Houze Robert A., Jr. Cloud Dynamics

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11 Precipitating Clouds in Extratropical Cyclones 439

surface and are characterized by predominantly convective clouds at some point

in their lifetimes. Comma clouds, such as the smaller cloud system to the west of

the main frontal cyclonic cloud system in Fig. 1.29, are the larger of the polar

lows, and they themselves may exhibit a surface low center and fronts, which are

the products of cyclogenesis and frontogenesis associated with a baroclinic wave

of especially short wavelength. Polar lows occurring farther from major frontal

cloud systems tend to be the smallest members of the spectrum and sometimes

appear to have a hurricane-like appearance (Fig. 1.30).

In this chapter, we focus on the clouds of extratropical cyclones (large frontal

cyclones, comma cloud systems, and smaller polar lows). We have already con-

sidered aspects

ofthese

clouds in Chapters

1,5,

and 6. In particular, we examined

the cloud streets, which form as the equatorward-moving cold-air stream of a

cyclone moves out over a warmer ocean surface (Figs. 1.11b, 1.30, and 5.21).

Although the individual clouds comprising the cloud streets may occasionally

precipitate, they are not the primary precipitation-producing clouds of the cy-

clones. Rather it is the broader bands of clouds with extensive cirriform cloud

tops, seen in satellite pictures such as those in Figs. 1.29 and 1.30 to be curling

around and extending outward from cyclonic storm centers, that produce most of

the precipitation in extratropical cyclones. In Chapter 6, we considered some

basic structural aspects of these larger, precipitating clouds. In this chapter, we

seek a dynamical and physical understanding of them.

Of all the cloud systems considered in this text, the precipitating clouds in

extratropical cyclones have the widest range of scales of phenomena contributing

to the cloud formation processes. The plan of this chapter is to proceed from the

largest of these scales to the smallest. On the largest scale, the dynamics of the

synoptic-scale baroclinic wave itself contribute to producing the clouds and pre-

cipitation, since it is the area of upward air motion within the wave that consti-

tutes the broad-scale region in which clouds can form. Therefore, we will begin in

Sec. 11.1 by considering the dynamics governing the vertical air motions of the

wave. We will see also, in that discussion, how the wind field in the baroclinic

wave favors the development of the low-level cyclone. In Sec. 11.2, we will

investigate how the vertical motions associated with the wave become concen-

trated in the vicinity offrontal zones by examining the dynamics offrontogenesis.

This discussion will serve to indicate the nature of the vertical circulation associ-

ated with the frontal zones that occur in a baroclinic wave.

is these vertical

circulations that are of primary importance in focusing the cloud formation pro-

cesses in the cyclone at the frontal zones. In Sec. 11.3, we will investigate how the

frontal zones, produced by frontogenesis, are distributed spatially within a devel-

oping cyclone, since this spatial pattern determines, to a first approximation, the

distribution of precipitating clouds in extratropical cyclones. Superimposed on the

clouds produced by the frontal air motions are still smaller mesoscale

rainbands

and convection, within which the precipitation-producing cloud microphysical

processes are most concentrated. In Sees. 11.4 and 11.5, we examine this super-

imposed structure by reviewing the observed structure of clouds and precipitation

in extratropical cyclones. Sections 11.4.1 and 11.4.2 concentrate on the more

440 11 Precipitating Clouds in Extratropical Cyclones

general aspects of the observed pattern of clouds and precipitation in a developing

frontal cyclone. Then Sees. 11.4.3-11.4.6 describe in more detail the clouds and

rainbands that occur within specific parts of this pattern. Finally, Sec. 11.5 exam-

ines the clouds of polar lows.

11.1 Structure and Dynamics of a Baroclinic Wave

11.1.1 Idealized Horizontal and Vertical Structure

A baroclinic wave can develop in the atmosphere when the wind at a given

pressure level is blowing parallel to the temperature (or potential temperature)

isotherms and there is a nonzero temperature gradient across the flow. If certain

other conditions are met, a small initial wave perturbation superimposed on the

flow will amplify, and the situation is referred to as a state of

baroclinic instabil-

ity.283

To obtain a qualitative understanding of the baroclinic wave-growth pro-

cess, consider a region of strong

north-south

temperature gradient in the North-

ern Hemisphere. Suppose that a weak wave-like perturbation (e.g., Fig. 11.1)

appears in a geostrophic flow that is otherwise uniform and purely zonal (i.e.,

west-east) at some particular pressure level.

Let

the wave perturbation be of

sufficiently large scale that its wind field is also geostrophic. For simplicity, as-

sume that the level under consideration for this wave-like perturbation is moving

at exactly the same velocity as the zonal flow. Then, if we were to view the air

motion in a coordinate system moving with the wave, the zonal flow would van-

ish, and the only horizontal motion remaining would be the meridional (i.e.,

northward and southward) velocity components associated with the wave-like

perturbation. As shown in Fig. 11.1, the meridional motions distort the isotherms,

which are assumed here to have been initially perfectly straight, with an

east-west

orientation and cold air lying to the north. The northerly wind in the vicinity of

point A advects cold air southward, while the southerly wind at B carries warm air

northward. These motions tend to produce a trough in the temperature isotherms

that lags the trough in the geopotential field by a quarter wavelength. In the

283 See Holton (1992) for a more detailed treatment of baroclinic instability.

Figure 11.1 Distribution of geopotential height

(<I»

and temperature

on a constant-pressure

surface in a developing baroclinic wave in the Northern Hemisphere. The pressure surface is located

near the level where the speed of the wave is the same as the speed of the mean zonal flow. (From

Wallace and Hobbs,

1977.)

11.1 Structure and Dynamics of a Baroclinic Wave

441

absence of other influences, the horizontal temperature advection by the perturba-

tion flow will continue to distort the isotherms from their original east-west

configuration, thus causing the

east-west

temperature contrasts in the wave to

amplify further.

In a growing wave, in which the temperature perturbations are becoming

larger, the kinetic energy associated with the wave disturbance increases. The

mechanism by which the kinetic energy increases is a thermally direct circulation,

in which the warmer air at B rises and the cooler air at A sinks, thus lowering the

center of gravity of the fluid. The air at B is then on an upward-sloping poleward

trajectory, while the air at A is on a downward-sloping equatorward trajectory.

Implications for cloud formation are then clear: clouds are favored east of the

trough of the baroclinic wave, while they are suppressed on the west side of the

trough.

The range of slopes of parcel trajectories is limited.

For

example, the rise of air

that we have postulated to occur at B cannot be too great, or else the effect of

warm advection will be completely counteracted by the decrease of temperature,

which must occur as the air rises. The allowable flow configurations can be deter-

mined theoretically, and the ones for which the disturbance grows most rapidly

can be calculated.P' All of these details are not necessary for our present discus-

sion. We will confine ourselves to some general features of the developing wave

that will enhance our appreciation of the vertical motions, which determine the

regions of the wave where clouds may be favored or suppressed.

To gain a qualitative picture of the nature of the vertical motions, we may

consider some aspects of the vertical structure of the idealized baroclinic wave

that are implied by the out-of-phase relationship of the temperature and geopoten-

tial height contours. The location of the temperature trough to the west of the

trough in the geopotential field (Fig. 11.1) indicates that the trough of the baro-

clinic wave tilts toward the west with height, as shown in Fig. 11.2. The tilt of the

trough toward the cold air is required in order to maintain hydrostatic balance

throughout the disturbance.P" The ridge in the height field (or high in the pressure

field) tilts westward for the same reason. Since the wave disturbance is in geo-

strophic balance, the winds between the ridges and troughs alternate between

northerly and southerly as indicated in Fig. 11.2. Since the disturbance is both

geostrophically and hydrostatically balanced, the thermal wind relation (Sec.

2.2.5) applies. Recall, however, that the northerly geostrophic winds advect

colder air into the plane of the cross section, and the southerly geostrophic winds

bring in warmer air. In the absence of other effects, these advective effects would

destroy the thermal wind balance. The upward motion of the warm air and the

downward motion of the cold air are needed to restore the balance. As the warm

air ascends, its temperature is lowered as a result of the expansion. The tempera-

ture of the cold air increases as it sinks. These temperature changes thus mitigate

the effect of the advection by lessening the

west-east

thermal contrast across the

284 See Hoskins (1990) or Holton (1992).

285 See Wallace and Hobbs (1977, Fig. 2.3c).

442

Precipitating Clouds in Extratropical Cyclones

cold

warm

(a)

cold

.--

warm

'Qj

--+

(b)

cold

<:==

warm

(c)

~wes

eas

--+

Figure 11.2 Longitude-height cross section through a growing two-dimensional baroclinic

disturbance. Features indicated include poleward

«(8)

and equatorward (circled dot) meridional

geostrophic air motions, zonal and vertical components of ageostrophic circulation

[(ua,lii) thin

arrows], regions of warm and cold air, and Q-vectors (double arrows). To determine the Q-vectors, it

is assumed that the geostrophic wind does not vary in the y-direction, that the zonal component of the

base-state geostrophic current

does not vary in the x-direction, and that by (not shown) is a negative

constant. Then the Q-vector reduces to

Q = -vgxbyi, which changes direction depending on the local

sign of

Vgxo (Adapted from Hoskins, 1990. Reproduced with permission from the American

Meteorological Society.)

disturbance. By continuity, the vertical motions must be compensated by horizon-

tal ageostrophic motions. If we assume the idealized disturbance depicted in Fig.

11.2 to be two-dimensional, with no variability in the meridional direction, then

the vertical motions must be compensated by ageostrophic motions in the zonal

direction, as indicated in Fig. 11.2b. These motions also act to reduce the effect of

the geostrophic temperature advection. The dynamical role of the ageostrophic

circulation

w) is in fact to keep the disturbance in thermal wind balance. This

fact can be demonstrated more formally, extended to three dimensions, and made

quantitative by the following mathematical arguments.

11.1.2 Dynamics Governing Large-Scale Vertical

Air Motion

Consider the flow to be Boussinesq, inviscid, dry adiabatic, geostrophic, hydro-

static, and at constant Coriolis parameter

f. Then the governing equations are the

dry adiabatic thermodynamic equation (2.11), the inviscid quasi-geostrophic equa-

11.1 Structure and Dynamics of a Baroclinic Wave 443

tion of motion with constant

f (2.23), the hydrostatic equation (2.38), and the

Boussinesq continuity equation (2.55).

In this chapter we will view these relationships in a reference frame in which

vertical variations are expressed in terms of the

pseudoheight

0,286

defined as

0==

[1-(p/P)IC](I/K)H

(11.1)

where A indicates a constant reference state, taken to be representative of condi-

tions near the

earth's

surface, K = Rd/c

and H, = p/(pg). This vertical coordi-

nate very nearly approximates the physical height z.

allows one to work with all

the advantages

a pressure coordinate system, while maintaining the ability to

visualize processes in a height framework. Taking

a(11.1)/

Bp; noting from the

hydrostatic relationship (2.38) that

d/dp =

_ag-

dldz

(11.2)

and making use of the definition of potential temperature (2.8), we find that the

pseudoheight is related to the geopotential height z by

9~0

= 0&

We define a new thermodynamic variable b, given by

b==g"

(11.3)

(11.4)

This quantity is simply the potential temperature expressed as an equivalent

buoyancy [as defined by (2.48)]. The use of this variable along with the pseudo-

height 0 gives a simplified form to the basic equations.

The horizontal equation of motion (2.23) for quasi-geostrophic flow is unaf-

fected by the introduction of

band

However, by combining (11.3) with (2.38)

and making use of (11.4), we find that the hydrostatic relation may be rewritten in

terms

band

0 as

del>

b=-

(11.5)

(11.6)

where

eI>

is the geopotential. By taking k x V (11.5) and making use of the

definition of the geostrophic wind as expressed in (2.68), we obtain a convenient

form of the thermal wind equation:

kxVb

This form could have also been obtained by substituting from (2.3), (11.2), (11.3),

and (11.6) in the thermal wind equation (2.41). The relations (11.3) and (11.4)

furthermore imply that the Boussinesq continuity equation (2.55) in o-coordinates

286 Following Hoskins and Bretherton (1972).

(11.7)

444

11 Predpitating Clouds in Extratropical Cyclones

takes the form

alnb

UX+VY+W

+waa=O

where w =

DalDt.

From the definition (11.4), it is evident that as long as the

potential temperature does not deviate too much from the near-surface potential

temperature

the Boussinesq continuity equation preserves its form in a-coordi-

nates:

+ v

+ w

'" 0 (11.8)

Under near-adiabatic and quasi-geostrophic conditions, the dry adiabatic form of

the First Law of Thermodynamics (2.11) may be written with the aid of (11.3) in

terms of

b and a as

Dgb

-2

- + N W '" 0 (11.9)

where

is defined as in (2.98), except with a replacing z. According to (2.25), the

subscript

g in

Dg/Dt

indicates that the only advection terms included in the total

derivative are horizontal advection terms and that they are calculated with the

geostrophic wind components

and v

•

We may now use the basic equations written in terms of b and a to see the role

of the ageostrophic circulation in maintaining the thermal wind balance. First take

k x V (11.9), to obtain

x Vb) = k x Q - R2

x Vw

where we have made use of the Q-vector, defined as

(-ugxb

vgxby)i

(-ugix

Vgiy)j

i + Q

(11.10)

(11.11)

Now, taking the a-derivative of the horizontal equation of motion (2.23), substitut-

ing from (11.6), and noting from (2.68) that the geostrophic wind is nondivergent

(i.e.,

-v

) ,

we obtain

(11.12)

From the first terms on the right-hand sides of (11.10) and (11.12), it is evident that

the action of the geostrophic wind upon the temperature field, as expressed quan-

titatively by the Q-vector, tends to destroy the thermal-wind balance by changing

the two sides of the thermal wind equation (11.6) by equal but opposite amounts.

From the second terms on the right-hand sides of (11.10) and (11.12), it is evident

that the role of the three-dimensional ageostrophic motion

(va,w) is to maintain

the thermal-wind balance. This result expresses more quantitatively the conclu-

sion drawn qualitatively from the idealized two-dimensional disturbance sketched

11.1 Structure and Dynamics of a Baroclinic Wave 445

in Fig. 11.2b and extends the conclusion to three dimensions. The role of the

ageostrophic circulation in restoring the thermal wind balance at the same rate

that it is being destroyed by the action of the large-scale wind on the thermal field

can be seen most clearly by subtracting (11.12) from (11.10), which leads to

vao

= 2Q

(11.13)

where VH is the horizontal gradient operator. Taking VH • (11.13) and making use

of the continuity equation (11.8), we obtain the

omega

equationV'

N2V~w+

woo

= 2V

· Q (11.14)

The solution of this equation for a given Q (i.e., for a given temperature and

geostrophic wind field) determines the vertical velocity field that is required to

maintain thermal wind balance. The omega equation is easily generalized to in-

clude additional sources of vertical motions arising from friction, diabatic heating,

and the variation of

f with latitude.

Since the left-hand side of (11.14) behaves like a three-dimensional Laplacian

lV,

we conclude that at a given pressure level (constant 3), convergence of Q is

associated with upward air motion and divergence of

Q with downward motion.

This relation is readily applied to the schematic disturbance in Fig. 11.2. The

Q-vectors associated with the middle-level geostrophic north-south flow and tem-

perature field are shown as double arrows. They converge in the warm-air region

and diverge in the cold region. Therefore, the vertical motion, which we previ-

ously postulated to be upward in the warm region and downward in the cold

region, is seen via (11.14) to be necessary to keep the disturbance in thermal wind

balance. Hence, a wave disturbance such as that shown in Fig.

11.1

must have

warm air rising and cold air sinking if it is to remain in geostrophic and hydrostatic

balance.

11.1.3 Application of the Omega Equation to a Real

Baroclinic Wave

The omega equation is a

diagnostic relation, meaning that it contains no time

derivatives and is readily applied to a real weather map showing the distribution of

temperature and geostrophic wind. A 700-mb chart showing a developing baro-

clinic wave is shown in Fig. 11.3. The geostrophic wind, indicated by the geopo-

tential height contours, and the temperature isotherms are shown in Fig. 11.3a.

The temperature trough lags the trough in the height field, as in the idealized

example in Fig. 11.1. A developing frontal cyclone at lower levels was associated

with the wave, as indicated by the superimposed surface frontal positions. The

Q-vectors computed from the 700-mb temperature and geostrophic wind fields are

indicated in Fig. 11.3b. The divergence of

Q is shown by dashed contours, which

indicate large-scale subsidence of cold air on the southwest side of the trough in

287 The name omega equation refers to the fact that this relation is often written in terms of the

"vertical motion" in pressure coordinates. w

Dp/Dt.

446 11 Precipitating Clouds in Extratropieal Cyclones

-(a)

100

50:N

100

...

;..'

...

(b)

/~foo,N'

, ,

, \

I ' ... _

C")

/-r

___

(-~~>:'t>

'---'"

--:;.-u

Figure 11.3 Example of a 700-mb chart showing a developing baroclinic wave. (a) Geopotential

height contours (30-m interval) and temperature (2°Cinterval). Direction of geostrophic wind indicated

by arrowheads. (b) Q-vectors computed from the temperature and geostrophic wind fields. Isopleths of

(2g/6)V .

Q for every 1 x 10-

m-1s-

with the zero isopleth being solid. Sea-levelfrontal positions are

indicated by standard symbols in (a) and by heavy dashed lines in (b). (From Hoskins and Pedder,

1980. Reprinted with permission from the Royal Meteorological Society.)

11.1 Structure and Dynamics of a Baroclinic Wave

447

the height field, centered well to the rear of the surface cold front. General ascent

of the warm air is indicated on the east and northeast sides of the trough. The

largest and most intense region of ascent is ahead of the surface warm front.

However, the region of ascent extends southward over the surface cold front, so

that altogether the region of upward motion (convergent

Q) forms a pattern that

resembles a giant comma. In view of this pattern of general ascent in the baro-

clinic wave, it is not surprising that cloud systems associated with baroclinic

waves (not only comma cloud systems per se, but also the larger frontal cyclones

and smaller polar lows) are typically in the form of a large comma shape (e.g., Fig.

1.29 and 1.30).

11.1.4 Low- Level Cyclone Development

The region of ascent at 700 mb in Fig. 11.3 is also seen to encompass an area lying

over the low-pressure center at low levels (where the cold front and warm front

meet). Upward motion at this level implies horizontal velocity convergence at

lower levels, since the vertical motion vanishes at the lower boundary. Taking

k·

V x (2.23) and applying the mass continuity relation (11.8), we obtain the

geostrophic vorticity equation

Dig

(11.15)

which is the form taken by (2.59) under quasi-geostrophic conditions at constant/.

At the center

ofthe

low-level cyclone the right-hand side is positive because of the

convergence. Therefore, since there is no vorticity advection at the center of a

vortex, the geostrophic vorticity in the low center must increase. According to

(2.69), the geostrophic vorticity is proportional to the Laplacian of the geopoten-

tial. The increase in vorticity then implies strengthening of the low-level depres-

sion in the geopotential field. Since the strongest ascent implied by the Q-vectors

in Fig. 11.3b is actually northwest of the surface low center, there is an indication

that the low-level cyclone is in the process of shifting its position to be more in line

with the upper-level trough.

The simultaneous evolution of the low-level cyclone and upper-level wave is

pictured schematically in Fig. 11.4. The connection between the lower- and upper-

level flow is found in the hydrostatic relation (11.5). Since the large-scale air

motion is in hydrostatic balance, the geostrophic flows at upper and lower levels

are related through the thermal field. Integration of

(11.5) with respect to ashows

that the geopotential height difference (or

thickness) between any two pressure

levels is directly proportional to the mean potential temperature of the intervening

layer of air.

288 Figure 11.4 shows the geopotential height contours, geostrophic

flow direction, and thickness pattern for the

1000-

and

SOO-mb

levels in an ideal-

ized developing baroclinic wave. As noted in discussing Fig. 11.1, continued

thermal advection in the incipient baroclinic wave leads to further distortion of the

288 The mathematical expression of this relation is called the hypsometric equation. See Chapter 2

of Wallace and Hobbs (1977).

448 11 Precipitating Clouds in Extratropical Cyclones

Figure 11.4 Idealized model of a midlatitude baroclinic wave disturbance with fronts (shown by

standard frontal symbols) in three stages of development. Geopotential height contours at 1000 mb

(thin solid) and 500 mb (heavy solid) are shown together with fronts at 1000 mb. Geastrophic flow

direction indicated by arrowheads. Contours of the 1000-500-mb thickness (dashed). (From Palmen

and Newton, 1969.)

6.'

4,2

,I'

'.........

..'

,,,

....

' ,

....

.....

•••

~-~.-i::%~

·..

:·~w

(a)

Figure 11.5 Schematic model showing stages in surface cyclone development that occurs when an

upper-level baroclinic wave (whose trough is indicated by the heavy dashed line marked with a T)

moves over a pre-existing low-level front. Thick arrows in (a) show velocities of trough and front. New

cyclone appears in (b). Further development is shown in (c). Isobars are labeled in departure in

millibars from a sea-level pressure of 1000 mb. Lighter dashed lines are isotherms. C and W indicate

cold and warm areas, respectively. (Adapted from Petterssen, 1956.Reproduced with permission from

McGraw-Hill, Inc.)

isotherms in time.

The

time sequence in Fig. 11.4 illustrates this distortion.

The

wave in the thickness field increases in amplitude as the storm ages. As antici-

pated in the discussion of Fig. 11.3, the low-level cyclone moves back toward a

position more in line with the upper-level trough. By the time of Fig.

II.4c,

the

intensification of the upper-level trough has led to the formation of a closed

cyclonic circulation,

and

the low-level cyclone is located almost under it. Eventu-

ally, the

upper

and lower parts of the disturbance become aligned. The thickness

field, which according to (11.5) is obtained by graphical subtraction of the upper-

and lower-level geopotential fields, becomes similarly aligned. Thermal advection

then ceases,

and

the

wave and frontal cyclone

can

no longer intensify.

Symbols are used in Fig. 11.4 to indicate the positions

fronts at the

lOOO-mb

level. All of the fronts are found at the warm edges of frontal zones, which are

zones of strong thermal gradient, identifiable in the figure by close packing of the