Houze Robert A., Jr. Cloud Dynamics

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5.3 Cirriform Clouds 179

occurs in a dry adiabatic layer bounded by stable layers above and below. Vertical

air motions in the head are

1 m s

-1.

When the wind shear is in the sense shown in

Fig. 5.29, the head contains a cloud-free hole. Ice particles generated and carried

up to the top of the updraft are advected over the top of the hole. The fallstreak to

the left of the hole contains downdraft associated with evaporation and/or drag of

the falling ice particles. Evaporation of the particles continues as they fall through

the stable layer below the base of the head. When the wind shear in the layer of the

head is of the opposite sense, the hole and fallstreak are to the right of the updraft

column (Fig. 5.30a). When there is no wind shear in the layer containing the head,

the ice particles occur within the updraft, and downdraft occurs around the pe-

riphery

ofthe

updraft (Fig. 5.30b). Two ideas on how new cirrus uncinus elements

could be generated by the older elements are indicated (Fig. 5.31). Either cooling

by evaporation causes turbulence in the stable layer, which in turn perturbs the

unstable layer above (Fig. 5.31a); or, because of the shear in the environment,

horizontal momentum transported downward in the downdraft produces conver-

gence at the bottom of the unstable layer on one side or the other of the downdraft

(Fig. 5.31b and c). In either case, a new cell could be triggered in the vicinity

ofthe

older uncinus element.

5.3.3 Ice-Cloud Outflow from Cumulonimbus

As shown in Fig. 1.5, a layer of ice cloud often emanates from the upper levels of a

cumulonimbus cloud. This cloud layer may take the form of cirrus spissatus, if

thinner and patchier, or cirrostratus cumulonimbogenitus, if thicker and more

widespread. If the base of the cloud layer is in the middle etage, it may take the

form of altostratus cumulonimbogenitus.

For

simplicity, we will refer to these

phenomena as ice-cloud outflows from cumulonimbus. The dynamics of these

outflows can be analyzed theoretically by considering the air intruding into the

environment from the cumulonimbus to be laden with ice, uniformly buoyant,

isotropically turbulent, and flowing into a stably stratified environment (Fig.

5.32).147

This outflow is envisioned to undergo a two-stage process as it moves

downstream.

In the first stage, the outflow undergoes a collapse similar to that of the wake of

a body moving through a stratified fluid (e.g., a submarine moving through the

oceanic thermocline). During the collapse, the wake is flattened and spread later-

ally by the environment (Fig. 5.33). The upper part of the outflow, being denser

than the environmental fluid, subsides, while the lower part of the outflow, being

lighter than the ambient air, rises. At the same time that this external collapse is

taking place, an internal collapse occurs, in which the turbulence in the plume is

147 The treatment of this problem follows Lilly (1988), who, 20 years after his landmark paper (Lilly,

1968), which established a theoretical framework for the radiatively driven boundary-layer stratus,

applied similar concepts to the turbulent layer of ice cloud emanating from the top of a cumulonimbus.

This section is based on his development of the problem.

180 5 Shallow-Layer Clouds

Figure5.32 Idealized outflow of cirriform cloud from cumulonimbus. Vertical lines AA and BB

indicate positions of cross sections in Fig. 5.33. (Adapted from Lilly, 1988. Reproduced with

permission from the American Meteorological Society.)

'-..

--/

Figure5.33 Collapse of a cumulonimbus anvil. Plume cross sections are shown for positions AA

and BB in Fig. 5.32. The horizontal lines indicate potential temperature surfaces in the environment;

the arrows indicate the motion field induced by the buoyancy difference between the outflow plume

and the environment. (From Lilly, 1988. Reprinted with permission from the American Meteorological

Society.)

slowly dissipated, its energy being transferred to waves and larger-scale two-

dimensional turbulence.l"

In the second stage of development of the outflow plume, the radiative destabi-

lization of the outflow layer becomes important, and the ice cloud is maintained by

a process similar to that of low-level mixed-layer stratus. The energy to maintain

148 Lilly (1988) considered the external collapse quantitatively by assuming it consists of the con-

version of the available potential energy to the kinetic energy of the spreading motion. In a simplified

example, he found the plume radius tripled in just under 10 min and suggested that the maximum

aspect ratio is achieved after about 20 min.

(5.41)

5.3 Cirriform Clouds

181

the turbulence is provided by radiative destabilization of a layer of

cirrus-as

indicated for example by the calculations shown in Fig. 5.27. The major difference

from the case of boundary-layer stratus is that the unstable layer is bounded by

stable layers both above

and below.

Radiation is the only significant heat source in the ice-cloud outflow layer since

latent heat release is negligible at these altitudes. The problem thus simplifies to

that of a dry mixed layer, in which case the mean potential temperature

()

constant with respect to z, and

)

- =--a w'o' + m =constant in z

dt z

which is the same as (5.21), except that the difference between ()e and

()

is now

ignored. The radiative flux divergence is assumed to be a linear function

- = B

(z-z

)-A

0 m 0

(5.42)

where

and B; are positive constants and Zm is the height of the middle of the

cloud layer.

149 Integration from cloud base Zb to cloud top ZI shows that the value

of the constant in (5.41) is

where

(w'O') - (w'O')

-=A

_ I b

dt 0

(5.43)

(5.44)

The values of the fluxes at ZI and Zb are both considered to be determined by the

rate of turbulent entrainment across the boundary of the mixed layer. Accord-

ingly, we let the fluxes at the top and bottom of the cloud layer be

(w'Ol

= -We,(dO), (5.45)

and

(W'O')b

Web(dO)b

(5.46)

respectively, where

(d()1

and (d()b are the change in

()

across the top and bottom

of the layer and

WeI

and

Web

are entrainment velocities across the top and bottom.

The changes in

()

across the top and bottom of the ice-cloud layer are defined as

positive numbers:

(5.47)

149 Lilly (1988) based this assumption on the calculations of Ackerman et al. (1988), who considered

radiative transfer theory in the context of cirrus near the equatorial tropopause.

182 5 Shallow-Layer Clouds

(,10), =

-0

+ 0

z = z,

(5.48)

The sounding of potential temperature in the environment ()O, including the spe-

cific values

()Or

and

BOb

at the top and bottom of the ice-cloud mixed layer, is

assumed to be known.

is readily seen that the expressions (5.45) and (5.46) are analogous to (5.27),

which represents the mixing across the top of the boundary layer in the case of

stratus and stratocumulus. The ice-cloud layer represented here differs from the

boundary-layer cloud, however, in having entrainment across

both the top and

bottom of the cloud layer.

In the case of boundary-layer cloud, the entrainment velocity is inferred from

the buoyant production of kinetic energy

integrated over the depth of the mixed

layer. Here we follow a similar procedure, in which (5.40) is simplified to

gIzl

\~)

W'7?

o Zb

(5.49)

The integral in this expression can be evaluated since

al1R/az

is the known linear

function of height (5.42), which, together with the assumption that

(W'()')b is

known, implies that

w'()' is a known quadratic function of z, which is obtained by

integrating (5.41) from

Zb to z:

When this expression is substituted into (5.49), we obtain

g2H[

(dOJ

B H

]

(~\

-=-

(w'O')

+_0_

/ B b 0 dt 3

(5.50)

(5.51)

To complete the mathematical description of the ice-cloud layer, some assump-

tion must be made to relate the net buoyancy generation of kinetic energy

(~)

the entrainment velocities at the top and bottom of the cloud layer. Just as in the

case of the stratus-topped mixed layer, the entrainment velocity at the top of the

mixed layer must be related to

(~)

[as discussed in relation to Eq. (5.40)]. To

establish a relationship for the ice-cloud outflow layer, the buoyancy generation of

kinetic energy given by (5.49) is subdivided into positive and negative contribu-

tions

(5.52)

5.3 Cirriform Clouds

183

where

(5.53)

(5.54)

and

(.N')

= -

~fZt

(w'lr),N'

8 Zb

Symbol <§ indicates generation of eddy kinetic energy [(w'O') > 0] while .N'indi-

cates consumption (i.e., negative generation) of eddy kinetic energy

[(w'O')

< 0].

This decomposition is an approach sometimes used in considering a dry (non-

cloud-topped) mixed layer

just

above the ground. The negative fluxes are associ-

ated with entrainment and are envisaged as being most strongly felt at the top of

the mixed layer, across which all the engulfed plumes must pass, while near the

surface only a slight effect is felt since only the most penetrative plumes reach

down to these levels (Fig. 5.34a).

is typically assumed that

(w'O')~

and

(w'O')x

both vary linearly, with

(w'

O')~

decreasing from w' 0' at the bottom of the layer to

zero at the top and

(w'

0' )x decreasing from 0 at the bottom to w' 0' at the top (Fig.

5.34b). The net flux then decreases linearly from a positive value at the surface to

a negative value at the top of the turbulent layer (Fig. 5.34c).

The procedure followed for the case of ice-cloud outflow is an extension of this

dry mixed-layer problem. The effect of the

cloud-top entrainment (wet) is assumed

to decrease linearly to zero at cloud base while the effect of the

cloud-base

entrainment (Web) is assumed to decrease linearly to zero at cloud top (Fig. 5.35).

Then

(5.55)

In the case of the dry mixed planetary boundary layer, it is assumed that

(5.56)

(5.57)

where

a is a positive constant.150 Extending this assumption to the ice-cloud

outflow layer, we can combine (5.43), (5.51), (5.52), (5.55), and (5.56) to obtain

(

- )

(-)

-1)2H

-------"-

b t 3

and

(5.58)

150 This approach is attributed to Stage and Businger (l98Ia,b), who based it on empirical studies of

the dry boundary layer. Lilly (1988) used a value of a = 0.8 for the cumulonimbus outflow calculation.

184

5 Shallow-Layer Clouds

(a)

entrainment

turbulence

(b)

(w'01t

(c)

~W'O')o

O'w'

w'o'

w'8'

Figure 5.34 Schematic of a dry (non-cloud-topped) mixed layer just above the ground.

(W'O)t

(W'e)b

w'(J,---..

Figure 5.35 Eddy fluxes associated with cloud-top entrainment and cloud-base entrainment in

ice-cloud outflow from cumulonimbus. The flux associated with cloud-top entrainment is assumed to

have a maximum magnitude

(w'O'),

at z. and decrease to zero at Zb. The flux associated with cloud-top

entrainment is assumed to have a maximum magnitude

(w'

0' h at Zb and decrease to zero at

Z,.

which show that the sum of the boundary entrainment fluxes

[(w'(j'h

(w'(j')rl

and the net integrated average eddy heat flux

(\?Jil)

are both driven by the vertical

gradient of radiative heating

)

in the ice-cloud layer.

Since (5.57) gives only the sum of the values of the boundary fluxes, a way is

needed to relate

(w'(j'h

and

(w'(j')r

in order to close the problem.

For

simplicity,

the two fluxes are arbitrarily made proportional to each other, such that

(5.59)

(5.60)

(5.62)

5.3 Cirriform Clouds 185

where a

is simply an adjustable parameter. Then (5.43) with substitution from

(5.57) and (5.59) becomes

(

)[(1-a)B

oH]

- = A +

1-2a

dt 0 0 3

This equation predicts the mean potential temperature in the ice-cloud mixed

layer. The parameter

represents the net radiative heating, which if positive

causes the layer to warm. The jump in

B at the top of the cloud layer decreases,

while that at the bottom of the layer increases. The second term is proportional to

which represents the vertical gradient of the radiative heating, which drives

the turbulence and entrainment. If the adjustable parameter

a" > 0.5, then the

temperature rise in the mixed layer is less than that provided by the net radiation

because more potentially cool air is entrained at the bottom than warm air at the

top of the cloud. The opposite situation applies if

a" < 0.5.

Substitution of (5.57) and (5.59) into (5.45) and (5.46) gives

(1 - a)B

= (5.61)

3(d8)b

and

dz,

2(1-

ao)(l-

a)B

-=-----'--------"--"------"----

3(d8)t

which, together with (5.60), form a closed set of equations for calculating ii,

z»,

and Zt as functions of time.

For

values of A

= 0 (no net radiative heating) and a =

0.5 (equal effect of entrainment at cloud top and cloud base), the change in the

depth of the mixed layer is found to be

slow-rather

similar to boundary-layer

stratus. If

a is again 0.5 but A

> 0 (i.e., radiation is warming the cloud layer in the

mean), then

ii increases until the jump (dB)t at cloud top becomes small. Then,

according to (5.62), the rate of increase in the cloud-layer depth becomes large."!

5.3.4 Cirriform Cloud in a Thin Layer Apart from a

Generating Source

The final type of cirriform cloud we will look at is that which occurs in a thin

unstable layer aloft, apart from a generating source such as a cumulonimbus

cloud. The dynamics of this type of cirriform cloud have been examined via a

numerical model.

152 The model applies in the two-dimensional

(x-z)

spatial do-

main indicated in Fig. 5.36. The domain is a shallow layer, in which the two-

dimensional Boussinesq vorticity equation (2.61) may be used to describe the air

motions, and nothing varies in the y-direction. If turbulent mixing in the layer is

lSI See Lilly (1988) for further discussion of these results.

152 This numerical model is described in a pair of papers by Starr and Cox (l985a,b). The present

discussion is based on that work.

186 5 Shallow-Layer Clouds

mean

current~

Random

buoyant

elements

I,J...<::::-

---.y

Figure 5.36 Design of a model used to study cirriform cloud that forms in a thin layer. (Adapted

from Starr and Cox, 1985a. Reproduced with permission from the American Meteorological Society.)

represented by K-theory (Sec. 2.10.1), with a constant mixing coefficient KI;' then

the mean-variable form of (2.61) may be written as

- - - -

-u~x

W~z

- B

+ Kg'\'

(5.63)

where, for this two-dimensional situation, '\'2 = a

2/ax

+ a

2/az

•

The overbars are

averages over one finite grid element. Thus, the barred quantities indicate the

model-resolvable scale of motion. The mean vertical velocity is assumed to have

the form

(5.64)

where WB is a background vertical motion, which is constant throughout the layer,

and

is the deviation from the background value. The mass-continuity equation

for the mean flow [obtained by averaging (2.55)] is satisfied by a stream function

defined such that

U = lfI

' W =

-lfI

The vorticity is then given by

= ,\,2lf1

Substituting (5.64) and (5.65) into (5.63), we obtain

(5.65)

(5.66)

(5.67)

5.3 Cirriform Clouds

187

If (5.64) and (5.65) are used in the advection terms of the Boussinesq version of

the mean-variable form of the First Law of Thermodynamics (2.78) and the eddy-

flux convergence of

()

is represented by K-theory with a constant mixing coeffi-

cient

Ks,

then (2.78) becomes

- _ - - - LsC

--lfIzOx+lfIxOz-wBOz+-----=---+K(jV'

° (5.68)

cpIl

where mis the net radiative heat flux and

LsC

cpn is the release of latent heat as

vapor is deposited on ice. The last three terms are the same as in (5.4), except that

the latent heat of sublimation

appears in place of the latent heat of vaporization

L and the deposition rate C« appears instead of the condensation rate C. To

complete the model, the Boussinesq versions of the mean-variable water-continu-

ity equations (2.81) are written in a form analogous to (5.6) and (5.11) as

(5.69)

(5.70)

(5.71)

and

qHt =

-lfIzqH

+ lfIxqH

- w

(VHqH) + K

HV'2

where

is the mixing ratio of hydrometeor (i.e., ice) mass, K" and K

are

constant mixing coefficients, and

VH is taken to be the mass-weighted mean-

particle fall speed,

(D)m

(D)Vk(D)dD

A k 0

-"------------

fNk(D)

ffi

(D)dD

k 0

where k indicates a particular type (habit) of crystal, D represents the diameter of

the crystal,

Nk(D)

the size distribution function for crystal type k, m, the mass of a

crystal of type

k and diameter D, and V

the fall speed of crystal type k as a

function of size. The expression in (5.71) is like that in (3.69), except that the

falling particles are ice particles of type

k instead of raindrops. Data on ice crystals

in cirriform clouds can be substituted in (5.71) to obtain an empirical value of

Similarly the measured

Nk(D)

may be substituted into an integral like that on the

right-hand side of (4.9) to obtain a value of

qH. Repeated measurements of

Nk(D)

can then be used to obtain an empirical correlation between VHand

(Fig. 5.37).

The deposition/sublimation

Cd is calculated by adjusting the ambient humidity to

values consistent with supersaturation with respect to ice. Radiative transfer

equations are used to calculate the radiative heating term in (5.68). An absorption

coefficient appropriate for cirrus is used.

If (5.66) is substituted into (5.67) to eliminate

t,then (5.67)-(5.70) can be solved

for

t/J,

ii, a.. and

iiH

as functions of time with appropriate boundary and initial

conditions. The two-dimensional wind field can be diagnosed from (5.65). Most

important of the boundary and initial conditions is that a O.5-km-thick cloud layer

is assumed to exist initially (Fig. 5.36). The layer is assumed to be saturated, with

188 5 ShaDow-Layer Clouds

1.5

1.0

<.;:

0.5

Figure 5.37 Relationship between mass-weighted particle fall speed VH and total ice water mixing

ratio

qH used in cirriform cloud model. (From Starr and Cox, 1985a. Reprinted with permission from

the American Meteorological Society.)

6.5

--10

min

-,-20

-----30

-40

10 20 30

pqH(mg m-

)

Figure 5.38 Results of model of cirriform cloud formation in a layer of air in which the background

vertical motion

WB = 2 em

S-I.

Vertical profiles of horizontally averaged ice water content

iiH

at various

times. (From Starr and Cox, 1985a. Reprinted with permission from the American Meteorological

Society.)

a constant ()e, and to contain random pulses of slightly buoyant air, which turn

into convective cells as the cloud layer begins to evolve. Thus, the thin cirriform

cloud layer that develops is not horizontally uniform but contains resolvable-scale

convective

elements-possibly

like cirrus uncinus (Fig. l.15c), cirrus floccus

(Fig. l.15b), or cells and rolls of cirrocumulus (Fig.

l.15a)-in

addition to the

small, subgrid-scale turbulence parameterized through the eddy mixing terms in

(5.67)-(5.70).