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In Eqs. (19) and (20), the accommodation coefﬁcient, a, characterizes the transfer

of heat by molecules arriving at and leaving the interface between the liquid and

vapor phase, and the condensation coefﬁcient, b, is the fraction of vapor molecules

hitting the droplet surface that actually condense. In these equations, C

is the

speciﬁc heat at constant volume, p is pressure, and R, the gas constant of dry air.

The ter ms l

and l

can be considered length scales. As r becomes large such that

r l

, l

, f (a) and g(b) approach unity. This essentially is the case by the time a

droplet reaches r ¼5 mm, as is apparent from Figure 11, which compares growth by

condensation with and without kinetic effects.

Equation (18) relates the radius of a droplet growing by vapor diffusion to the

supersaturation in the atmosphere, and the environmental pressure and temperature.

It is clear that the rate of growth of the radius of a droplet is inversely proportional to

its size. Solution and curvature effects are most important when the droplets are very

small, as indicated by the 1=r and 1=r

dependence. Table 1 gives calculated growth

times in seconds for droplets with different nuclea r masses of NaCl at T ¼273 K,

P ¼900 mb and a supersaturation of 0.05%. Note that for a droplet with a large

nucleus, growth to r ¼20 mm takes 5900 s or 1.63 h. Growth to a radius of 50 mm

takes 41,500 s, or approximately a half-day. A very small drizzle droplet is about

50 mm radius. Individual cloud parcels are unlikely to exist in a supersaturated

environment for half a day. Clearly, diffusional growth alone is inadequate to explain

the formation of rain in clouds.

Equation (18) is valid for a stationary droplet. As droplets grow, they fall through

the air. Air ﬂowing around the droplet causes the ﬁeld of vapor molecules to interact

differently with the droplet surface than for a still droplet. The net result is that the

growth rate (or evaporation rate if the relative humidity <100%) of the droplet

increases. The effect of ventilation has been studied using both numerical simula-

tions and by making direct measurements in wind tunnels. For droplets with

Figure 11 Comparison of condensation growth, with and without kinetic effects, for

droplets of initial radii of 1 and 5 mm (Rogers and Yau, 1989).

276

MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

r 50 mm, ventilation can be ignored, but for droplets of greate r sizes, the ventilation

effect increases with droplet size. In the growth equation, ventilation is accounted for

by multiplying the numerator on the right by f

, the ventilation coefﬁcient.

Growth of Population of Droplets by Condensation

Cloud droplets will activate on CCN when a parcel of air rises through cloud base.

The larger, more soluble CCN activate ﬁrst, followed by the more numerous, but less

effective CCN. The new droplets each extract water vapor from the parcel. The

maximum supersaturation occurs, and the droplet concentration is determined,

when the rate of vapor deposition on the drops equals the rate at which vapor is

supplied through cooling of the rising parcel. Above this point in a cloud, the

supersaturation remains below its peak value and no additional droplets are acti-

vated. The cloud droplet concentration therefore depends on the spectrum of CCN

entering the cloud, the updraft strength, and the cloud base temperature.

Equation (18) shows that the growth rate of activated cloud droplets decreases

with radius. Thi s equation predicts that the width of a spectrum of different sized

droplets narrows with time, since smaller droplets in a rising parcel grow faster than

larger droplets. This is illustrated by the measurements and computations reported

by Fitzgerald (1972). Fitzgerald found close agreement between cloud droplet spec-

tra measured 200 to 300 m above cloud base with that predicted by condensation

theory includ ing kinetic effects (see Fig. 12).

For collisions between cloud droplets to occur, the droplet spectrum must evolve

so that droplets possess a range of sizes and fall speeds. Calculations suggest that the

collision efﬁciency of two falling drops only becomes appreciable (>10%) when the

radius of the larger collector drop (R) and smaller collected droplet (r) reach

R ¼25 mmandr ¼7.5 mm. Equation (18) predicts that a droplet with a large nucleus

will require 1.6 h to grow to r ¼20 mm, yet many tropical maritime clouds produce

TABLE 1 Time (s) required for Droplet to Grow from

Initial Radius of 0.75 mm to Speciﬁed Size at 273 K, Pres-

sure of 900 mb, and Supersaturation [100(S

w,v

7 1)] ¼0.05

for Salt Nuclei with 3 Nuclear Masses

Radius (mm) 10

14

g10

13

g10

12

2 130 7 1

5 1,000 320 62

10 2,700 1,800 870

15 5,200 4,200 2,900

20 8,500 7,400 5,900

25 12,500 11,500 9,700

30 17,500 16,000 14,500

35 23,000 22,000 20,000

From Best (1951).

4 FORMATION OF RAIN IN WARM CLOUDS 277

rain in about 20 min. Continental clouds, which have much higher droplet concen-

trations, also produce rain in short times compared to the prediction of Eq. (18).

Although continued condensation within a parcel may eventually generate droplets

larger than about 25 mm radius to initiate coalescence, the time for this process to

produce rain is generally much too long.

Proposed Mechanisms for Broadening of Droplet Spectra

Several mechanisms have been suggested to explain the production of large cloud

droplets capable of initiating coalescence. These include (1) favorable CCN spectra,

such as low concentrations of smaller nuclei and high concentrations of larger

nuclei; (2) mechanisms that enhance condensation growth of a few larger cloud

droplets through mixing of air parcels; (3) stochastic condensation, the mixing of

droplets (rather than parcels) that have different supersaturation histories; and (4)

ﬂuctuations in supersaturation or collision rates induced by turbulence.

One of the earliest mechanisms hypothesized to explain the onset of precipitation

is that very large (>10 mm radius) wettable or soluble nuclei exist below cloud base

Figure 12 Computed and measured cloud droplet distributions at a height of 244 m above

cloud base (Fitzgerald, 1972).

278

MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

that, when carried through cloud base, can almost immediately begin to collect cloud

droplets. Measurements in marine environments show that giant salt particles can

often occur in signiﬁcant concentrations. Within the moist region below cloud base

such salt particles will deliquesce into much larger solution droplets [e.g., a NaCl

particle of 1 ng (5 mm) has a 25-mm equilibrium radius at 99% relative humidity].

Thus, there are often signiﬁcant concentrations of coalescence nuclei that can enter

the base of maritime clouds. Numerical parcel models initialized with CCN spect ra

from observations have been used to evaluate the role of giant CCN in the produc-

tion of warm rain. Calculations show that raindrops can form on these CCN in time

scales comparable to observed cloud lif etimes. Drop trajectory calculations also

suggest that accretion of cloud water on giant and ultragiant nuclei can account

for the formation of rain in observed time scales when these large particles are

present at cloud base. Nevertheless, questions remain as to how important these

large salt particles really are to the warm rain process in maritime clouds. Uncer-

tainties about the growth rate of these particles in updrafts remain, since their growth

is highly nonequilibrium. Also, the residence time of larger salt particles in the

atmosphere is short, with the highest concentrations located close to the ocean

surface. Much less is known about the role giant par ticles play in initiating coales-

cence in continental clouds. Measurements show that 50-mm-diameter particles exist

in concentrations of 100 to 1000 m

3

in the boundary layer over the High Plains of

the United States. These are typically insoluble soil particles that will not grow as

they pass through cloud base. They are large enough, however, to make effective

coalescence nuclei without undergoing growth by condensation.

A second mechanism to explain the onset of precipitatio n involves broadening of

the drop size distribution during the mixing process. Early studies of parcels

subjected to velocity ﬂuctuat ions were unable to reproduce the appreciable broad-

ening of the cloud droplet spectra that occurs in warm cumulus. These studies

invoked the process of homogeneous mixing in which dry air is mixed into a

cloudy parcel of air such that all droplets are exposed to the dry air and evaporate

until the parcel is again saturated. This process has been shown to produce a constant

modal size, but a decrease in mean size because of a broadening toward smaller

sizes. Absent in this process is the large drop tail to the spectr um required for

coalescence growth.

Laboratory experiments by Latham and Reed (1977) raised questions about the

applicability of the homogeneous mixing process. Their experiments suggested that

the mixing process occurs in such a way that some portions of a cloudy parcel are

completely evaporated by mixing with unsaturated air while other portions remain

unaffected. Broadening of the droplet spectrum toward larger sizes will occur by

subsequent condens ation of the parcel if it continues to ascend, since there is

reduced competition for vapor among the remaining droplets. Further studies have

showed that an idealized inhomogeneous mixing process will produce appreciable

broadening compared to homogeneous mixing. However, other studies have found

that special dynamical sequences of vertical mixing can achieve the same spectral

broadening without invoking inhomogeneous mixing concepts. Observations in

clouds over the High Plains of the United States found that the large drop peak

4 FORMATION OF RAIN IN WARM CLOUDS 279

was more consistent with a closed parcel environment than a mixed one and that

there was no evidence that mixing or cloud age increased the size or concentration of

the largest drops. Considerable uncertainty remains concerning the importance of

mixing to the broadening of the d roplet spectrum.

The stochastic condensation mechanism invokes the idea that droplets can have

different supersaturation histories. This process considers the mixing of droplets,

rather than parcels. Cooper (1989) describes the theory of stochastic condensation.

Regions of clouds having ﬁne-scale structure, such as mixed regions of clouds

outside of adiabatic cores, would best support this type of process. Measurements

within warm, orographic clouds in Hawaii have shown littl e broadening in the

laminar clouds, but appreciable broadening in the breaking wave regions of the

clouds near cloud top, suggesting the importance of stochastic condensation in

producing large droplets.

A fourth hypothesis is that the breadth of the droplet spectra in clouds can be

increased by turbulent ﬂuctuations. Conceptually, turbulence may be thought to

induce vertical velocity ﬂuctuations that induce ﬂuctuations in supersaturation.

These ﬂuctuations, in turn, lead to the product ion of new droplets at locations

throughout the cloud. Turbulent ﬂuctuations can also lead to drop clustering,

which can create opportunities for favorable supersaturation histories and=or

enhanced collision rates. A general approach has been to examine the evolution

of a distribution of droplets exposed to a supersaturation that varied with a known

distribution, such as a normal distribution, and to derive analytic expressions for the

droplet size distribution as a function of time. This approach predicts continued

dispersion of the spectra with time. This approach has been criticized on the basis

that updrafts and supers aturation are highly correlated. A droplet that experiences a

high supersaturation is likely to be in a strong updraft and will arrive at a given

position in a cloud faster, which means there will be less time for growth. Conver-

sely, a droplet that experiences a low supersaturation will grow slower but have a

longer time to grow. The net result is that the droplets arrive at the same place with

approximately the same size. Supersaturation ﬂuctuations in clouds can arise not

only from ﬂuctuations in the vertical velocity, but also from ﬂuctuations in integral

radius (mean radius multiplied by the droplet concentration) of the droplet spectra.

Realistic droplet spectra can be obtained in model simulations when the ﬂuctuations

in mean radius are negatively correlated with vertical motions. Unfortunately, experi-

mental data in cumulus show the opposite behavior—the largest droplets occur in

upward moving parcels.

Recent studies have provided conﬂicting evidence regarding the role of turbu-

lence in spectral broadening. Studies show that turbulence can lead to signiﬁcant

trajectory deviations for smaller droplets in clouds, leading to larger relative vector

velocities for droplets and collector drops and enhanced collision rates. Results

indicate that cloud turbulence supports spectral broaden ing and more rapid produc-

tion of warm rain. Studies have also examined whether turbulence can create regions

of preferential concentration in conditions typical of cumulus clouds, and whether

nonuniformity in the spatial distribution of droplets and=or variable vertical velocity

in a turbulent medium will contribute to the broadening of the drop size distribution.

280 MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

These studies suggest that turbulence severely decreases the broadening of the size

distribution compared to numerical experiments performed in the absence of turbu-

lence. The importance of turbulence in drop spectral broadening is still a matter of

debate. Villiancourt and Yau (2000) provides a review of recent understanding of the

role of turbulence in spectral broadening.

Growth by Coalescence

When sufﬁciently large (>25 mm radius) cloud droplets exist in a cloud, their growth

is accelerated by collision and coalescence with neighb oring droplets. The nonlinear

behavior of the ﬂow around a drop affects both the drop’s terminal velocity and the

manner in which a smaller drop is swept around a larger drop as the two approac h in

free fall.

Single Drop Hydrodynamics The Reynolds number, given by N

¼ 2V

r=n,

where V

is the drop terminal velocity, and n is the kinem atic viscosity, is the ratio of

inertial to viscous forces and is typically used as a scaling parameter to describe the

ﬂow regimes around a drop. The characteristics of ﬂow around a sphere as a function

of the Reynolds number are well documented. When N

is very small (<0.4), the

droplet is in the so-called Stokes ﬂow regime where viscous forces dominate and the

ﬂow ﬁeld around a droplet is symmetric and laminar. At about N

¼2 (about

r ¼50 mm at standard temperature and pressure), a wake appears behind the droplet.

At about N

¼20 (about r ¼150 mm), an eddy of rotating ﬂuid appears behind the

drop. This eddy grows as N

increases to 130 (r ¼350 mm), at which point the wake

becomes unstable. At N

¼400 (r ¼650 mm) eddies are carried downstream by the

turbulent wake.

The problem of determining the ﬂow around a drop is complicated by shape

deformation due to hydrodynamic and sometimes electrical forces. The nature of

the deformation is primarily related to the droplet size. Observations in wind tunnel

experiments and model calculations have provided information on the nature of drop

deformation. Numerical modeling has further clariﬁed the nature of drop deforma-

tion under the inﬂuence of external hydrostatic pressure. Chuang and Beard (1990)

summarize the effect of hydrodynamic and electric forces on both charged and

uncharged drops. Figure 13 from their study shows the shapes drops can assume

as they fall in various electric ﬁelds characteristic of thunderstorms.

The drag force on the droplet and the force of gravity determine the terminal

velocity of a droplet. For Stoke’s ﬂow, determining the balance between these forces

is relatively straightforward, but in the more complicated ﬂow regimes, a single

accurate analytical expression for the terminal velocity is not available. Rogers

and Yau (1989) suggest the following general expression:

¼ ar

ð21Þ

for drops that remain spherical, where the terms a and b have different values

depending on the Reynold’s number. Gunn and Kinzer (1949) provide the most

4 FORMATION OF RAIN IN WARM CLOUDS 281

accurate observational data over a large range of drop sizes. Their measurements,

made at 1013 mb and 20



C, are shown in Figure 14.

Collection Efﬁciency When two droplets interact in free fall, the outcome of

the interaction will depend on the droplet’s sizes and the offset, x, between the drop

centers. Inside some critical offset, x

(see Fig. 15), the outcome will be a collision,

while outside x

the outcome will be a miss. For drops acting as rigid spheres, the

Figure 13 Modeled drop shapes and axis ratios (a ¼vertical=horizontal dimension) for 5-

mm diameter drops with various distortion effects: (a) stationary drop (surface tension only),

(b) drop resting on a ﬂat surface, (c) raindrop falling at terminal velocity, (d) stationary drop in

strong vertical electric ﬁeld, (e) raindrop falling in a strong vertical electric ﬁeld, ( f ) highly

charged drop falling at terminal velocity, (g) stationary drop in the maximum vertical electric

ﬁeld before disintegration, (h) charged raindrop falling in maximum thunderstorm electrical

ﬁeld with upward directed electric force, and ( j) same as (i), but with downward electric force

(Chuang and Beard, 1990).

282

MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

Figure 14 Terminal velocity of a raindrop as a function of its diameter at 1013 mb and 20



(adapted from Gunn and Kinzer, 1949).

Figure 15 Collision geometry for two droplets.

4 FORMATION OF RAIN IN WARM CLOUDS 283

collision efﬁciency, E

col

, is deﬁned as

col

ðx

ðR þ rÞ

ð22Þ

the ratio of the collection cross section to the geometric cross section for the large

and small drops. Theoretical calculations for collector drops with R < 75 mm are

shown in Figure 16. Typically most newly formed cloud droplets will range in

size from about 4 < r < 8 mm. A 25-mm-radius collector drop will have a collection

efﬁciency of about 10% for these droplets. For this reason, the 25-mm radius is

generally considered a threshold size for initiation of coalescence. Theoretical colli-

sion efﬁciencies for larger R and r=R ratios rapidly approach unity and can even

exceed unity when r=R approaches 1 and R > 40 mm due to the possibility of capture

Figure 16 Collision efﬁciencies for small spheres as a function of the ratio of their radii.

Solid lines from Schlamp et al. (1976), dashed lines from Klett and Davis (1973), dashed-dot

lines from Lin and Lee (1975). (Adapted from Schlamp et al., 1976.)

284

MICROPHYSICAL PROCESSES IN THE ATMOSPHERE

of the small droplet in the large droplet wake. Unfortunately, no reliable laboratory

studies exist for R < 40 mm, so the critical efﬁciencies for the coalescence threshold

remain untested.

The calculations of E

col

in Figure 16 assume that the drops are rigid spheres.

Drizzle-sized drops (R > 100 mm) deform upon approach to another droplet as the air

ﬁlm between the drops drains. Studies of small precipitation-sized drops in the

laboratory have shown that the outcome of collisions can be complicated, with the

possibility for coalescence, temporary coalescence, and satellite production. These

outcomes depend on R and r, the drop separation, charge, and the ambient relative

humidity. Coalescence depends upon the drop–droplet interaction time and their

impact energy. Formulas to calculate coalescence efﬁciencies, E

coal

, over a wide

range of drop sizes are now available. The coll ection efﬁciency, E, is the product

of E

col

and E

coal

. Beard and Ochs (1984) show that as E

col

approaches unity,

coal

decreases, leading to a maximum E for drop–droplet pairs in the size range

R ¼100, r ¼12 mm (see Fig. 17).

Figure 17 Contours of collection efﬁciency (in percent) as a function of drop and droplet

radius (Beard and Ochs, 1984).

4 FORMATION OF RAIN IN WARM CLOUDS 285