, soluble p articles are typically salts or

other chemicals that dissociate into ions when they dissolve. For solutions in which

the dissolved molecules dissociate, the number of moles of solute, n

, must be

multiplied by the factor i, the degree of ionic dissociation. Unfortunately, the

factor i, called the van’t Hoff factor, has not been determined for many substances.

Other quantities such as the rational activity coefﬁcient, the mean activity coefﬁ-

cient, and the molal or practical osmotic coefﬁcient have been used as alternate

expressions to the van’t Hoff factor and are tabulated in many chemical reference

books (Pruppacher and Klett, 1997).

The Kelvin equation (8) describes the equilibrium vapor pressure over a curved

water surface, e

, and Raoult’s law (12) describes the reduction in vapor pressure

over a solution droplet. Multiplying (8) by (12) to eliminate e

, gives

s;v

s;1

¼ 1 þ

Þm

1

exp

w;v



ð13Þ

266 MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

which describes the equilibrium saturation ratio over a solution droplet. For a sufﬁ-

ciently dilute solution, this equation can be simpliﬁed by approximating e

1 þ x

and (1 þ y)

1

1 7 y. Ignoring the small product x y, and m

compared to the

mass of water, one obtains

s;v

¼ 1 þ



ð14Þ

where a ¼2M

w,v



¼3.3 10

5

=T, b ¼3im

=4pM

¼4.3im

cgs units with T in kelvins. Equations (13) and (14) are forms of the Ko¨hler equation,

ﬁrst derived by the Swedish meteorologist H. Ko¨hler in the 1920s. Curves 2 to 6 in

Figure 6 show solutions of the Ko¨hler equation for droplets containing ﬁxed masses

of NaCl and NH

, common components of CCN. Curve 1 shows the solution for

a pure droplet, the Kelvin equation. For a given S

s,v

, droplets on the left side of the

maximum in the Ko¨hler curves are in stable equilibrium. If a droplet in equilibrium

on the left side of the curve experiences a small increase in radius due to chance

collection of vapor molecules, the droplet would ﬁnd the vapor pressure around itself

less than that required for equilibrium at its new radius. Physically, less water

molecules would be striking the droplet from the vapor ﬁeld than would be evapor-

ating from the droplet. As a result, the droplet would shrink, returning to its position

on the equilibrium curve. The opposite would happen if the droplet lost water

molecules—it would grow back to its equilibrium size. Note that when S

s,v

1,

all droplets growing from CCN remain on the left side of the curves. The small

droplets in stable equilibrium on the left side of the curve are called haze droplets.

Over some cities, where soluble particles are abundant and relative humidities high,

haze droplets can severely restrict visibility.

Figure 6 Variations of the relative humidity and supersaturation of air adjacent to droplets of

(1) pure water and solution droplets containing the following ﬁxed masses of salt: (2) 10

19

of NaCl, (3) 10

18

kg of NaCl, (4) 10

17

kg of NaCl, (5) 10

19

kg of (NH

)

, and (6)

18

kg of (NH

)

(Wallace and Hobbs, 1977).

3 FORMATION OF INDIVIDUAL CLOUD DROPLETS AND ICE CRYSTALS 267

A droplet at the peak of a curve, gaining a few molecules by random collisions,

would ﬁnd the vapor pressure around it greater than that required for equilibrium.

More molecules would strike the droplet than evaporate. Vapor would rapidly

deposit on the dropl et and it would quickly grow into cloud droplet. At the peak,

and on the right side of the curves, droplets are in unstable equilibrium. Droplets that

reach the peak in their Ko¨hler curves are said to be activated. Such droplets are

called cloud droplets. The peak of a curve (the critical radius) is the transition point

between the effect of the solute and drop curvature. When the supersaturation

s,v

7 1) in the atmosphere exceeds a critical value, droplets containing solute of

a critical mass will rapidly grow from haze droplets into cloud droplets. As indicated

in Figure 6, larger nuclei, which produce stronger solution droplets, are much

more likely to become cloud droplets because they require lower supersaturation

to activate.

Homogeneous Nucleation of Ice Crystals in Humid Air

Equations analogous to (8) and (10) can be derived for the homogeneous nucleation

of ice in humid air by assuming that the embryonic ice particle is spherical. In (8),

for example, S

w,v

, s

w,v

, and r

are replaced by S

i,v

¼e

r,i

si, 1

, the saturation ratio

with respect to ice, s

i,v

, the surface tension at the ice surface, and r

, the density of

ice. Numerical evaluation of the Kelvin equation for ice embryos shows that the

required supersaturations exceed those for water droplet nucleation. Calculations of J

for ice nucleation and the Kelvin equation both show that this process does not occur

in the atmosphere.

Homogeneous Nucleation of Ice Particles in Supercooled

Water Droplets

Homogeneous nucleation of ice in a water droplet requires that a stable icelike

molecular structure form within the droplet through statistical ﬂuctuations in the

arrangement of the water molecules. The development of such an ice embr yo is

favored compared to homogeneous nucleation of ice in humid air because the water

molecules in the droplet will be in direct contact with any icelike molecular struc-

tures created through statistical ﬂuctuations. Unlike homogeneous nucleation of ice

in moist air, the process of homogeneous nucleation in water involves two energy

barriers. The ﬁrst, DF

, is associated with the increase in free energy at the ice

crystal–liquid surface interface. The second, DF

, exists because energy is required

to break the bonds between individual water molecules before they can realign

themselves to join the ice embryo. The nucleation rate of ice in water therefore

depends on the number of liquid molecules per unit volume per unit time that will

contact an ice structure of critical size, the probability that an icelike structure will

exist in a droplet, and the probability that one of these liquid molecules will over-

268 MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

come the energy barrier and become free to attach to the ice structure. The rate

equation becomes

J ¼ 2N



w;i



1=2

exp 





ð15Þ

where h is Planck’s const ant, s

w;i

is the interface energy per unit area between the ice

surface and the water, and N

is the number of water molecules in contact with a unit

surface area of ice. Figure 7 shows measurements of J from several experiments. J

increases from about 1 to 1020 cm

3

=s as the temperat ure decreases from 32 to

40



C. Until recently, insufﬁcient information was available on the properties of

supercooled water and on DF

. Pr uppacher (1995) showed that incorrect extrapola-

tions of data for these properties to large supercoolings, and a lack of understanding

of the behavior of water molecules at large supercooli ngs led to the disagreement

between J calculated from classical theory and experimental data (Fig. 7).

Pruppacher (1995) resolved the discrepancies between the classical nucleation

equation, laboratory data, and the results of an earlier molecular theory. Pruppacher

noted that because water molecules become increasingly bonded at colder tempera-

tures, DF

might be expected to increase with decreasing temperature. However,

earlier cloud chamber experiments found that DF

decreases sharply at temperatures

Figure 7 Variation of the rate of homogeneous nucleation in supercooled water. Data are

from different experiments. The solid line (1) is from early classical theory. The solid line (2)

is from the revision of classical theory discussed by Pruppacher (1995). The dashed line is

from molecular theory (Pruppacher, 1995).

3 FORMATION OF INDIVIDUAL CLOUD DROPLETS AND ICE CRYSTALS 269

colder than 32



C. Pruppacher used nucleation rates measured in laboratory

experiments and inferred from ﬁeld observations, and recent measurements of the

physical properties of supercooled water, to calculate DF

and showed using the

classical nucleation equation that DF

indeed decreases at temperatures colder than

32



C. This behavior was attributed to the transfer of clusters of water molecules,

rather than individual water molecules, across the ice–water interface. With clusters,

the only hydrogen bonds that must be broken are those at the periphery of the cluster.

Pruppacher (1995) summarized the experiments of many investigators to deter-

mine the homogeneous nucleation temperature threshold for ice in water droplets.

He reasoned that (1) the larger the volume of a droplet, the larger is the probability of

a density ﬂuctuation in the droplet and the larger the probability that an ice embryo

will be produced and (2) the probability for ice formation in a given sized droplet

increases with increasing time of exposure of the droplet to a given range of

temperatures. Reexamining the available experimental data, he showed that the

lowest temperature at which virtually all pure water droplets froze was a function

of droplet diameter, with the spread in the data attributable to the different tech-

niques used in the experiments to support the drops (Fig. 8). Figure 8 shows that

freezing occurs at 40



C for 1-mm droplets and 35



C for 100-mm droplets.

Pruppacher’s results apply to cloud droplets, which are nearly pure water droplets,

their solute concentrations typically less than 10

3

mol=L.

Unactivated haze droplets consist of much stronger solutions whose chemical

consistencies depend on the parent CCN. Recent interest in understanding the

importance of polar stratospheric clouds in ozone depletion and the role of cirrus

in climate change has fueled investigations of the homogeneous nucleation of haze

droplets. The homogeneous nucleation tem perature in strong solution haze droplets

Figure 8 Lowest temperature to which extra pure water drops of a given size and exposed to

cooling rates between 1



C=min and 1



C=s have been cooled in various laboratory experi-

ments, indicated by different letters. Lines 1 and 2: Temperature at which 99.99% of a

population of uniform-sized water drops freezes when exposed to cooling rates of 1



C=min

and 1



C=s, respectively (Pruppacher, 1995).

270

MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

is depressed in proportion (nonlinearly) with increased solution molality. The equi-

librium size of a solution haze droplet increases and solution molality decreases with

increasing ambient relative humidity. As a result, the homogeneous nucleation

temperature is depressed most at low relative humidities. Two factors of interest

are the ice supers aturation where homogeneous nucleation occurs and the extent to

which a sulfate solution can be supercooled above ice saturation over the solution

drop before ice nucleation is possible. Figure 9 shows calculations of homogeneous

nucleation of droplets con taining sulfate aerosols made by Tabazadeh and Jensen

(1997). Their results, which correspond to J ¼1cm

3

=s, show that supercoolings of

about 3 K (below the equilibrium freezing temperature of a strong H

solution)

and ice supersaturation between 40 and 50% are required for homogeneous nuclea -

tion at ambient temperatures between 33 and 63



C (240 and 210 K). These

conditions can exist in upper tropospheric clouds, suggesting that the high ice

particle concentrations observed in some cirrus clouds are due to homogeneous

nucleation of haze droplets.

Heterogeneous Nucleation of Ice Crystals

Ice particles form in the atmosphere through homogeneous nucleation of super-

cooled water droplets, heterogeneous nucleation processes that involves aerosol

particles called ice nuclei (IN), and shattering of existing ice particles during

Figure 9 Critical ice supersaturation and the extent of supercooling required to achieve a

nucleation rate J ¼1cm

3

=s for strong H

solution droplets. The ice supersaturation is

deﬁned as the ratio of the ambient water vapor pressure over the ice saturation vapor pressure

when J ¼1cm

3

=s. Supercooling is deﬁned as the critical nucleation temperature minus the

temperature of ice that has a vapor pressure equal to the ambient water vapor pressure

(Tabazadeh and Jensen, 1997).

3 FORMATION OF INDIVIDUAL CLOUD DROPLETS AND ICE CRYSTALS 271

collisions or droplet freezin g events. Homogeneous nucleation of ice particles in

supercooled water is limited to temperatures colder than about 33



C. At warmer

temperatures, primary ice particle formation requires ice nuclei.

Most ice nuclei are composed of clay particles such as vermiculite, kaolinite, and

illite and enter the atmosphere during wind erosion of soils. Combustion, volcanic

eruptions, and airborne microorganisms are also sources of IN. Ice nuclei funct ion in

four modes: (1) deposition or sorption-nuclei adsorb water vapor directly on thei r

surfaces to form ice; (2) condensation-freezing nuclei act ﬁrst as CCN to form drops

and then as IN to freeze the drops; (3) immersion nuclei become incorporated into a

drop at T > 0



C and act to initiate freezing after the drop has been transported into a

colder region; and (4) contact nuclei initiate freezing of a supercooled drop on

contact with the drop surface. The fact that IN can function in these different

modes has made their measurement difﬁcult. Instruments designed to measure IN,

which include rapid and slow expansion chambers, mixing chambers, thermal preci-

pitation devices, membrane ﬁlters, and other devices, typically create conditions

favoring one mode, and often only measure the dependence of IN concentration

on one variable, such as temperature [see reviews by Vali (1985) and Beard (1992)].

In addition, the time scales over which ice nucleation can occur in natural clouds

often differs substantially from those characterizing the measurements. Measure-

ments made with different instruments at workshops in controlled conditions with

air samples drawn from the same source have shown considerable scatter. Absolute

values of concentrations of ice nuclei should therefore be viewed with some caution.

The concentration of ice nuclei in the atmosphere is highly variable (Fig. 10). In

general, ice nuclei concentrations increase by an order of magnitude with each 4



decrease in temperature but can vary by nearly an order of magnitude at any

temperature. The equation

¼ A expðb DTÞð16Þ

where A ¼10

5

per liter, b ¼0.6=



C, N

the number of active ice nuclei per liter

active at temperatures warmer than T and DT ¼T

7 T is the supercooling in degrees

centigrade, reasonably approximates this behavior. Ice nucleation can occur at rela-

tive humidities below water saturation, provided that the air is supersaturated with

respect to ice. Experiments have shown that at a given temperature, N

increases

with increasing supersaturation with respect to ice (s

) according to the relationship

¼ C

ð17Þ

where the values of C and k depend o n the air mass. The concentration of IN can

undergo orders of magnitude variations at a single location from day to day. Expla-

nations forwarded for these rapid changes include advection of dust from desert

windstorms, downward transport of stratospheric IN created from meteor bombard-

ment, IN production from evaporation of cloud and precipitation particles, and

preactivation of IN in the cold upper troposphere followed by transport to the

surface. Studies of the vertical proﬁle of IN concentrations have found that

272 MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

concentrations generally decrease with height in the troposphere above the surface

mixed layer, although evidence exists for higher concentrations of IN in the vicinity

of the jet stream.

Because ice nuclei must provide a stable solid substrate during the growth of ice

embryos, they are normally water insoluble. Nucleation occurs when an ice crystal

with a speciﬁc lattice structure ﬁrst forms o n a substrate that has a different lattice

structure. The probability of an ice nucleation event increases when the lattices of

atoms composing the ice crystal and ice nuclei closely align. For the IN atoms and

ice atoms to bond, a strain in the bonds must be accommodated between the out-of-

place atoms in the IN and ice lattices. The surface free energy of the interface

increases in response to increased elastic deformation. The greater the lattice

mismatch, the higher the surface free energy and the less likely the ice crystal

will form. Particle surfaces normally contain con taminants, cracks, crevices, and

particular growth patterns and may possess speciﬁc electrical properties due to

ions or polar molecules. Certain of these locations are much more effective at

adsorbing water molecules onto the surface of the aerosol and enhance the nucleat-

ing capability of the substance. Since the water molecule is polar, and the ice lattice

is held together by hydrogen bonding, substances that exhibit hydrogen bonds on

their surfaces act as more effective nucleants. For this reason, some organics exhibit

strong ice nucleating behavior. Ice nuclei are large aerosols, typically having radii

larger than 0.1 mm.

Figure 10 Mean or median worldwide measurements of ice nuclei concentrations from Bigg

and Stevenson (1970) (vertical gray bars), compilations of 11 studies by various authors in

Go¨tz et al. (1992) (thick lines), and compilations of 10 additional studies by Pruppacher and

Klett (1997) (thin lines). Bigg and Stevenson’s data at 10, 15 and 20



C are spread over a

small temperature range for clarity. The heavy dashed line is N

¼10

5

exp(0.6DT).

3 FORMATION OF INDIVIDUAL CLOUD DROPLETS AND ICE CRYSTALS 273

The theory governing the nucleation rate, J, of an ice embryo in a saturated vapor,

or of an ice embryo in a supercooled water drop, proceeds in a similar way to the

theory concerning heterogeneous nucleation of a drop. The theory assumes an ice

embryo forms a spherical cap with a contact angle with the surface substrate.

Hydrophobic substances have large contact angles and act as poor ice nucleants.

The equations are analogous to those used for nucleation of a water droplet, except

that all terms applying to water and vapor now apply to ice and vapor, or ice and

water. A discussion of this theory, and more complicated extensions of the classical

theory, is presented by Pruppacher and Klett (1997). The theory predicts that at

5



C, particles with radii smaller than 0.035 mm and a contact angle of 0 will not be

effective as ice nuclei. The threshold is 0.0092 mmat20



C. Few if any particles

will exhibit contact angles of zero, so actual ice nuclei will have to be somewhat

larger than these values.

Inside a water droplet, nuclei sizes can be somewhat smaller, with the threshold at

least 0.010 and 0.0024 mmat5 and 20



C, respectively. Experiments have shown

that the exact value of the cutoff is also dependent on the chemical composition of

the particle and on its mode of action (deposition, freezing, or contact). In the case of

deposition nuclei, it also depends on the level of supersaturation with respec t to ice.

Some organic chemicals have been found to have somewhat smaller sizes and still

act as ice nuclei.

There is considerable experimental evidence showing that atmospheric ice nuclei

can be preactivated. Preactivation describes a process where an ice nucleus initiates

the growth of an ice crystal, is subjected to an environment where complete subli-

mation occurs, and then is involved in another nucleation event. The particle is said

to be preactivated if the second nucleation event occurs at a signiﬁcantly warmer

temperature or lower supersaturation. Ice nuclei can also be deactivated, that is lose

their ice nucleating ability. This effect is due to adsorption of certain gases on to the

surface of the nucleus. Pollutants such as NO

,SO

, and NH

have been found to

decrease the nucleation ability of certain aerosols. There has also been evidence

from laboratory and ﬁeld experiments that the nucleating ability of silver iodide

particles decreases when the aerosols are exposed to sunlight.

The very poor correspondence between ice nucleus measurements and ice particle

concentrations in clouds has yet to be adequately explained. Hypotheses forwarded

to explain these observations generally focus on more effective contact nucleation,

particularly in mixed regions of clouds where evaporation can lead to the formation

of giant ice nuclei, and secondary ice particle production, which involves shattering

of existing ice particles during collisions or droplet freezing events. The relative

importance o f each of these mechanisms in real clouds is still uncertain.

4 FORMATION OF RAIN IN WARM CLOUDS

Raindrops form through one of two microphysical paths. The ﬁrst occurs when ice

particles from high, cold regions of clouds fall through the melting level and become

raindrops. In cloud physics literature, clouds that support this process are called

274 MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

‘‘cold’’ clouds. ‘‘Warm’’ clouds are clouds that develop rain in the absence of ice

processes. In the tropics, shallow clouds such as trade wi nd cumulus and stratocu-

mulus produce rain entirely through warm rain processes. Deep tropical convective

clouds and summertime midlatitude convection also support active warm rain

processes, although melting ice also contributes to rainfall.

The warm rain process occurs in three steps: (1) activation of droplets on CCN,

(2) growth by condensation, and (3) growth by collision and coalescence of droplets.

The rapid production of warm rain in both maritime and continental clouds remains

one of the major unsolved problems in cloud physics. The centr al problem lies in the

transition from step 2 to 3. Theory predicts that g rowth by condensation will create

narrow drop spectra in clouds. Growth by coalescence, on the other hand, requires

large and small cloud droplets within droplet spectra. Determining how broad

droplet spectra develop in clouds has been a central theme of cloud physics research.

Growth of a Single Droplet by Condensation

Cloud droplets form when the supersaturation in the atmosphere becomes sufﬁ-

ciently large so that haze droplets reach their critical radii a nd begin to grow in

the unstable regime to the right of the Ko¨hler curves (Fig. 6). The equation for the

growth of a pure water droplet, ﬁrst derived by Maxwell in 1890, is obtained by

assuming that (1) heat transfer and vapor concentration in the vicinity of the droplet

both satisfy the diffusion equation, (2) an energy balance exists at the surface of the

droplet such that the latent heat added during condensation balances the diffusion of

heat away from the droplet, and that (3) the concentration of droplets in the vapor

ﬁeld is independent of direction outwards from the droplet. Modiﬁcations must be

made to account for curvature and solute effects. Additionally, one must account for

kinetic effects near the drop surface. These include vapor mol ecules striking the

surface of the droplet and rebounding rathe r than condensing, and the direct heat

transfer that occurs at the droplet–vapor interface as water molecules cross the

interface between the liquid and gas phases.

Accounting for these effects, the equation describing the diffusional growth of a

droplet is

w;v

 1  a=r þ b=r

ððL

TÞ1ÞðL

=KT

f ðaÞÞ þ ðr

T=De

s;1

gðbÞÞ

ð18Þ

where a and b are deﬁned in Eq. (14), L

is the latent heat of vaporization, R

is the

gas constant for water vapor, D is the diffusion coefﬁcient, and K, the thermal

conductivity. The quantities f (a) and g(b) are given by:

f ðaÞ¼

r þ½ðK=apÞð2pR

TÞ

1=2

=ðC

þðR

=2ÞÞ

r þ l

ð19Þ

gðbÞ¼

r þðD=bÞð2p=R

TÞ

1=2

r þ l

ð20Þ

4 FORMATION OF RAIN IN WARM CLOUDS 275