Houze Robert A., Jr. Cloud Dynamics

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2.9 Instabilities 59

Equation (2.154) contains the positive buoyancy, which attempts to restore the

perturbation to its rest state. We represent the vertical velocity perturbation in

terms of a velocity potential

<P',

which is defined such that

al/>l

---

I -

az'

al/>2

= - -

(2.156)

The use of the velocity potential implies that the flow is irrotational (i.e., has no

vorticity) in both the upper and lower layers, which is consistent with the assump-

tion that all of the vorticity is concentrated in the infinitesimally thin interface. We

also make use of the fact that at the interface we must have

pl(h)

(h)

(2.157)

(2.158)

Integration of (2.153)-(2.155) with substitution from (2.156) and (2.157) yields

(:t

+ u

~}i

+ (pz - PI)gh =

(:t

)l/>2

A point on the material interface surface must obey the following two relations:

al/>l

= dh z

+ U

~)h

al/>2

= dh z

~)h

Equations (2.158)-(2.160) have wave solutions of the form

where

k is a positive number and

= pzu

+ Plul +

-PIPZ(UZ

UI)Z

+ g(pz - PI)

+ PI - (pz +

PI)Z

k(pz +

(2.159)

(2.160)

(2.161)

(2.162)

The coefficients of z in (2.161) are obtained by assuming that the.magnitude of the

vertical velocity at the peak of the disturbance pictured in Fig.

2.5 is equal in

magnitude and opposite in sign to the vertical velocity at the trough, by applying

the continuity equation

(2.100), and by eliminating physically unrealistic cases

that have no possible stable solutions. The physically unrealistic cases would arise

if the coefficients of z in

(2.161) were allowed to have the same sign. An expres-

sion would be obtained in place of

(2.162) that would have a number under the

radical that was always negative. Thus, the solutions would always be exponen-

tially growing. The solutions given by

(2.161) and (2.162) also have unstable

(exponentially growing) solutions when the expression under the radical in

(2.162)

60 2 Atmospheric Dynamics

is negative. However, these unstable solutions occur only when

g(pZ

_ pZ)

k > Z 1 (2.163)

PIPZ

(ii

- iid

pz = PI + Ap, where Ap is small compared to

PI,

then (2.163) becomes

2gp-

1Ap

k »

(2.164)

(Aii)z

where Au

- UI.

is thus apparent that there are more unstable modes the

larger the wind difference (i.e., the greater the vorticity available to the perturba-

tions) and the smaller the buoyancy (i.e., the weaker the restoring force).

The infinitesimally thin interface is rather idealized. However, an instability

condition similar to (2.164) can be derived for a continuously stratified, incom-

pressible (or Boussinesq) fluid by considering two parcels of fluid, each of

unit mass, initially separated. by a small distance

sz in a two-dimensional fluid

(no variation in the y-direction). The lower parcel is denser by an amount

8p =

-(aplaz)8z, where p is the mean state density of the fluid.

the parcels switch

positions in the vertical, the increase of potential energy per unit volume of fluid is

g8p8z

(2.165)

The only source of energy for the exchange is the shear of the mean wind

aulaz.

Let VI and V

be the initial velocities

ofthe

upper and lower parcels, respectively.

Since momentum must be conserved in the exchange, the final velocities can be

represented by

VI + A and Ui - A. The change of kinetic energy per unit mass

that occurs in the exchange is

(2.166)

which is maximum for

-U

A = (2.167)

Substitution of

= V and VI = V +

as well as (2.167) into (2.166) shows that

the maximum kinetic energy that can be lost in the exchange, per unit volume of

fluid, is approximately

(2.168)

(2.170)

From (2.165) and (2.168), it is evident that for energy to be released in the ex-

change (in the form of an instability), we must have

g8p&

< tp(8U)Z

(2.169)

Or, since

= (aulaz)8z, this may be restated as

. _ -(g/p)(dpldZ) 1

Ri =

(diildZ)Z

2.9 Instabilities 61

Figure 2.6 A laboratory example of waves resulting from Kelvin-Helmholtz instability. (From

Thorpe, 1971. Reprinted with permission from Cambridge University Press.)

where Ri is called the Richardson number. Thus, again, we see that the vertical

shear of the horizontal motion must be strong enough that the vorticity will be able

to overcome the static stability.

An example of waves resulting from Kelvin-Helmholtz instability are shown in

Fig. 2.6. A long covered trough was filled with a lower layer of salt water and an

upper layer

offresh

water. As the trough was tilted, salt water flowed downward,

while fresh water flowed upward. Strong shear was thus produced at an interface.

Waves formed on the interface, at first sinusoidal, then curling over and breaking,

leading to turbulent mixing. The development of Kelvin-Helmholtz waves in time

is further illustrated by Fig. 2.7. An example of Kelvin-Helmholtz waves on the

top edge of a small cloud is shown in Fig. 2.8.

is not uncommon to observe

Kelvin-Helmholtz waves on the borders of clouds, where there is often a discon-

tinuity in density and wind.

2.9.3 Rayleigh-Benard Instability

Another kind of instability that is relevant to cloud dynamics arises when a thin

layer of fluid is subjected to heat fluxes at the top and/or bottom of the layer. The

motion that results from this type of instability is referred to as

Rayleigh-Benard

convection, and it is manifest in certain types of stratocumulus, altocumulus, and

cirrocumulus, which exhibit substructure in the form of rolls and cells (e.g., Figs.

l.l1b,

l.14a and b, l.ISa). An instability theory to explain the phenomenon was

provided by

Rayleigh.f

His theory is based on the Boussinesq equations (Sec.

2.3) applied to an incompressible fluid that expands as its temperature is in-

creased.

For

such a fluid the buoyancy is given by

B =

gaT'

(2.171)

where a is the expansion coefficient, defined such that p' /

-aT'.

Friction and

heat conduction are parameterized in terms of a constant viscosity

and thermal

conductivity R. The equations are linearized about a state of zero mean motion

37 This type of convection was described by Thomson

(1881),

who observed a pattern of cells of

overturning fluid in a barrel of warm soapy water behind an inn used for cleaning glasses. The top

surface of the water was evaporating into the cool air. Benard

(1901)

devised a laboratory experiment

to reproduce the cells and pointed out the analogy between the laboratory phenomenon and "mackerel

cloud," a name sometimes used to describe altocumulus rolls. Lord Rayleigh

(1916)

proposed the

instability theory summarized here to explain the cells.

62 2 Atmospheric Dynamics

...J

-c

o 0.5 1.0 1.5 2.0

HORIZONTAL

DISTANCE

Figure 2.7 Development of Kelvin-Helmholtz waves in time. The shape of the disturbed fluid

interface is shown at a succession of times,

(1-15'

(From Rosenhead, 1931.)

and horizontally uniform temperature. Perturbations are then governed by the

equation of motion,

av'

1 A

- =

--Vp'

gaT'k

2v,

the continuity equation,

V·

= 0

and the thermodynamic equation

er:

--wy

2T'

(2.172)

(2.173)

(2.174)

Figure 2.8 An example of Kelvin-Helmholtz waves on the edge of a small fragment of stratus

cloud. (Photo by Brooks E. Martner, NOAA Wave Propagation Laboratory.)

2.9 Instabilities 63

where K

R/pc, cis the specific heat

ofthe

homogeneous fluid, and yis the basic-

state lapse rate

(==

-sttea maintained by heating below and/or cooling above.

Substitution of solutions of the

form"

w',T'

sinmzei(kx+IY)e

u',

v',

oc cosmz e

kx+ly)eVI

(2.175)

(2.176)

(2.181)

(2.182)

into (2.172)-(2.174) leads to the dispersion relation

+ [2 + m

) +

v(i

D)(k

+ [2 + m

Di(

+ [2 + m

yga(

= 0 (2.177)

Solving this quadratic equation leads further to the conclusion that unstable solu-

tions

(v positive and real) occur when

-yga(k

[2)

Di(

+ [2 + m

)3 < 0 (2.178)

If there is no friction

(l)

= 0), this relation reduces to simply

y > 0 (2.179)

That is, the lapse rate must be positive to get unstable growth. If both friction and

conduction are finite, then terms in

(2.178) can be rearranged to

)\ 4

Ra>

2 2 (2.180)

where Ra is the Rayleigh number, defined as

yga

-A-A-

and h is the depth of the fluid. Ra contains all of the prescribed characteristics of

the fluid.

it is assumed that the vertical wave number is related to h by

m=-

then it is clear that (i) instability can occur for any number of combinations of k

and I, including both cells (k = l) and rolls (k = 0, I

01=

0); and (ii) the value which

Ra must exceed, according to

(2.180), is a function of horizontal wave number

'Vk

+ [2, as shown in Fig. 2.9. In order to be unstable at all, the Rayleigh number

characterizing the fluid must exceed the minimum value:

(2.183)

38 The vertical form assumed by Rayleigh is actually rather unrealistic when the fluid has a rigid

bottom or top, since horizontal velocity perturbations are not required to disappear at z

= 0 or D. With

the requirement of zero horizontal velocity at the top and bottom, the analysis becomes more difficult.

64 2 Atmospheric Dynamics

(2.184)

2 3 4 5

WAVE

NUMBER-

Figure 2.9 Value which Ra must exceed for Rayleigh-Benard instability. (Ra), is called the

critical Rayleigh number.

which is found by differentiating the right-hand side of (2.180) with respect to

+ [2). The most unstable solution for the special case of K = Dis obtained by

differentiating

(2.177) with respect to (k

+ [2 + m

)

[or (k2 + [2)] and setting

dvld(k2 + [2 + m

)

= 0 (to obtain the condition of maximum v). A second equation

v is thus obtained, which may be combined with (2.177) to eliminate v and

obtain

Ra(

+ [2 +

= 1- /if k

+ P + m

which is the relation among the wave numbers when v has its maximum value. If

we again assume that

m =

'TTl

h and that we have square cells such that k = I =

2'TTI

S, where S is the spacing of the cells, then (2.184) implies an expected ratio of

S to

h. Under laboratory conditions this ratio is about 3 : 1. No simple relation like

(2.184) is obtained for the more general case of K

2.10 Representation of Eddy Fluxes

We have seen that deviations from mean motions on the scale of interest (defined

by the overbar) can be important if the deviations are systematically correlated,

thus constituting fluxes [recall eddy flux terms in

(2.78), (2.81), and (2.83)-(2.84)].

This is usually the case where clouds are involved and in the planetary boundary

layer (to be discussed in Sec.

2.11 below). An important problem thus arises

because the eddy fluxes

ov'

sl", where

S'l

represents any of the variables

(),'TT,

qi.u,v, or w, become additional variables in the equations for the mean variables

[e.g.,

Pov'()' in (2.78) and

Pov'u'

in (2.83)]. There must, therefore, be some rational

way to represent these fluxes in order for the system of equations governing

atmospheric motions to remain a closed set. Representation of the fluxes for

purposes of closure is difficult. The method adopted for this purpose depends,

first, on the scale of the region represented by the average variables. This scale, in

2.10 Representation of Eddy Fluxes 65

turn, determines the scale of motions represented as eddy fluxes. When the aver-

ages represent very large scales of motion (e.g., the synoptic scale relevant to

weather forecasting and analysis), entire clouds and cloud systems are part of the

perturbation motions. When the averaging is done on smaller scales (e.g., meso-

scale or convective scale), the mean variables explicitly describe the general flow

in and around a cloud, while the eddy fluxes represent turbulence on only those

scales smaller than basic mesoscale or convective-scale motions within the cloud.

These eddy fluxes are always important in cloud dynamics, even in the most

apparently quiescent clouds such as fog and stratus.

There are several approaches to representing eddy fluxes in the mean-variable

equations. Three of these approaches are summarized briefly below.'?

2.10.1 K-Theory

(2.188)

(2.187)

(2.186)

(2.189)

turbulent motions are sufficiently small, the turbulent fluxes of a quantity .'il can

be considered as analogous to molecular diffusion, which proceeds down-gradient

at a rate proportional to the spatial gradient of

.sa;

that is,

pv'.'il'

= -KAPV.'il

(2.185)

In the case of atmospheric fluid motions, the exchange coefficient K

is pre-

sumed to be proportional to the product of eddy size and velocity. This coefficient

depends strongly on wind shear and thermodynamic stratification (i.e., on the

Richardson number Ri). This dependence is understandable, since we have seen

(2.170) that any layer of fluid is unstable and subject to small-scale waves and

turbulence when Ri

< 1/4. Conversely, such breakdown of the fluid motion is

highly suppressed when Ri is sufficiently large. Near the ground,

becomes a

function of other variables such as height above ground and roughness of the

underlying surface. When

is expressed quantitatively in terms of all these

parameters, and they are known, then the basic equations for the mean variables

are said to be closed. This approach is called

K-theory or first-order closure.

For

many atmospheric situations, including the planetary boundary layer, the

vertical fluxes are predominant. Some of the most frequently referred to are

{

Momentum

flUX}

-,-,

ali

r ==puw

=-pK

in x - direction x m

{

Momentum

flUX}

rr::

ali

r ==pvw

=-pK

in y - direction y m

{

Sensible heat}

~==

pcpw'(J' =

-cppK

u z

{

Water vapor}

or.

aqv

pw qv =

-pK

-a

ux z

For

more extended discussions of the representation of eddy fluxes, the books by Panorsky and

Dutton (1984), Arya (1988), and Sorbjan (1989) are recommended, especially Chapter 6 ofSorbjan.

66 2 Atmospheric Dynamics

2.10.2 Higher-Order Closure

A more detailed approach, called higher-order closure, is sometimes used to close

the equations when turbulent fluxes are important. Equations for covariances

(i.e.,

w'(J',

w'u',

etc.) are derived from the basic equations by procedures similar

to that used to obtain the turbulent kinetic energy equation (2.86) (which is the

equation for the variance of the velocity). The derived covariance equations pre-

dict the components of any flux

pv'SIl'. However, these new equations involve

triple-product terms [analogous to

r:ztJ

in the turbulent kinetic energy equation

(2.86)], which must in turn be dealt with, and the approach is generally very

complicated.

2.10.3 Large-Eddy Simulation

A third approach to achieving closure of the equations, when turbulence is a

factor, is called large-eddy simulation. The largest eddies, which account for the

largest portion of the turbulent fluxes, are resolved by the grid of a numerical

model. Smaller-scale eddies are parameterized, for example by K-theory.

2.11 The Planetary Boundary Layer

All of the low-cloud types (fog, stratus, stratocumulus, cumulus, and cumulonim-

bus) either occur within or interact strongly with the planetary boundary layer

(PBL), which is the layer of the atmosphere in which the air motion is strongly

influenced by interaction with the surface of the earth. The interaction occurs in

the form of vertical exchanges of momentum, heat, moisture, and mass. These

exchanges are largely effected by atmospheric turbulence. Since the wind velocity

must vanish at the

earth's

surface, the PBL is always characterized by shear

production of turbulent kinetic energy

[<:<6

in (2.86)]. Buoyant production may

either enhance or reduce the amount of turbulence

in (2.86)]. However, there is

always at least a small amount of mixing present, and the mixing, whether small or

large, can have a substantial effect on low-cloud development. The clouds, in

turn, can have important feedbacks on the PBL through condensation, evapora-

tion, radiation, downdrafts, and precipitation.

2.11.1 The Ekman Layer

The balanced flow regimes, quasi-geostrophic, semigeostrophic, and gradient

flow, discussed in Sec. 2.2 all apply in

the/ree

atmosphere, which is the part of the

atmosphere lying above the PBL. The forces associated with molecular friction

and turbulence [terms

F and

in (2.83)] were ignored to obtain these balanced

flows. Turbulence is indeed negligible in large-scale flow in the free atmosphere

(although it is not negligible in the mesoscale and convective-scale motions within

clouds in the free atmosphere). However, in the PBL, the turbulence is not negli-

2.11 The Planetary Boundary Layer 67

0.4

bl)

0.2

0.8

1.0

u/Ug

Figure 2.10 Hodograph of the Ekman spiral solution. Points marked on the curve are values of

"YEZ, which is a nondimensional measure of height. (From Holton, 1992.)

gible. In fact, the

PBL

can be thought of as a layer in which the turbulence force is

comparable in magnitude to the pressure-gradient and Coriolis forces. Let us

assume that these three forces are in balance in the horizontal components of the

Boussinesq version of the equation of motion (2.83). Under the Boussinesq as-

sumption the density terms

p., disappear from (2.84). If we take the horizontal

average of the equations, assume that the horizontal fluxes are small compared to

vertical fluxes, and express the eddy flux of momentum in terms of K-theory, with

a constant value of

we obtain

Kmii

f(v

- v

)

=0 (2.190)

Kmv

f(ii

-u

) = 0

(2.191)

is further assumed that the geostrophic wind components are constant, that

v = Oat z = 0, and that

as Z

co, For convenience, the coordinate

axes are oriented such that

= 0. Then (2.190) and (2.191) have the solutions

ii=Ug(l-e-rEzcosrEz),

v=uge-rEzsinrEz (2.192)

where

'YE

f/2K

•

The vector wind composed of these components describes a

hodograph in the shape of a spiral (Fig. 2.10), and these solutions are referred to as

the Ekman spiral.

The depth of the Ekman layer is

(2.193)

which is the height at which the wind is parallel to and approximately equal to the

geostrophic value.

Ideal Ekman layers are not usually observed in the PBL. There are several

reasons for this fact. First,

is not a constant. Second, the K-theory approxima-

tion is often not accurate for momentum fluxes. Third, the Ekman solution is

actually unstable.

the solutions in (2.192) are assumed to be the mean state, the

40 Named after the Swedish oceanographer V. W. Ekman, who first discussed this spiral to explain

the change of current with depth in the ocean boundary layer (Ekman,

1902). See Sverdrup et al. (1942)

for further discussion.

68 2 Atmospheric Dynamics

time-dependent perturbation equations have unstable solutions, which grow to

finite size and alter the mean

state."

2.11.2 Boundary-Layer Stability

Although the

Ekman

layer is an idealization, it provides a useful way to associate

the depth of the

PBL

hE with the degree of mixing in the layer.

is evident from

(2.193) that the important factors in determining this depth are the mixing coeffi-

cient for momentum flux

and the Coriolis parameterf. Values of

= 5 m

S-I

andf=

10-

S-1

imply he = 1 km, which is a fairly common value for the depth of

the PBL. However, this depth can vary greatly, primarily as a result of the

variation of

with the Richardson number (Ri).

When the Richardson number is sufficiently high, the buoyant production

the turbulent kinetic energy equation (2.86) constitutes a large negative effect. The

shear production of turbulence

C£5

is thus counteracted by 00, as a result of the

buoyancy restoring force suppressing the shear-induced motion perturbations.

Under these conditions, the value of

Ki; may be greatly lowered. In extreme

cases, such as where strong radiational cooling of the ground is occurring, the

PBL top may drop to as little as 10 m or so. Fog and stratus clouds are often

associated with such stable boundary-layer conditions.

In contrast, when the

PBL

stratification is tending toward unstable, Ri is low,

and the buoyant production

of turbulent kinetic energy reinforces or completely

dominates the

shear

production

<:(6.

In this case,

may be greatly increased and

the turbulent layer

can

become very deep, say

2-3

km. In these cases, the bound-

ary-layer depth may be controlled by conditions in the free atmosphere above.

For

example, large-scale subsidence aloft may produce a stable layer in the low tropo-

sphere, through which the turbulent eddies generated in the

PBL

cannot pene-

trate. This way

establishing the boundary-layer depth is particularly important

in the tropics, where

f is small and (2.193) loses its meaning.

Unstable boundary layers capped by large-scale subsidence are often marked

by stratus or stratocumulus at the top of the mixed layer, where the buoyant,

rising eddies

ascend

above their condensation level. Once formed, the cloud layer

produces radiative feedback to the

PBL

structure. Clouds in both stable and

unstable boundary layers are discussed in more detail in Chapter 5.

2.11.3 The Surface

Layer

The amount of turbulence in the PBL, of course, generally decreases with height.

In the lowest 10% of the

PBL,

however, it is a reasonable and useful approxima-

tion to consider the turbulent fluxes to be nearly constant with height. As we will

see in Chapter 5, this view turns

out

to be helpful in analyzing fog, which occurs

next to the ground. One of the useful simplifications

the near constancy

the

wind stress

vector

)

is that the wind direction does not vary with height.

41 See the reviews of Brown (1980, 1983)for further discussion of this topic.