Houze Robert A., Jr. Cloud Dynamics

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7.3 Entrainment

239

observed quantity provided by visual observation from research aircraft

flying

near the clouds.

For

simplicity, b was assumed to be a constant for a particular

cloud. The thermodynamic structure of the environment was provided by ra-

diosonde data taken in the vicinity of the clouds.

was found that the most

accurate cloud-top heights were calculated for an entrainment rate of approxi-

mately 0.2/

b, which corresponds to either the

jet

or the starting plume analogy.

178

Cloud-top height, however, is not an especially good test of the cloud dynamics,

since it is basically the stability of the environment that controls cloud height

rather than any assumed model properties. Extensive experimentation has shown

that one-dimensional Lagrangian models are incapable of simultaneously predict-

ing cloud-top height and liquid-water content.

179 This realization has led to the

modern notion of discontinuous, inhomogeneous entrainment, to be considered in

the next section.

7.3.3 Discontinuous, Inhomogeneous Entrainment

The concept of rapid, continuous, homogeneous, lateral entrainment discussed

above is useful for visualizing and predicting some of the grosser aspects of

cumulus and cumulonimbus clouds. However, this view of entrainment does not

hold up well under close scrutiny. As already mentioned, these models are not

capable of predicting accurate hydro meteor content and cloud height simulta-

neously.

has also been found that droplet size spectra in cumulus cannot be

calculated accurately if continuous entrainment is assumed. The calculated drop-

let spectrum becomes narrower as a continuously entraining parcel rises. How-

ever, observations suggest that the spectrum width in cumulus either increases or

does not change very much with height.P"

These observed differences between real cumulus clouds and clouds character-

ized by continuous entrainment has led to a reexamination of the basic premises of

entrainment theory (i.e., that entrainment is rapid, thorough, continuous in space

and time, and lateral). The discrepancies between real clouds and those described

in terms of continuous entrainment theory can be traced to the fact that

none of

these assumptions are well met in real convective clouds.

Attempts to resolve this dilemma have led to the concept of discontinuous,

inhomogeneous entrainment. The basic idea is that entrainment occurs not in a

continuous stream but in pulses that are intermittent in both time and space. The

turbulence in a cumulus updraft is now thought to be similar to a laboratory plume

such as that illustrated in Fig. 7.12. Turbulent entrainment is seen to occur in

strands or gulps. Data obtained on flights through cumulus are consistent with this

178 See Weinstein and MacCready (1969) and Simpson and Wiggert (1971).

179 This fact was first demonstrated in a seminal paper by Warner (1970), who made careful compar-

isons of model simulations of small nonprecipitating cumuli for which aircraft measurements of both

microphysical and kinematic variables were available.

180 This result is found in aircraft observations of cumuliform clouds obtained in a wide variety of

locations, for example, Australia (Warner, 1969aand b), the High Plains of the northern United States

(Rodi, 1978, 1981; Blyth and Latham, 1985), and Hawaii (Raga, 1989).

240 7 Cumulus Dynamics

Figure 7.12 Turbulence in a laboratory plume. (Photos courtesy of Shenqyang Shy.)

o 5 10 15 20

TIME

(seconds)

Figure 7.13 Data obtained on aircraft flights through cumulus showing intermittent regions of

homogeneous properties. Here a region of highly variable droplet concentration

NT (bracketed; units

are number per sample volume, which was variable but typically =0.3 ern') was found adjacent to a

region of uniform number, characteristic of a steady, nonentraining cloud. (From Austin

et al., 1985.

Reprinted with permission from the American Meteorological Society.)

7.3 Entrainment

241

view in showing intermittent regions of homogeneous properties. An example of

data in Fig. 7.13 shows a region of highly variable droplet content (bracketed)

adjacent to a region of uniform maximum number, characteristic of a steady,

nonentraining cloud. Thus, both laboratory and field data suggest that parcels

rising in a cloud are only occasionally subjected to mixing with blobs of entrained

air (i.e., the occurrence of mixing is

discontinuous in time). The mixing events are

characterized by an engulfed strand of environmental fluid. When the mixing

takes place, it is not instantaneous but rather slow enough that it remains localized

to the region of the engulfed air. Thus, the degree of mixing across an updraft of a

cloud is expected to be spatially quite

inhomogeneous.

This radically different view of entrainment can be illustrated mathematically

first by considering the effect of the entrainment on the drop size spectrum in

cumulus. The lateral mixing in the traditional view of entrainment has the prop-

erty that it occurs instantly

(7.50)

where

'Tm is the time scale of mixing and 'Te is the time scale of evaporation. That is,

air is immediately mixed across cloud, then drops evaporate into the uniform air

mixture. In inhomogeneous entrainment,

just

the opposite situation holds:

(7.51)

Thus, drops evaporate locally in ingested blobs, while in other areas drops are

unaffected.

This view of entrainment was developed in the context of a one-dimensional

Lagrangian cloud model.

181

The model consisting of Eqs. (7.22), (7.24), (7.25),

(7.38), (7.41), and (7.42) was first simplified by prescribing a constant updraft

motion

(we = constant). The momentum equation (7.42) is thus eliminated. The

continuous entrainment relationship (7.38) and its associated continuity equation

(7.41) are disregarded in favor of the concepts to be discussed below. The equa-

tions remaining from the former model are then the thermodynamic equation

(7.22) and the water-continuity equations (7.24) and (7.25). We may illustrate the

important features of the discontinuous, inhomogeneous model by focusing atten-

tion on the hydrometeor equations (7.25). They are posed in terms of the explicit

water-continuity scheme represented by (3.52). Only the early evolution of the

cloud-droplet spectrum is considered; thus, the stochastic collection

C£;,

breakup

Q3;, and fallout

15;

terms in (3.52) are assumed to be negligible.

is further assumed

that there is no horizontal air motion, so that with

= constant, the three-

dimensional wind divergence is zero in (3.52). With these assumptions, together

with the assumption that entrainment affects the drop size spectrum, (3.52) may

be rewritten as

+ E

i = 1,

...

, k

18\ See the innovative paper of Baker et al. (1980).

(7.52)

(7.53)

242

7 Cumulus Dynamics

where, as in (3.52), k is the number of drop size categories,

~~;

represents the

change in the number concentration

N; as a result of nucleation of drops of mass

m, to m,

+ dm,

stl;

represents the change in N; owing to vapor diffusion, while e,

represents the effect of entrainment in diluting N;. When this equation is solved

simultaneously with equations for the temperature and water vapor, the drop size

distribution on a parcel is determined as a result of interplay of the three terms on

the

right-nucleation,

vapor diffusion, and entrainment.

The vapor diffusion term

stl;

can be written as

stl.

I Ie ie

where the subscripts c and e represent contributions associated with condensation

and evaporation, respectively. Consistent with

(3.54), the condensation term is

given by

stl;e = -

[tltcondN]!m=m

(7.54)

where

mcond

is the rate of change of mass of a drop of mass m by condensation of

vapor onto the drop, as given by the right-hand side of

(3.21) under conditions of

supersaturation (8

> 0). The term stl

will be formulated in a special way to

represent the effect of entrained air.

The entrainment term in

(7.52) can be expressed as

e, = - :£N; (7.55)

where :£

(S-I)

is the entrainment rate. Three cases can then be considered.

In the first, there is no entrainment, so :£

To follow the evolution of the

initial spectrum of drops in the rising parcel of air, it is assumed that no nucleation

of new drops occurs. Thus,

is also zero. Since the parcel is always saturated in

this case,

stl

too is zero. Equation (7.52) then reduces to

DN;

stl.

(7.56)

Since small droplets grow faster than larger ones, this relation leads to narrowing

of the drop size spectrum, which is counter to observations.

In the second case, an instantaneous, continuous lateral entrainment is as-

sumed, like that considered in previous sections. Accordingly, we set

:£

;£

= constant> 0 (7.57)

The time scale of the mixing is assumed to obey

(7.50), and the value

of;£

is low

enough that the entrainment does not dry out the cloud. Consequently, the cloud

is everywhere at all times saturated, and

stl

is therefore again zero.

follows

that

I =

+:!).

-:£N

i = 1,

...

, k

Dt I

(7.58)

which is of the form of the standard entrainment equation (7.14), where

in this

case is

N; and the source terms are

and stl

To make this case comparable to

7.3 Entrainment

243

the first case, the total concentration of drops

per

unit volume of air NT is held

constant:

NT =

N, = constant

i=1

(7.59)

(7.61)

(7.60)

Since condensation does not affect the total number of drops, it follows from

(7.58)

that

the total nucleation rate must

just

offset the net entrainment rate. Since

the nucleated particles are small, they should be placed in the smallest size cate-

gory. Thus, we may write the nucleation rate for this case as

{

-±

Cj'

for i = 1

j=1

0, for i = 2,

...

, k

Calculations

N; using (7.54) and (7.60) in (7.58) show a drop size distribution

again narrowing as the parcel rises, though not as much as in the zero-entrainment

case. 182

The third case assumes

that

the entrainment is both discontinuous in time and

inhomogeneous in space. To represent the intermittency in time, it is assumed that

the rising parcel of volume

is subjected to pulses of entrainment such that

:£=

~8(t-tj)

At~

where

8 is the entrained volume, t, represents the discrete times at which the

parcel entrains a volume

environmental air, 8 is the Dirac delta function, and ar

is the time step in the numerical calculation. The entrainment term is then ob-

tained by substituting (7.61) into (7.55) to obtain

c. =

8(t

- t

.)]N

(7.62)

MY:

J I

P J

The inhomogeneity of the entrainment in space is represented by assuming that

when a volume of air is entrained, all of the drops in a localized subregion of the

parcel are evaporated. Thus, the total number of drops in each size category is

reduced in proportion to the total reduction of liquid water mass in the parcel.

Then

where

is the fraction

~ie

-~Ni

8(t

(7.63)

(7.64)

= 1i[pqvs(T) -

Peqve]

At~pqL

where the subscript e as usual indicates values characteristic of the environment

of the cloud. Nucleation is again parameterized by requiring that (7.59) be satis-

182 See Baker et al. (1980) for details.

(7.65)

for

i = 2,

...

, k

for

i = 1

244 7 Cumulus Dynamics

fied. In this case the net drop concentration is reduced by both

entrainment-

dilution and the local evaporation episodes brought on by the slow, inhomoge-

neous mixing.

particles of the smallest size category are nucleated to offset

these effects, we have

; (

e + ej

Calculations of N

using (7.53), (7.54), (7.62), (7.63), and (7.65) in (7.52) lead to the

finding that the width of the drop size spectrum remains roughly constant with

height. This result is more consistent with observations and, when it was origi-

nally carried

out,183

it was one of the first indications that discontinuous, inho-

mogeneous entrainment is a more satisfactory conceptual model of entrainment

than the earlier continuous, rapid, homogeneous entrainment models. More so-

phisticated models which predict the cloud dynamics and thermodynamics more

exactly have confirmed the results of the calculations with the simple model

described here.

184

Evidence

discontinuous, inhomogeneous entrainment in cumulus is also

found in certain analyses ofthermodynamic

data

collected during aircraft penetra-

tions of cumulus clouds. These data are exhibited in so-called Paluch diagrams'P

in which two conservative properties are plotted.

For

example, if F and 0 repre-

sent two such properties and two parcels (labeled

1 and 2) are mixed, then

(7.66)

and

C= (1- nCI +

(7.67)

where f is the fraction of a unit mass of the final mixture constituted by fluid

originally contained in parcel 2.

follows that

F(C)

is a straight line with slope

F1)/(Oz

- 0

)

(see Fig. 7.14).

For

warm cloud elements that do not contain

precipitation, it is convenient to use the total water mixing ratio

(=qv

+ qc) and

the equivalent potential temperature

(Je'

Two types

data

may be plotted in a Paluch diagram: observations from a

vertical sounding in the environment and measurements from an aircraft penetra-

tion across the cloud at a given altitude. On some occasions, the aircraft data fall

on a straight line between the cloud base and cloud top, thus indicating that the

data at flight level are mixtures, in various proportions, of cloud-base air and air

entrained downward from cloud top. This situation is illustrated schematically in

Fig.

7.15, where

data

from an imaginary aircraft penetration through a cumulus

cloud at a fixed altitude have been plotted. Also shown are the data from a balloon

sounding in the environment

the cloud. The aircraft data are shown lying on a

183 Ibid.

184

For

example, Hill and Choularton (1986).

185 Introduced by Paluch (1979).

7.3 Entrainment 245

straight line connecting the sounding data for cloud-top and cloud-base levels. The

schematic of the cloud indicates that parcels passing through flight level in this

case must be either undiluted parcels from cloud top or cloud base, parcels origi-

nating at cloud top but mixing with a parcel from cloud base at some time before

reaching flight level, or parcels originating at cloud base but mixing with a parcel

from cloud top before reaching flight level. Data from a real case exhibiting a

mixing line of this type are shown in Fig. 7.16. Such cases indicate that, at least on

some occasions, vertical entrainment and mixing can be a dominant process. This

picture is a clear departure from the traditional notions of lateral entrainment.

On other occasions, the aircraft data fall close to a straight line between the

cloud-base point and a particular level below cloud top and near flight level. The

situation resulting in such a mixing line is illustrated by the schematic in Fig. 7.17,

and a real example of this type of mixing line is shown in Fig. 7.18. As shown in

the schematic, parcels mix in various fractions, with entrained environmental air

from near flight level, lose their buoyancy, and come to rest near this level. Of

course, some parcels, by chance, come to rest at other levels. Some parcels never

encounter any entrained air.

is these undiluted parcels that determine the height

of the cloud top. This conceptual model thus resolves one of the major dilemmas

of the traditional continuous-entrainment

ideas-it

allows for the average liquid

water content at a particular level in a cloud to be far from undiluted even though

the cloud top

is high, since the cloud-top height is determined by the maximal

ascent

only the least diluted parcels.

The model in Fig. 7.17 is essentially a modification of the old bubble model

discussed in Sec. 7.3.2. Just as in the bubble model, the parcels which form the

cloud towers defining the very top of the cumulus arrive there without entraining

environmental air. The new model, however, differs from the old one in that those

parcels that do not ascend to cloud top have their upward journey truncated by

entrainment rather than by erosion.

Another type of observation obtained by aircraft is illustrated schematically in

Fig. 7.19 and by a real example in Fig. 7.20.

In this case, the data show a lot of

scatter, indicating that the parcels at flight level are mixtures of cloud-base air

with environmental air entrained from a multiplicity of altitudes. Again, cloud top

would be determined by the rise of the few undiluted or nearly undiluted parcels.

Of course, the possibility exists that the parcels could undergo more than one

entrainment

event-as

used in the above analysis of the drop size spectrum.

However, it has been shown that a good reproduction of observations can be

obtained by considering a cumulus cloud to consist of a collection of parcels that

each undergo mixing with entrained air at one and only one level.

186 As illustrated

in Fig. 7.21, equal parcels are considered to be released from cloud base to several

discrete levels above. Upon reaching its designated level, each parcel is split into

several subparcels, each mixing with a different fraction of environmental air.

Each subparcel then rises or sinks to its level of neutral buoyancy, where it is

detrained to the environment. This simple model of discontinuous, inhomoge-

186 See Raymond and Blyth (1986).

Figure 7.14

--------------------:~··r

. ,

•••

«4 :

••••••

-----~

G-

Balloon

"Mixingline"

Figure 7.15

Figure 7.16

315

330

Balloon

Figure 7.17

+PB

++-Aircraft

+ Data

usually

within

±1 kmofpF

----f"P--,-

"Lucky"

(undiluted)

parcel

7.4 Vorticity

247

neous entrainment produced vertical profiles of detrainment similar to detrain-

ment profiles inferred from aircraft observations of continental cumulus.

The above discussion illustrates that discontinuous, inhomogeneous entrain-

ment accounts qualitatively for several properties of cumulus not adequately

accounted for by lateral continuous entrainment with instantaneous and homoge-

neous mixing across the cloud. The drop size spectrum, total water content, and

cloud height are all better accounted for by discontinuous, inhomogeneous en-

trainment. Considerable physical insight into the process of entrainment is thus

gained. Much work remains, however, to make the understanding of this process

quantitatively more tractable.

7.4 Vorticity

7.4.1 General Considerations

One of the important and fascinating aspects of convective clouds is that they

develop internal vortical motions

We have already seen that buoyant parcels tend

to overturn in the manner of a Hill's vortex (Fig. 7.9) as they

ascend-like

a rising

smoke ring. A ring of horizontal vorticity is generated by the symmetric horizontal

gradient of buoyancy centered on the buoyant parcel. We will see that the down-

wardly accelerating, negatively buoyant parcels of downdrafts are also bounded

by a ring of vortical motions and that this ring of vorticity is a particularly danger-

ous part of intense downbursts (see Sec. 8.10). As downdraft air spreads out along

the ground, the leading edge of the colder air continues to be characterized by the

horizontal buoyancy gradient, which maintains a roll circulation just behind the

gust front (Sec. 8.9). When horizontal vortex tubes associated with convective

Figure 7.14 Paluch diagram. F and Gare conservative properties. Straight line

F(G)

represents

mixing of parcels of air in varying proportions.

Figure 7.15 Paluch diagram of the total water mixing ratio

q, +

and "equivalent potential

temperature"

(li,) for an idealized aircraft flight in a cumulus cloud characterized by mixing of air

parcels from cloud base and cloud top and balloon ascent in the environment.

Figure 7.16 Paluch diagram showing aircraft measurements (small crosses) and environmental

sounding data (heavy solid line) for a real cumulus cloud characterized by mixing of air parcels from

cloud base and cloud top. The aircraft data were obtained at the 488-mb level. Thin solid line is

theoretical mixing line for parcels of air from cloud base (heavy cross) and cloud top (375 mb). (From

Jensen, 1985.)

Figure 7.17 Paluch diagram of the total water mixing ratio

qr(=

q, + qc) and "equivalent potential

temperature"

(li,) for an idealized aircraft flight in a cumulus cloud in which air parcels rising from

cloud base to flight level have mixed in various fractions with air entrained from near flight level and

balloon ascent in the environment. Parcels entrain environmental air from near flight level, lose their

buoyancy, and come to rest near this level. Cloud top is determined by the maximum height achieved

by the

"lucky"

parcels, which suffer no entrainment and may rise undiluted to their level of zero

buoyancy.

325

320315

OL-

...L-

.......L

-----l

310

8r-------,--------,-------=------,

Figure 7.18

Balloon

"Lucky" (undiluted) parcel

Figure 7.19

10r-------.------,---------,

levelj

'Lucky' (undiluted) parcel

11--4---4--1-

level i

Figure 7.20

Figure 7.21