Houze Robert A., Jr. Cloud Dynamics

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3.2 Microphysics of Cold Clouds 89

lost to the warm hailstone by shedding. However, a considerable portion of the

collected water becomes incorporated into a water-ice mesh forming what is

called spongy hail. This process is called wet growth. During its lifetime, a hail-

stone may grow alternately by the dry and wet processes as it passes through air

of varying temperature. When hailstones are sliced open, they often exhibit a

layered structure, which is evidence of these alternating growth modes.

3.2.6 Ice Enhancement

When the concentrations of ice particles are measured in natural clouds, it is often

found that there are far more ice particles present than can be accounted for by the

typical concentrations of ice nuclei activated by lowering the temperature of air in

expansion chambers.

60 Figure 3.11 compares some measurements of ice particle

concentrations in cumuliform clouds with the concentration of ice nuclei expected

from expansion-chamber measurements to be active at the cloud-top temperature.

The latter concentration is calculated from (3.38). The cloud-top temperature is

the lowest temperature anywhere in the cloud and hence provides an estimate of

the maximum possible ice nucleus concentration in the cloud according to (3.38).

The actual particle concentration is seen typically to exceed the maximum possi-

ble nucleus concentration by one or more orders of magnitude.

These high concentrations are found in many cold clouds. They do not, how-

ever, occur in uniform spatial and temporal patterns within a cloud. A common

characteristic is that they occur in older rather than newly formed portions of

clouds, and they are found in association with supercooled cloud droplets. They

are most likely to be found when the size distribution

ofthe

droplets is broad, with

largest drops exceeding about 20

/-tm

in diameter. The high concentrations de-

velop initially near the tops of clouds, and the high concentrations may develop

suddenly (e.g., the concentration may rise from 1 to 1000

t:'

in less than 10 min).

The microphysical process, or processes, by which the concentrations of ice

particles become so highly enhanced relative to the number of nuclei which would

appear to be active according to (3.38) are not certain. Some hypotheses that have

been suggested

are"

(i) Fragmentation

ice crystals. Delicate crystals may break into pieces as a

result of collisions and/or thermal shock.

(ii) Ice splinter production in riming.

has been found in laboratory experi-

ments that when supercooled droplets >23

/-tm

in diameter collide with an ice

surface at a speed of

;:::1.4

S-I

at temperatures of - 3 to

-8°C,

small ice splinters

are produced.P

For

a more complete account of the observations and hypotheses regarding the occurrence of

high ice particle concentrations in clouds, see Hobbs and Rangno (1985, 1990)and Rangno and Hobbs

(1988, 1991).

For

a more full discussion and many references to the literature. see the Hobbs and Rangno

papers mentioned in the previous footnote.

62 The laboratory experiments were performed by Hallett and Mossop (1974). This ice-enhance-

ment process is often called the Hallett-Mossop mechanism.

90 3 Cloud Microphysics

10'

-35

iI-

: -

-5

-10 -15

-20

-25

-30

CLOUD TOP TEMPERATURE (0C)

Q'lI

Q;l

Figure 3.11 Maximum ice particle

concentrations observed in mature and

aging maritime (open humps), continen-

tal (closed humps), and transitional (half-

open humps) cumuliform clouds. The

line represents the concentrations ex-

pected at the cloud-top temperature.

(From Hobbs and Rangno, 1985. Re-

printed with permission from the Ameri-

can Meteorological Society.)

(iii) Contact nucleation.

is thought that when certain aerosol particles come

into contact with supercooled droplets they can cause nucleation at higher tem-

peratures than they would through other forms of nucleation.

(iv)

Condensation or deposition nucleation. There is evidence that the ice-

nucleating activity of atmospheric aerosol particles by either condensation or

deposition nucleation is greatly increased when the ambient supersaturation rises

above 1

% with respect to water. Ice nucleus counters whose data lead to the

expression in (3.38) are usually operated near water saturation. A pocket of high

supersaturation in a cloud might be favorable for the sudden appearance of a large

number of ice particles.

The latter two mechanisms, (iii) and (iv), do not require the pre-existence of ice

particles and may thus help to account for the sudden appearance of high concen-

trations in cloudy air at relatively high temperatures.

3.2.7 Fall Speeds of Ice Particles

The fall speeds of ice particles encompass a wide range. Observations show that

these speeds depend on particle type, size, and degree of riming and that the more

heavily rimed a particle, the more its fall speed depends on its size. Individual

snow crystals (lower curves of Fig. 3.12) and unrimed to moderately rimed aggre-

3.2 Microphysics of Cold Clouds 91

2.5

2.0

t::

1.5

-l

1.0

• x

0.5

•

••

. .

•

I I ! ! !

12345678

MAXIMUM DIMENSION OF CRYSTAL (mm)

Figure

3.U

Terminal fall speeds of snow crystals as a function of their maximum dimensions.

Open circle (uppermost curve) indicates graupel fall speeds. Other curves are for rimed crystals (dot in

circle), needles (filled circle with slash), spatial dendrites (triangle), powder snow

(x),

and dendrites

(filled circle). (From Nakaya and Terada, 1935.)

gates of crystals (Fig. 3.13) drift downward at speeds of 0.3-1.5 m

S-I,

with the

aggregates showing a tendency to increase slightly in fall speed as they approach

12 mm in dimension. Graupel fall speeds increase sharply from 1 to 3 m

S-1

over a

narrow size range of 1-3 mm (upper curve of Fig. 3.12 and Fig. 3.14). Empirical

formulas for fall speeds of snow and graupel for the data set represented in Fig.

3.12 and Fig. 3.14 are listed in Table 3.2. These formulas apply near the earth's

surface. They do not take into account the fact that the fall speed also depends on

the density of the air through which the particles are falling. The dependence on

air density has been determined experimentally and theoretically.

Hailstone fall velocities are an order of magnitude larger than those for snow

and graupel. At a pressure of 800 mb and a temperature of

O°C,

they obey the

empirical formulas'

(3.46)

63 See, for example, Bohrn (1989).

64 From Pruppacher and Klett (1978, p. 345), based on data of Auer (1972).

2.0

1.5

•

-----

..J

1.0

..J

•

0.5

0.0

12 14

MAXIMUM

DIMENSION

(mm)

Figure 3.13 Terminal velocity and maximum dimension measurements for unrimed to

moderately rimed aggregates. Combinations of: sideplanes (a type of branched crystal), bullets and

columns (circles, dotted curve); sideplanes (triangles, solid curve); radiating assemblages of dendrites

(asterisks, dashed curve); dendrites (squares,

dash-dot

curve). (From Hobbs, 1974, based on data

from Locatelli and Hobbs, 1974. Reprinted with permission from Oxford University Press.)

3.0

•

2.5

I 4

, 1

•

2.0

• / I

•.

•

/4/

I·

·"~I

1.5 / I

• . L

/ /* : *

..J

,. *

* 4

1.0

./~

.*.**

/ 4 *

. t

••

***

...

I * '\ * *

* *

0.5

51 2 3 4

MAXIMUM

DIMENSION

(mm)

O.O~-----'------:'---~-----'----~

Figure 3.14 Terminal velocity and maximum dimension measurements for graupel and

graupel-like snow. Cone-shaped graupel (circles,

dash-dot

curve); hexagonal graupel (triangles,

dashed curve); graupel-like snow of hexagonal type (asterisks). (From Hobbs, 1974, based on data

from Locatelli and Hobbs, 1974. Reprinted with permission from Oxford University Press.)

3.2 Microphysics of Cold Clouds 93

Tahle 3.2

Empirical Relationships between the Terminal Velocities v (in m

S-I)

of Solid Precipitation Particles

and Their Maximum Dimension D

(in mm)

Graupel and graupel-like

snow

Unrimed aggregates

Densely rimed aggregates

Densely rimed columns

Type of particle-

Cone-shaped graupel

Hexagonal graupel

Graupel-like snow of lump type

Graupel-like snow of hexagonal type

Combination of sideplanes, plates, bullets,

and columns

Sideplanes

Radiating assemblages of dendrites or den-

drites

Radiating assemblages of dendrites or den-

drites

u-Di; relationship

v = I ·2Do·

V =

D£'

V =

l'ID~'28

V =

O'86D~25

V = O'

69D~'41

v =

O'82D~'12

V = O'

8D~'16

v=O'79D~27

v = I

'ID~'56

Source: Locatelli and Hobbs (1974).

a Based on Magana and

Lee's

(1966) classification.

where D

is the diameter of the hailstone in cm. This formula was obtained for

hailstones in the size range

0.1-8

em. Over this range, the fall speeds indicated by

(3.46) are roughly 10-50 m s

These large values imply that updrafts of compara-

ble magnitude must exist in the cloud to support the hailstones long enough for

them to grow.

Hence,

hail is found only in very intense thunderstorms, of the

types considered in Chapters 8 and 9.

3.2.8 Melting

Ice particles can change into liquid water when they come into contact with air or

water that is above

O°C.

A quantitative expression for the rate of melting of an ice

particle of mass m can be obtained by assuming heat balance for the particle

during the melting. According to (3.45), this balance may then be written as

-Lfrit

me1

4nRIC

a[T(oo)-273

K]V

ritcolcw(Tw

-273

K)+Q

(3.47)

where

mmel

is the rate

change of the mass of the ice particle as a result of

melting.

The

first term on the right is the diffusion of heat toward the particle from

the air in the environment.

The

second term is the rate at which heat is transferred

to the ice particle from water drops of temperature

Tw that are collected by the

melting particle.

The

third term,

c.,

is the gain or loss of heat by vapor diffusion

[given by (3.43) in the case of a spherical hailstone]. If both the air and drop

temperatures exceed 273 K, then both the first and second term in (3.47) contrib-

ute to the melting.

94 3 Cloud Microphysics

3.3 Types of Microphysical Processes and Categories

of Water Substance in Clouds

From the foregoing review

cloud microphysics (Sees. 3.1 and 3.2), it is evident

that water substance can take on a wide variety of forms in a cloud and that these

forms develop under the influence of

seven basic types

microphysical pro-

cesses:

1. Nucleation of particles

2. Vapor diffusion

3. Collection

4. Breakup of drops

5. Fallout

6. Ice enhancement

7. Melting

These individual processes may sometimes be isolated for study in numerical

models or in the laboratory. However, in a natural cloud several or all of these

processes occur simultaneously, as the entire ensemble of particles comprising

the cloud forms, grows, and dies out. Thus, the various forms of water and ice

particles coexist and interact within the overall cloud ensemble.

is the behavior

of the overall ensemble that is

primary interest in cloud dynamics, and it is

generally unnecessary to keep track of every particle in the cloud in order to

describe the cloud's gross behavior. At the same time, it is also impossible to

ignore the microphysical processes and accurately represent the cloud's overall

behavior. To retain the essentials of the microphysical behavior, it is convenient

to group the various forms of water substance in a cloud into several broad

categories

water substance:

• Water vapor

is in the gaseous phase.

• Cloud liquid water is in the form of small suspended liquid-phase droplets

(i.e., drops that are too small to have any appreciable terminal fall speed

constitute cloud liquid water and therefore are generally carried along by

the air in which they are suspended).

• Precipitation liquid water is in the form of liquid-phase drops that are large

enough to have an appreciable fall speed toward the earth. This water may

be subdivided into drizzle (drops 0.1-0.25 mm in radius) and rain (drops

>0.25 mm in radius), as defined in Sec. 3.1.3.

• Cloud ice is composed

particles that have little or no appreciable fall

speed. These particles may be in the form of pristine crystals nucleated

directly from the vapor or water phase, or they may be tiny particles of ice

produced in some form of ice enhancement process.

• Precipitation ice refers to ice particles that have become large and heavy

enough to have a terminal fall speed

-0.3

S-I

or more. These particles

may be pristine crystals, larger fragments of particles, rimed particles, ag-

gregates, graupel, or hail. To simplify the description of the gross behavior

(3.48)

3.4 Water-Continuity Equations 95

of a cloud these particle types are sometimes grouped into categories ac-

cording to their density or fall speed. Such groupings are arbitrary; how-

ever, a commonly used scheme (employed in discussions below) is to divide

these particles into

snow,

which has lower density and falls at speeds of

-0.3-1.5

S-1

(see Figs. 3.12-3.13);

graupel,

which falls at speeds of

-1-3

S-I

(see Figs. 3.12 and 3.14); and hail, which falls at speeds of

-10-50

m s" [see Eq. (3.46)].

According to these categories of particles, the water substance in a sample of

air may be represented by eight mixing ratios:

mass of water vapor/mass of air

mass of cloud liquid water/mass of air

mass of drizzle/mass of air

mass of rainwater/mass of air

mass of cloud ice/mass of air

mass of snow/mass of air

mass of graupelltnzss of air

mass of hail/mass of air

The evolution of a cloud can be characterized in terms of fields of these mixing

ratios of water substance. The various categories are interactive. For example,

drops grow at the expense of vapor during nucleation and condensation, precipita-

tion ice grows at the expense of cloud and precipitation liquid water during riming,

rainwater is produced at the expense of precipitation ice during melting, and so

on. The many conversions of water substance from one form to another in a given

sample of air are illustrated in Fig.

3.15 for a cloud whose water substance is

divided into water vapor, cloud liquid water, rainwater, cloud ice, snow, and

graupel. Also indicated are the possible gains or losses by precipitation fallout.

The six categories in Fig.

3.15 are a subset of the eight listed above, drizzle and

hail being omitted. This six-category scheme is frequently used in cloud dy-

namics. Sometimes hail rather than graupel is used as the sixth category. To be

perfectly general, all the categories should be included.

is easy to see from Fig.

3.15 that the number of interactions to be considered would increase greatly by

expanding to a seven- or eight-category scheme.

is for this reason that the

number of categories is often limited to six or less.

3.4

Water-Continuity Equations

In cloud dynamics one is concerned with the overall development of a cloud, in

which any combination of the categories of water substance defined above and

any combination of the microphysical processes linking the various categories (as

shown in Fig.

3.15) may be present simultaneously in the context of a particular

set of air motions.

is therefore necessary to have a way to keep track of all of the

96 3 Cloud Microphysics

WATER

Evaporation

during

melting

Condensation

VAPOR I

Initiation

Deposition

Riming

CLOUD

Melting

CLOUD

<Il

WATER

I I

ICE

<Ii

"Cl

of ice bv

rain

Collection

<Ii

''8

<Ii

...

<Ii

:::I

irl

::l

:::I

L..j

Riming

I-

RAIN

I I

jE.~,SNOW

I Collection

Collection

~he<1<1ing

I GRAUPEL

Melting

lPRECIPITATION

FALLOUT

Figure 3.15 Conversions of water substance from one form to another in a bulk water-continuity

model in which there are six categories of water substance. Dashed lines indicate various interactions

leading to production of graupel: collection of cloud ice by rain freezes the ice by contact nucleation

and produces either snow or graupel; or riming of snow by collection of either cloud water or raindrops

can also lead to production of graupel. In the model, the mass of ice produced by these two processes

passes temporarily through the snow category. (Adapted from Rutledge and Hobbs,

1984. Reproduced

with permission from the American Meteorological Society.)

processes in some systematic way.

For

this purpose, the water-continuity equa-

tions (2.21) are used. They allow one to account numerically for the amount of

water contained in the form of vapor and in the form of particles of different types

and sizes throughout a cloud, as it evolves. This system of equations is referred to

as a

water-continuity model. In (2.21), each category of water is assigned a mixing

ratio

qi, where the subscript i refers to a particular category of water and the

mixing ratio is defined as the mass of water of category

i per unit mass of air. The

total water substance in a parcel of air is then given by the sum of the water

contained in each of the categories:

(3.49)

where n is the total number of categories into which the total water qT in the air

parcel has been divided.

There is no limit to the number of categories into which the total water in a

parcel of air can be divided. To begin with, we can divide the water into the eight

3.5 Explicit Water-Continuity Models 97

categories listed in (3.48).

Each

of these categories, however, can be further

subdivided. The cloud liquid water, drizzle, and rainwater categories can be sub-

divided according to drop size.

Each

drop size category can be further subdivided

according to such factors as the type of nucleus on which the drops formed,

chemical composition of the drops, etc. The snow category can be subdivided by

type of particle (columns, plates, aggregates, ice splinters, etc.), and these particle

types can be further subdivided according to particle size, density, or other fac-

tors.

The

graupel may be subdivided by size and shape (some graupel particles are

cone shaped while some are lumpy), and hail may be subdivided by particle size,

whether it is spongy or hard, and

other

factors. Obviously, calculations can be-

come highly complex if all of these subdivisions are employed. Generally the

strategy in cloud dynamics is to identify and use only the categories and subcate-

gories of water substance that are essential to keep account of in the particular

problem being considered. Hence, practically every water-continuity model em-

ployed is tailored to the problem at hand.

Once the n categories to be used in a particular water-continuity model have

been established, the source terms on the right-hand side of (2.21) must be formu-

lated to allow for all the possible interactions among the different categories of

water (e.g., the interactions shown for the six-category model in Fig.

3.15). The

source terms are formulated in terms of the seven basic cloud microphysical

mechanisms mentioned earlier (nucleation, vapor diffusion, collection, particle

breakup, fallout, ice enhancement, and melting). Two general strategies have

been

employed to formulate the source terms. In bulk models, the liquid and ice

water mixing ratios are grouped into categories according to particle type only. In

explicit models, the hydrometeors are subdivided according to size within each

particle-type grouping.

The

following sections summarize the salient features of

these two types

models.

3.5

Explicit Water-Continuity Models

3.5.1 General

Hydrometeors may be grouped into categories according to particle type as in

(3.48). In an explicit water-continuity model, one or more of these categories are

subdivided according to particle size. Since the mass of water contained in parti-

cles of different sizes is calculated, the size distributions of particles are able to

evolve naturally in

each

air parcel associated with the cloud. The only disadvan-

tage of the explicit model is computational. A large number of size categories

(-10-100)

and associated interactions have to be included to represent the size

distribution of the particles of a given category of water substance accurately.

From

a physical standpoint, the explicit method is the more direct approach. The

microphysical principles reviewed in previous sections can be applied directly to

the calculation

the size distributions within a given category of water sub-

stance.

(3.50)

(3.51)

98 3 Cloud Microphysics

3.5.2 Explicit Modeling of Warm Clouds

3.5.2.1 Drops Subdivided by Size

We consider first clouds without ice. Therefore, none of the ice categories in

(3.48)are relevant;

q/ =

= qg = qh =

The cloud and precipitation liquid water

categories are combined into a single total liquid water mixing ratio

ai.

qc + q« +

qr. The total liquid water content is thus viewed as a continuous spectrum of

drops, ranging from small, freshly nucleated cloud droplets to large raindrops.

The size distribution is represented by

N(m), where

N(m)

dm is the number of

particles per unit volume of air in mass range m to m

+ dm, in which case

qL = -

mN(m)dm

p 0

For computational purposes the size distribution is approximated by k discrete

mass categories so that

qt. is approximated by

1 k

qL "" P

mjNj(Am)j

The water continuity in the cloud may then be expressed by k + 1 equations'<

and

DN.

-N;'\'

. v +

91;

+ ;t)j +

C£j

+ m

+ il, i = 1,

...

, k

(3.52)

(3.53)

Dqv 1 k

m,(in

+ ;t)i)(Am)i

i=1

The first term on the right-hand side of (3.52) represents changes in N, associated

with air motions.

expresses the decrease (increase) in concentration associated

with the expansion (contraction) of the volume of air within which the population

of drops is located. The remainder of the terms in (3.52) represent the microphysi-

cal processes affecting the drop size distribution. They express the changes in the

concentration of drops in size range

i resulting from nucleation from the vapor

phase

in;,

vapor diffusion (condensation or evaporation) ;t);, collection

C£;,

drop

breakup

m;,

and sedimentation (or fallout)

15;.

These microphysical processes

correspond to the first five of the seven basic microphysical mechanisms listed in

Sec. 3.3. The remaining two on that list, ice enhancement and melting, do not

apply in a warm cloud.

The nucleation term

in (3.52) is calculated by first making assumptions about

the characteristics of the condensation nuclei present in the air. Then the appro-

65 This type of water-continuity model was used in early studies by Takeda (1971), Ogura and

Takahashi (1973), and Soong (1974).