Houze Robert A., Jr. Cloud Dynamics

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4.4 Doppler Velocity Measurements

129

principle, it can be used to provide a third equation of the form (4.30) at each data

point and thus eliminate the use of the continuity equation. However, it is desir-

able to retain the use of the continuity equation so that the inferred winds will

satisfy mass continuity. If the continuity equation is retained, the problem can be

solved by using statistical (variational) techniques to find the fields of

u, v, and w

that satisfy mass continuity and give the best fit to all the observed radial veloci-

ties. According to this procedure, there is no limit to the number of radar observa-

tions (either from additional radars or by the radars increasing their scanning rate)

that can be incorporated into the synthesized wind field to reduce its uncer-

tainty.l'" Despite its difficulties, multiple-Doppler synthesis is by far the most

important and widely used method for determining the air motions within precipi-

tating cloud systems.P?

4.4.7 Retrieval of Thermodynamic

and Microphysical Variables

Once a field of

u, v, and w has been constructed from multiple-Doppler radar data,

the equations of motion, the First Law of Thermodynamics, and water-continuity

equations can be employed to diagnose (or

retrieve) the fields of thermodynamic

and microphysical variables that are consistent with the observed velocity fields.

The basic premise of this retrieval methodology is that by detecting the velocity

fields with high resolution in space and time, the radar measures the horizontal

and vertical components of the acceleration of the wind. The basic equations can

then be solved for the buoyancy and pressure gradient fields that produce the

accelerations.

There are various approaches and strategies for carrying out a retrieval. To

illustrate the basic idea, we consider a simplified case in which the storm being

observed is steady-state in time, two-dimensional in

x and z, and the microphysics

can be described by the bulk warm-cloud parameterization discussed in Sec.

3.6.1.

110

Molecular friction is ignored. Under these conditions, the horizontal

component of the anelastic equation of motion (2.53) can be written in mean-

variable form (see Sec. 2.6) as

where

arr*

--=A

(4.43)

(4.44)

108 Further discussion of the variational approach to multiple-Doppler radar synthesis can be found

in the paper by Kessinger

et al. (1987).

109

For

further discussion of multiple-Doppler synthesis techniques, see Ray (1990).

110 This example is based on a study by Hauser et al. (1988). Their paper cites other key papers on

the general technique of retrieval. Important among these are Gal-Chen (1978). to whom the concept is

usually attributed,

Hane

et al. (1981), and Roux (1985).

130 4 Radar Meteorology

and (}v

in (2.53) has been approximated by (}o in the denominator of (4.44). Except

for new terms defined here, the notation used in the equations of this section is the

same as that used in Chapters 2 and 3. As in Sec. 2.3, the asterisk indicates a

perturbation from the hydrostatic base state used in the anelastic form of the

dynamical equations, while the base state is indicated by the subscript o. II is the

nondimensional pressure, or Exner function, defined by (2.14). The symbol

;g;u

represents the turbulence term in the x-component of the equation of motion

[recall (2.83) and (2.84)].

If accurate Doppler synthesis of the wind field provides the horizontal compo-

nent of acceleration

Ax,

then the horizontal gradient of pressure perturbation is

determined, according to (4.43). The vertical component of the equation of motion

(2.53) can be written

(4.45)

where

(4.46)

and

(4.47)

(4.48)

The quantity (}a represents the buoyancy in (4.45); it is the apparent poten-

tial temperature perturbation that corresponds to the buoyancy term in (2.53).

;g;

w is the turbulent mixing term in the vertical component of the equation of

motion. The horizontal vorticity equation [similar to (2.61)] obtained by taking

a(4.43)/az - a(4.45)/ax provides an expression for the horizontal gradient of

Oa,

aOa

-=B

where

cpO;

(aA

)

x g

(4.49)

Thus, Doppler measurements

ofthe

acceleration components imply the horizontal

gradient of buoyancy according to (4.48) as well as the horizontal gradient of

pressure perturbation according to (4.43). The radar observations do not as read-

ily indicate the vertical gradient of II* since the buoyancy appears in the vertical

equation of motion (4.45).

Since the buoyancy, as expressed in the third term on the right in (2.53), is a

function of the thermodynamic properties and water content of the air, appeal is

made to the First Law of Thermodynamics and water-continuity equations. Under

(4.51)

(4.50)

4.4 Doppler Velocity Measurements

131

the assumed conditions, the First Law (2.13) can be written in mean-variable form

aijv - aijv J at:

u-+w-=---

u-+w--'?J'

+~8

cpIl

where

'?:F

and

'?:Fe

represent the turbulent mixing effects on

iiv

and

Ii.

The advection

[defined by (4.47)] can be written with the aid of (4.50) as

aOa

u-+w-=B

where

(a-

a- J

- qv - qv u;;, - 0 * - -

---

u-+w--~

+'?J'

-W-(1-0.61q

+q +q

)

cpIl

v 8

(4.52)

(4.53)

(4.54)

According to the bulk warm-cloud water-continuity model (Sec. 3.6.1), the rain-

water content of the air is governed by

(3.67), which under the present assump-

tions may be written as

_ aij, _ aij, a

)

u-+w---

V:q

-'?F.=

-E

r r c c r

where

'?:F,

is the turbulent mixing effect on ii,. Addition of (3.65)-(3.67) implies that

continuity of total water substance is governed by the continuity equation

_ aijT _ aijT a

)

u-+w---

Vij

= 0

r T

where

'?:F

is the effect of turbulent mixing on the total water mixing ratio, defined

qc + q, + qv (4.55)

Remembering that, in addition to (4.55), A

+ K; - E, = !(ii"ii",iiv)' V = !(ii,),

= iiv - qv

and (J* = ii - (Jo, and that

and A

are measured by radar, we see

that

(4.43), (4.45), (4.48), (4.51), (4.53), and (4.54) comprise a set of equations in

Il*,

ii,

ii"

and q., provided that the turbulent mixing terms

'?:Fa,

'?:F

'?:Fe,

'?:F"

and

'?:F

can be parameterized appropriately. There is not really an extra equation

since the information in

(4.48) is redundant with that in (4.43) and (4.45). The

relation is nevertheless valuable because in the solution of the set of equations it is

helpful to have the horizontal gradient of

(Ja

expressed directly in terms of the

observed quantities

and A

This set of relationships among the thermodynamic and microphysical vari-

ables is a very awkward set of differential equations. One way it has been solved is

132 4 Radar Meteorology

by iteratively taking first-guess values of

ii"

ii", and iiI' as given; determining

and 0from (4.43), (4.45), (4.48), and (4.51); and then taking

and 0as given and

determining

ii" iic, and

iiI'

from (4.53), (4.54), and the assumption that

{

: qr -

_q\,

:\,s'

qr_-

q,.

q\,s

> 0 (4.56)

- 0 and

q\,

- qr -

q,.

If qr -

q,.

q\,s

< 0

where

q\,S

is the saturation mixing ratio. These alternate solutions are repeated

until the five variables stabilize in value.

The values of

and if that are most consistent with the wind observations are

deduced by retrieving the fields of

and

that best fit the observational infor-

mation contained in the terms

Ax,

and B

•

Since the water fields are

regarded as given for this step of the procedure,

()

is readily determined from

Oa,

according to (4.47). Note also that

and

could be determined even if nothing

were known about the water fields, and this more limited thermodynamic retrieval

is often the only part of the retrieval that is possible to carry out. However, in this

partial retrieval,

cannot be decomposed to determine 0 or

O\,'

To retrieve the buoyancy

(Oa)

and pressure perturbation

(0*)

fields, the first

step is to seek a fit of the

field to the data, in the least-squares sense, by

minimizing the integral

I. =

{(d~

-B

-UBxlJ}dx<h (4.57)

where

refers to the domain of the radar observations. As a horizontal boundary

condition, it is assumed that the first squared term in the integral is zero on the

horizontal borders of the radar-echo region. Minimization of the first term in the

integral assures that the horizontal gradient of the retrieved

field is a least-

squares fit to the observed

data [recall (4.48)] and is thus simultaneously

consistent with the horizontal and vertical equations of motion at each level.

Minimization of the second term assures the field of

is vertically consistent by

requiring that the vertical gradient of

be as consistent as possible with the wind

observations and retrieved microphysical variables contained in

and B

[recall

(4.51)] and thus simultaneously with the equations of motion and thermodynamic

equations. The velocity

is a user-chosen weighting function that assures homo-

geneity of the dimensions in the integral and decides the relative weight to be

ascribed to the vertical and horizontal gradients in fitting

to the observations.

Once the buoyancy field

(Oa)

is determined, a fit of the pressure perturbation

(0*)

field to the data can be obtained by minimizing the integral

(4.58)

Horizontal boundary conditions are applied in the same manner as for (4.57).

Minimization of the first term in (4.58) assures that the horizontal gradient of the

4.4 Doppler Velocity Measurements

133

retrieved

TI*

field best fits the observed

values [recall (4.43)] and is thus

consistent with the horizontal equation of motion at each level. Minimization of

the second term assures the fields are vertically consistent by requiring that the

vertical gradient of

TI*

be as consistent as possible with the wind observations

contained in

[recall (4.45)] and thus with the vertical as well as the horizontal

equations of motion, as well as with the retrieved thermodynamic field.

The overall accuracy of thermodynamic and microphysical fields derived by

the retrieval technique is difficult to determine. The confirmatory data are hard to

obtain by aircraft or other direct means. When the retrieval technique is applied to

artificial fields of wind and reflectivity produced by numerical cloud models,

III

the

retrieved fields compare well with the model fields for steady-state, horizontally

uniform, and non-turbulent regions of convective systems. When the precipitation

is varying rapidly in time, the technique described above must be extended to

include time derivatives, which are very difficult to measure. In the future, more

rapidly scanning radars may provide the time resolution necessary to estimate

these terms.

111

For examples of applying the retrieval technique to model output, see Hane et al. (1981)and Sun

and Houze (1992).

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Part

Phenomena

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Chapter 5 Shallow-Layer Clouds

"The

cold mist hangs like a stretch'd

canopy"!"

In the remainder of this book, we will examine the dynamics of specific types of

clouds. We begin in this chapter with clouds confined to shallow layers of air in

which the rate of cooling resulting in cloud formation is rather slight. These clouds

include fog, stratus, stratocumulus, altostratus, altocumulus, cirrus, cirrostratus,

and cirrocumulus. The only type of layer cloud we do not consider in this chapter

is nimbostratus, which is very deep and produces substantial rain and snow.

Chapter 6 is devoted to the subject of nimbostratus. The clouds we consider in this

chapter are generally nonprecipitating, though they may occasionally produce

drizzle. Although there are certain important exceptions, water contents in shal-

low layer clouds are mostly

< I g kg

while mean vertical motions are generally

-1-10

em s

-I.

These values are very small compared to the convective clouds we

will consider in Chapters

7-9.

The cloud layers are generally

-I

km or less in

vertical extent, although altostratus may sometimes be several kilometers in

depth. Because the mean vertical air motions in shallow layer clouds are small,

the condensation cannot be accounted for by upward air motion alone. Other

physical mechanisms are also important, especially radiation and eddy mixing. In

this chapter, we will see how the vertical air motion, radiation, and turbulent

mixing all interact to give shallow layer clouds their particular character.

Shallow-layer clouds can occur quite locally; however, they can also be very

widespread, covering mesoscale or even synoptic-scale regions. Fog, for exam-

ple, can cover an area of the size of the entire central valley of California (Fig.

1.7b). Stratus occurs in huge persistent sheets covering the Arctic Ocean. Strato-

cumulus often covers the wide subtropical oceanic regions dominated by the

circulations of the semipermanent subtropical anticyclones off the west coasts of

continents (Fig. 1.11a).

also typically covers large midlatitude oceanic regions

off the east coasts of continents, when cold air flows out over the ocean behind

strong fronts (Fig. I.

lib).

Layers of shallow midlevel cloud (altostratus and alto-

cumulus) can also be quite widespread; they are among the most ubiquitous of

cloud types, providing vivid optical effects (e.g., the corona, mock sun, sun dog)

as well as frequent illustrations of atmospheric fluid dynamical behavior (rolls and

cellular structures of various types). Upper-level layer clouds also occur in large

sheets covering great expanses of the earth, as can be seen by viewing routine

112 Goethe's words to describe stratus.

137

138 5 Shallow-Layer Clouds

satellite pictures, which show wide expanses of high-cloud tops in both the tropics

and midlatitudes (e.g., frontispiece and Figs. 1.27-1.30). The propensity of layer

clouds at low, middle, and upper levels to cover great areas has a major impact on

the climate of the earth through the absorptive and scattering effects of these

cloud layers in the

earth's

radiation balance.

In examining the dynamics of layer clouds, we will begin in Sec. 5.1 with the

case of fog and stratus occurring under highly stable conditions, governed by the

mechanics of a stable boundary layer cooled from below (Sec. 2.11.2). We will see

that even when air is generally calm and stable, mixing by eddy motions is quite

important to the development and maintenance of fog and stratus. We will further

see that, as fog thickens, differential radiative heating can make the upper layer of

the fog less stable. Vertical mixing is thereby enhanced, and the cooling is shifted

aloft to deepen the fog layer or change the cloud from fog (in contact with the

ground) to elevated stratus. Once a layer of stratus has formed it produces green-

house heating effects below its-base. As examples of this type of fog and stratus

formation, we will examine briefly the life cycle of a nocturnal radiation fog over a

soil surface and the formation of widespread and persistent Arctic stratus, which

occurs as warm air from the south moves

over

the melting pack ice of the Arctic

Basin during summer.

After considering fog and stratus under essentially stable conditions, we pro-

ceed in Sec. 5.2 to the case of stratus and stratocumulus over a warm ocean,

where the boundary layer is so strongly heated from below that it is unstable and

convective and thus tends to be well mixed, governed by the dynamics of an

unstable boundary layer. Wind shear can provide a second source of mixing in

these more active boundary layers. In contrast to the stable case, in which the

cloud first appears as fog at the bottom of the boundary layer, the cloud stratum

appears at the top

the boundary layer when the strong convective mixing begins

to have upward plumes reaching above the lifting condensation level. The convec-

tive mixed layer continues to deepen as the plumes of convection push farther

upward and environmental air from above the boundary layer continues to be

mixed downward across the top

ofthe

mixed layer. The cloud layer evolves into a

more or less continuous layer of cloud, which no longer depends so much on the

surface heating as on the differential radiative heating between the top and bottom

of the cloud layer to drive its turbulence. Mesoscale organization of the active

eddy motions resulting from various combinations of differential heating and wind

shear in the boundary layer can give the mixed-layer cloud sheet a spatial texture.

The cloud then takes on a discrete stratocumulus structure, in which the cloud

elements consist sometimes of rolls and sometimes of cells. When the mixing is

especially vigorous the elements

the cloud layer are more of the form of small

cumulus than stratocumulus. Sometimes they even achieve the size of small cu-

mulonimbus.

After considering the physics and dynamics of boundary-layer fog and stratus

and cloud-topped mixed layers, we will proceed to Sees. 5.3 and 5.4, where we

will examine shallow-layer clouds that occur well above the boundary layer (i.e.,

altostratus, altocumulus, cirrus, cirrostratus, and cirrocumulus). These cloud