Houze Robert A., Jr. Cloud Dynamics

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5.1 Fog and Stratus in a Boundary Layer Cooled from Below

149

decreasing to l20e at a height of 0.1 m. The atmosphere is assumed to be isother-

mal at l20e up to a height of 0.5 km, decreasing to 11°e at 1 km and to - 3°e at

3 km. Initial soil temperature is assumed to increase to

15Se

at a depth of

-5

and then to decrease gradually to 15°e at a depth of

-1

m. Relative humidity at the

initial time is assumed to be 80% up to a height of 0.5 km, then to decrease linearly

to 60% at 3 km. The wind at the top of the domain (3-km altitude) is assumed to be

geostrophic and 2 m s

Atmospheric aerosol particles activated as a function of

relative humidity to form embryo droplets, and appropriate schemes for comput-

ing solar and infrared radiation are employed. Explicit equations are used for

predicting the soil temperature and liquid water content down to a depth of 1 m

below the ground.

From 1800to 2000LST in the example, the radiative cooling is seen to produce

a temperature inversion rapidly. Thus, the height of the boundary layer drops

quickly to near the earth's surface. The geostrophic wind layer, lying just above

the boundary layer, thus drops almost to the ground. After 2000 LST, the rate of

surface cooling decreases considerably, but the inversion continues to increase in

intensity.

begins to rise above the surface as turbulent mixing becomes stronger

(as indicated by the increase in exchange coefficient shown in Fig. 5.3d). The

increased mixing is related to the intensifying shear across the inversion layer.

The mixing in the stable layer shifts some of the surface cooling to higher levels,

just

below the inversion. By 2300 LST cooling below the inversion layer is

suffi-

ciently strong that the fog begins to form (as indicated by the appearance of liquid

water in Fig. 5.3b). At this time, the strong effect of the fog on the radiation and

thus on the generation of turbulence via cooling aloft is seen as the inversion layer

suddenly jumps

-10

m upward (Fig. 5.3a). From 0000 to 0200 LST, the inversion

layer continues to rise and the strongest cooling begins to occur at the top of the

fog layer, rather than at the surface (Fig. 5.3c). As the inversion layer continues to

rise, the turbulence continues to increase as the eddies are freer to move about in

the deepening boundary layer. The turbulence continues to transfer heat down-

ward, and by

just

after 0200 LST the minimum temperature lifts off the ground to a

position

just

below the inversion layer bounding the top of the fog. This point in

time marks the beginning of a series of oscillations in the fog, which are probably

the model's attempt to produce a convective mixed layer in response to the

radiative cooling at the top of the fog layer. The net effect of the mixing is to bring

liquid water up to higher levels, and the fog layer deepens through the night. Just

after 0600 LST, the upper portion of the fog decouples from the lower portion and

the oscillations cease. The upper part thus becomes a stratus layer, and only a

weak layer of fog is left below. As we will see in the next subsection, it is typical

for this layering to occur in fog and stratus in the stable boundary layer. Green-

house warming below the base of the upper stratus is apparently too great to allow

condensation in the layer between the stratus and the weaker fog below. At

sunrise, the fog layer dissipates to a thin ground fog, which, after 0750 LST,

somewhat surprisingly increases in depth, as dew and soil moisture are evapo-

rated. After 1030, as the insolation intensifies, only high relative humidity and

haze persist.

150 5 Shallow-Layer Clouds

5.1.4 Arctic Stratus

In the simulation

radiation fog development illustrated in Fig. 5.3, the fog

deepens as a result of the turbulent mixing, and the radiative cooling at the top of

the fog layer encourages stronger mixing, which further deepens the layer until

finally the main cloud layer separates from the lower layer of weak fog and

becomes a layer of stratus aloft. But soon after this occurs the whole process is

terminated by the rising of the sun, rapid solar heating of the boundary layer, and

dissolution

ofthe

cloud. Had there been no diurnal cycle and had the surface layer

been maintained at a constant temperature, the stratus layer and the lower fog

layer might have reached an equilibrium and persisted indefinitely. This latter,

hypothetical case is similar to what actually occurs over the Arctic Basin during

the summer months, when the melting pack ice comprises a widespread uniform

lower boundary (of temperature O°e) over which warm air from the south moves.

As the warmer air is chilled, condensation can occur in the lower boundary layer.

Stratus cloud evolves,

whichis

never dissolved because the sun remains at a

constant position, very low in the sky. As a result of this process, the Arctic Basin

is noted for its ever-present widespread summertime stratus. During the summer,

the monthly mean cloud amount is

over

80%, and low clouds are the dominant

type (Fig. 5.4).

The formation of arctic stratus has been simulated numerically from the same

basic equations [(5.2)-(5.7)] used to describe the nighttime radiation fog in Sec.

5.1.3.

121

Somewhat different parameterizations for the radiation, turbulence, and

condensation were used in the equations. The basic idea of the calculation is the

same, however, except that in this case there is a nonzero basic-state (geo-

strophic) horizontal flow v

in which the air is envisaged to be moving in the

x-direction

over

a lower surface of ice. The basic current is assumed to be con-

stant with respect to height in the layer of interest. The form of (5.2)-(5.7) then

remains unchanged, as long as the time derivative is simply considered to be

calculated in a coordinate system moving with the basic current. The equations

are integrated in time as before, but under rather simpler conditions. There is no

need for a complicated treatment

the lower boundary, which is considered to be

melting ice. The temperature is simply maintained at

ooe

at the surface. There is

also no diurnal variation of solar radiation; the sun is held at a fixed position low in

the sky (zenith angle 74°).

The

air moving

over

the surface is initially 4°C at the

surface and has a potentially stable lapse rate characterized by

aOelaz

= 1°C km"

and 90% relative humidity. To correspond to conditions

over

the polar region, a

mean downward vertical velocity (of the order of a few tenths of a millimeter

per

second) is assumed, and a vertical advection term is added to the thermodynamic

equation.

is found that this term is negligible compared to the radiative and

turbulent terms,

but

its inclusion is nonetheless illustrative of how stratiform

clouds need not depend on lifting for their existence. Indeed, they

can

thrive in an

environment of mean

downward motion.

121 For the detailed account of how the equations are set up, see the original paper by Herman and

Goody (1976), on which this discussion is based.

5.1 Fog and Stratus in a Boundary Layer Cooled from Below

151

M9an monthly frequencies of:

L--L

Low clouds

M-

-M

Middleclouds

H-

- H High clouds

0--0

"Clear" skies

*-- --

I/o:

Precipitation

100

(a)

(/)

Figure 5.4

West Eurasian Arctic

-_._-

East Eurasian Arctic

-----

Canadian Arctic

---

Central Arctic

ASOND

100

(b)

F M A M J A

SON

2000

o-----.----..---.--..,.....--r--....-......,

Solar

-IR

Solar-Diffusive

1600

g1200

w 800

:I:

400

Figure 5.5

2 4

TIME (days)

1600

Figure 5.6

IR -Cooling

2 4

TIME(days)

Solar·IR

Solar

·IR

Figure 5.4 Monthly mean cloud conditions over the Arctic Basin. (a) Total cloud amount.

(b) Frequency of various types of cloud conditions and precipitation. (From Huschke,

1969.)

Figure 5.5 Calculated development of arctic stratus. Liquid water mixing ratio as a function of

height and time in isopleths of g kg-I. (From Herman and Goody,

1976. Reprinted with permission

from the American Meteorological Society.)

Figure 5.6 The radiative and turbulent processes producing the structure seen in Fig. 5.5. (From

Herman and Goody,

1976. Reprinted with permission from the American Meteorological Society.)

Results of the calculations are shown in Fig. 5.5. The stratus, which develops

and reaches a steady state after about a week, has a decidedly layered structure,

characterized by a more dense upper layer and a more tenuous lower layer. The

radiative and turbulent processes producing this structure are illustrated in Fig.

5.6. The terms "diffusive" and "convective" in this diagram refer to the diver-

152 5 Shallow-Layer Clouds

gence of the turbulent flux under stable and unstable conditions, respectively. As

the warm air moves over the ice, the boundary layer is rapidly cooled by contact

with the surface, and turbulent mixing rapidly spreads the cooling upward. In

slightly more than a day's time, the diffusive cooling together with infrared cool-

ing leads to condensation. Once the cloud forms, the radiative regime is greatly

altered by the absorptive properties of the droplets. The upper, more dense cloud

layer becomes unstable by long-wave loss from the top of the cloud, and a radia-

tive-convective balance is established in the upper cloud layer. Convective

warming is approximately equal to radiative cooling at the top, while the convec-

tive cooling balances the heating by solar absorption in the interior. The upper

cloud retains this character as the steady-state configuration is approached toward

the end of the week. The lower, more tenuous cloud layer is characterized by a

general balance between the solar heating penetrating into

from above and the

diffusive cooling from below. The clear layer separating the upper and lower

clouds is in a state of purely radiative equilibrium. Intense heating produced by a

greenhouse mechanism precludes condensation in this layer.

122 Another such radi-

ative equilibrium layer exists

just

below the base of the lower cloud.

The fact that no fog occurs

just

above the surface during the first hour is a

feature of the turbulence parameterization used. When a different, weaker eddy

mixing was adopted, the result in Fig. 5.7 was obtained. In this case, fog formed

first at the surface and then lifted, as in the case of radiation fog (Fig. 5.3). In this

case the steady state is reached and maintained since there is no diurnal cycle to

terminate the cloud by the sun rising higher in the sky. When no turbulent fluxes

were allowed at all, the lower cloud layer never left the surface, but the cloud

nonetheless split into two layers after 3.5 days. These results suggest that in the

Arctic, generally persistent layered stratus should prevail, with fog occasionally

appearing when the mixing is especially weak. This inference from the calcula-

tions is borne out by experience.

For

example, observers aboard a drifting ice

station in the Arctic Basin during the International Geophysical Year observed

sky coverage to be over eight-tenths on average throughout the summer months of

1957

and

1958

(Fig. 5.8). These clouds have been described by Professor N.

Untersteiner, who was present on the ice station for 366 days, as almost continu-

ously present low stratus, of a

"boring"

character, interrupted only occasionally

by breaks in the cloud cover and occasional periods of fog.

5.2 Stratus, Stratocumulus, and Small Cumulus in a

Boundary Layer Heated from Below

5.2.1 General Considerations

Satellite pictures show that great sheets of stratus and stratocumulus are almost

always present over the oceans at subtropical latitudes to the west of continents in

122 See Appendix B of Herman and Goody

(1976)

for a theoretical treatment of the greenhouse

mechanism.

5.2 Stratus, Stratocumulus, and Small Cumulus in Boundary Layer Heated from Below

153

1400

1200

1000

I-

800

iii

600

400

200

TIME (days)

Figure 5.7 Calculated development of arctic stratus. Same as Fig. 5.5 except a weaker diffusivity

is assumed. (From Herman and Goody, 1976. Reprinted with permission from the American

Meteorological Society.)

1957

r---.

CTEJll'£RATURE

VAPOR

PRESSURE.

\/I

.....

nth

1111

CLOUDAMOUNT,NI10

11\1'0

'1/

IV'

JUN JUL AUG SEP OC NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT

-1

-2

-3

Figure 5.8 Observations taken aboard a drifting ice station in the Arctic Basin during the

International Geophysical Year. (a) Location of the station. (b) Observations of temperature (II-day

running means), vapor pressure (5-day means), and cloudiness (5-day means). (From Untersteiner,

1961.)

association with subtropical anticyclones (Fig.

la). In middle and high lati-

tudes, large regions of stratocumulus occur where cold air streams offshore across

the coastlines of cold continents or ice sheets and forms a layer oflow cloud as the

cold air suddenly comes in contact with the warm underlying ocean (Fig. 1.1lb).

These two situations are both examples of a cloud-topped boundary layer heated

from below. This type of shallow-layer cloud formation stands in stark contrast to

the radiation fog and arctic stratus considered in Sec. 5.1, which form in a bound-

ary layer cooled from below. As the boundary layer is heated from below, it can

become conditionally unstable, and sometimes the clouds take the form of a broad

field of small cumulus clouds rather than stratus or stratocumulus. In the present

discussion, we do not make a strong distinction regarding whether the shallow

layer of cloud is entirely stratus, stratocumulus, or cumuliform, as the form of a

particular cloud layer is probably a matter of the relative strength of the heating

and vigor of the mixing in the particular boundary layer in which the layer of cloud

154 5 Shallow-Layer Clouds

forms. The salient dynamics of the boundary layer leading to the formation and

maintenance of the cloud layer have much in common from case to case.

The cloud-topped boundary layer is such a widespread phenomenon over the

oceans that it is of great importance to the global climate. The maps shown in Fig.

5.9 illustrate this point. In summer, the maxima of low-cloud amount in the

Atlantic and Pacific Oceans near 30

and to°S to the west of the continents show

the major areas of low cloud layers associated with the subtropical anticyclones.

The maxima on the east coasts of North America and Asia and south of Greenland

in the boreal winter and north of Antarctica in the austral winter indicate the

regions of oceanic stratocumulus forming in response to outbreaks of cold air from

0°

300S

600S

120

ISooE

180°

ISOoW

120

w 60

0°

300S

600S

120

ISooE

180°

ISOoW

120

w 60

0°

Figure 5.9 Average cloud cover by stratus, stratocumulus, and fog. Data are from standard

surface observations and are expressed in percent of sky covered. The years 1952to

1981

are included

in the data set. (a) June, July, August. (b) September, October, November. (Analysis by C. Leovy of

data published in the atlas of Warren et al., 1988.)

5.2 Stratus, Stratocumulus, and Small Cumulus in Boundary Layer Heated from Below

155

the continents. The extent of these regions of oceanic low clouds is so great that it

has been estimated that

mere 4% increase in the area of the globe covered by

low-level stratus clouds would be sufficient to offset the 2-3 K predicted rise in

global temperature due to a doubling of CO

•

"123

5.2.2 Cloud-Topped Mixed Layer

5.2.2.1 Conceptual Models

A conceptual model of the development of a cloud-topped mixed layer is given

in Fig. 5.10. We will refer to this illustration throughout the following discussion.

Panel (a) of the figure represents the boundary-layer structure before any cloud

appears. The boundary layer is envisioned to be characterized by a population of

buoyant plumes rising from the warm sea surface.

becomes well mixed as a

result of the vigorous upward fluxes of sensible and latent heat. Unlike a radiation

fog layer, in which buoyancy is suppressed by thermodynamic stability, turbulent

kinetic energy is generated strongly by the buoyancy [term

in (2.86)], as well as

by wind shear. The mixing maintains a layer of constant mean equivalent potential

temperature

ii, in a layer of depth h. This mixed layer is bounded above by a layer

of distinctly different

ii"

usually higher. The ii, of the layer aloft is typically

maintained by large-scale subsidence in the lower troposphere. In the case of

subtropical and tropical oceanic stratus and stratocumulus, the subsidence is

associated with the descending branch of the Hadley cell, concentrated in the

eastern sectors of the subtropical oceanic anticyclones. In the case of cloud layers

in polar airstreams off continents in winter, the subsidence occurs in the cold-air

mass behind a front. The change of

ii,across h is represented by

!:J.(J,.

124 The higher

ii, air in the upper layer is entrained by turbulence into the lower layer, thus

diluting the boundary layer but at the same time increasing its depth

As the depth

ofthe

boundary layer increases, the tops

ofthe

turbulent elements

rise above the lifting condensation level, and small cloud elements form (Fig.

5.l0b). As the cloud elements gradually thicken to form a more continuous sheet

of cloud at the top of the mixed layer (Fig. 5.lOc), the physical picture changes

significantly, with radiation taking on an important role. The ocean surface is

shielded by the cloud layer and the turbulent heat flux in the subcloud layer

becomes weak, or even negative. The turbulent energy generation is concentrated

in the cloud layer and is produced by buoyancy as radiative cooling at cloud top

destabilizes the layer of cloud. Turbulence is also generated by wind shear in the

layer [term

C(6

in (2.86)]. The turbulent layer continues to increase in depth as a

result of entrainment.

Finally, the cloud layer breaks up. Observations indicate that when the bound-

ary layer is not extremely unstable, the cloud layer breaks up into stratocumulus

123

This estimate was made by Randall et al. (1984).

124 In this discussion A indicates the difference obtained by subtracting the lower-layer value from

the upper-layer value.

sea or

land

surface

(a)

slralDcumulus

cloud

element

.,L.J.-,~....,.ll.,Jlh~..J,J~';""""7-:

/ / / / / , 7 7 / / / .

l!Ji.o<O

(c)

Figure5.10 Conceptual model of the evolution of a cloud-topped mixed layer over the ocean. (a)

Unsaturated turbulent plumes are occurring in a well-mixed layer of depth h.

'!F,o

is the heat flux across

the ocean surface.

ii, is the mean equivalent potential temperature in the mixed layer. (b) Plumes

reaching above the lifting condensation level have stratocumulus elements in their upper portion.

Large arrows indicate entrainment across the top of the mixed layer. (c) Cloud layer becomes solid.

Infrared radiative flux

9Jl.

becomes important, especially at cloud top.

(b)

•

III".

III.

:://I!

fll/iI;:

_"11'"

,.,11'"

IbN!!:

1',/:::.

:i:;;!

I,'

)777777777777777777777

(a)

Figure 5.11 Breakup of cloud-topped mixed layer into (a) stratocumulus (Sc) elements and

(b) cumulus (Cu) and cumulonimbus (Cb) elements.

Droplets grow as

cloud top rises

Ice

enhancement

as large droplets

eVaPorate

at T $ -6"C

(a)

(b)

(c)

(d)

Figure5.12 Empirical model of the formation of ice in stratocumulus clouds. The model is based

on extensive probing of clouds by research aircraft. (From Hobbs and Rangno,

1985. Reprinted with

permission from the American Meteorological Society.)

5.2 Stratus, Stratocumulus, and Small Cumulus in Boundary Layer Heated from Below

157

or small cumulus elements, as suggested by Fig. 5.11a. However, when the

boundary layer is especially unstable, the elements in the breakup phase may take

the form of larger cumulus or even small cumulonimbus (Fig.

5.llb).

The mechanism of the breakup process is not well understood.

has been

suggested that it is related to increased entrainment across the top of the mixed

layer. The entrainment might be expected to become large and run out of control

when the

!i.()e between the boundary layer and the layer above is negative and

exceeds a certain large magnitude. According to this idea, a parcel of air entrained

into the mixed layer from above, upon mixing with the cloudy air, would become

denser than its surroundings. Because of the negative buoyancy, the air from aloft

would be rapidly mixed through a portion of the cloud, leaving patchy, dissipating

stratocumulus. Observational studies show, however, that the cloud layer persists

even when the

!i.()e would appear to be sufficiently negative for instability. The

problem appears to be that the potential energy expended in pulling down a tongue

of stable air from above exceeds the energy released in the mixing.

125 The cloud-

layer breakup problem will be discussed further in Sec. 5.2.3.

When the tops of mixed-layer stratocumulus elements extend to heights above

the

-6°C

level, they sometimes produce high concentrations of ice particles by

rapid ice enhancement (Sec. 3.2.6). When the drop size spectrum in the cloud is

broad, with largest particles

-20

J.tm

in dimension, high concentrations of ice are

observed to form in

-10

min. These particles fall out in the form of ice strands in

the cloud (Fig. 5.12).

has been argued that the most likely ice enhancement

mechanism near cloud top is contact nucleation [mechanism (iii) in Sec.

3.2.6].126

Once the droplets are frozen, they grow by vapor diffusion, while smaller super-

cooled droplets in their vicinity evaporate or are collected by the growing ice

particles. Thus, the cloud element is rapidly glaciated. Aircraft observations show

that the strands of ice particles extend down through the cloud, rapidly removing

the liquid cloud particles. Finally, all that remain are a few remnants of the

original cloud and streamers of falling ice particles extending below the original

cloud base.

5.2.2.2 Mathematical Modeling

The essential dynamics of the development of the cloud atop the mixed layer

(depicted schematically in Fig. 5. lOa-c) can be examined quantitatively through a

consideration of the thermodynamic and water-continuity equations integrated

over the depth

h of the boundary layer.!" For simplicity, we will consider the

cloud to be warm (no ice) and nonprecipitating, although in reality drizzle may

sometimes form and contribute significantly to the water budget of the stratus

125 For a discussion of this problem, see Siems et al. (1990).

126 See Hobbs and Rangno

(1985)

for details of the argument.

127 The treatment of the cloud-topped boundary layer presented here was introduced in a seminal

paper by Lilly (1968).

158 5 Shallow-Layer Clouds

layer. The total water-mixing ratio can be subdivided simply into liquid and vapor

form:

qT =qv + qL (5.18)

where qv is the vapor-mixing ratio and qL is the liquid water-mixing ratio. Since

the clouds are assumed to be nonprecipitating, there is no need to subdivide the

liquid water content further into cloud and precipitation drops, and

qT is a con-

servative variable. As a thermodynamic variable, we use the equivalent potential

temperature

0e s defined by (2.15) and (2.17).

The basic assumption of the mixed layer is that within the layer horizontal

averages of

and

are independent of height (Fig. 5.13). The average value ii

remains constant because the heating at the lower boundary (and/or cooling at the

top by radiation once the cloud layer forms) continually produces a conditionally

unstable lapse rate, which is practically instantly removed by vertical mixing and

release of the instability.

The

total water is constant with height as a simple result

of the mixing and the fact

that'

is a conservative quantity. The mean values ii

and

iiT

change suddenly at the top of the boundary layer, as subsidence in the

lower free atmosphere maintains both high potential temperature and low dew

point

just

above the boundary layer. There is a net increase in

from

just

below

just

above h.

In the development and maintenance of a cloud layer under a level where a

positive

!i.O

occurs, conservation of mass requires the rate of change of depth of

the mixed layer to be

- =

+w(h)

(5.19)

- -

.1..._---,

1---------

--[--

L10

Cloud layer

- - - - - -

LevelwhereqT

qvs

...

qT 0e

Constant with z

in mixed layer

Figure 5.13 Assumed variation of

and (J, with height in modeling of the cloud-topped mixed

layer.