Salby M.L. Fundamentals of Atmospheric Physics

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510

15 The General Circulation

-10

y o

-4

-5 0 5 10 15

-10 -5 0

-10 -5 0 5 10

I ~ I I

(el

5 10 15

Figure

15.10 Zonally asymmetric circulation produced by a deep heating anomaly over the

equator (stippled). (a) Planform view of surface motion. (b) Streamfield in a vertical cross section

over the equator. (c) Surface pressure distribution. Adapted from Gill (1980).

E1 Nifio is accompanied by a seesaw in surface pressure between the western

and eastern Pacific. Known as the

Southern Oscillation, this seesaw shifts low

pressure from the western Pacific and Indian ocean to the eastern Pacific, as

is evidenced by the anticorrelation of changes in those regions (Fig. 15.12).

These phenomena are referred to jointly as E1 Nifio Southern Oscillation or

ENSO. Because it reconfigures the convective pattern, ENSO introduces a

major perturbation to the tropical circulation.

ENSO is also felt at extratropical latitudes. By upsetting the distribution of

heating in the equatorial troposphere, ENSO alters planetary waves excited

thermally at low latitude that radiate poleward. Figure 14.20 shows the anoma-

lous upper-tropospheric height for winters during E1 Nifio, when convective

heating is positioned in the equatorial central Pacific. A planetary wavetrain

radiates poleward from anomalous convection, being refracted equatorward

along a great circle route. Known as the

Pacific North American (PNA) pat-

tern,

this perturbation to the normal height field alters the extratropical cir-

culation, in particular, by introducing a prominent ridge over western Canada

and trough to its east. The jet stream (wavy trajectory in Fig. 14.20a) is then

deflected northward over the western United States and Canada and south-

ward over the eastern United States. Since the jet stream marks the boundary

between tropical and polar air (Sec. 12.2.2), these changes produce abnor-

Figure

15.11

Anomalous

SST

(K)

during

Nifio.

After Philander

(1983).

3OoW

30OE

60"

90"

1200

1500

1800

ls00

1200

900

60O

mow

Figure

15.12

Correlation of monthly mean surface pressure in the Pacific with that at Djakarta. Oppositely phased variations

in the eastern and western Pacific characterize the

Southern Oscillation.

After Philander

(1983).

Problems 513

mally warm temperatures over the western United States and Canada and

abnormally cold temperatures over the eastern United States. By altering the

locations of the storm tracks, they also modify the development and movement

of baroclinic systems.

Suggested Reading

Atmospheric Science: An Introductory Survey

(1977) by Wallace and Hobbs

contains a nice introduction to moist static energy and other concepts figuring

in the global energy budget.

An historical overview of available potential energy is given in

Atmosphere-

Ocean Dynamics

(1982) by Gill, along with applications to wave motions. It

also includes a comprehensive discussion of tropical circulations and their

interaction with the oceans.

An Introduction to Dynamic Meteorology

(1992) by Holton contains a formal

treatment of the zonal-mean circulation and its relationship to eddies.

Introduction to Circulating Atmospheres

(1993) by James develops the observed

Hadley circulation from first principles and in light of the three-dimensional

structure of the general circulation.

Global Physical Climatology

(1994) by Hartmann includes a thorough descrip-

tion of the general circulation and its relationship to eddy transports of heat

and momentum and to regional climates.

Physics of Climate

(1992) by Peixoto and Oort is a comprehensive treatment

of the general circulation that surveys a wide range of observations.

Problems

15.1.

15.2.

15.3.

15.4.

15.5.

See Problem 1.18.

A national research announcement for a precipitation monitoring satel-

lite asserts that latent heating inside convection is the primary energy

source driving the atmospheric circulation. Discuss the accuracy of this

assertion (a) for the globe as a whole, in light of Problem 15.1, and

(b) inside regions of organized deep convection, in light of the discus-

sion in Sec. 9.5.1.

Derive expression (15.2.2) for conservation of moist static energy.

Relate the change of moist static energy E -

cpT + l~r + d~

for an air

parcel to its change of equivalent potential temperature 0e.

Consider an air parcel that moves equatorward at sea level, ascends

adiabatically in the tropics, returns poleward at a constant altitude of

15 km, and then descends in the extratropicsmas symbolized by the

idealized thermodynamic circuit in Fig. 6.7. If heat transfer is restricted

to upper and lower legs of the circuit and if, along the lower leg, the

514

15 The General Circulation

parcel's initial temperature and mixing ratio in the extratropics are

250 K and 1 g kg -1, respectively, whereas it assumes a temperature and

mixing ratio in the tropics of 300 K and 20 g kg -1, calculate the radiative

heat transfer that occurs along the upper leg of the circuit.

15.6. Demonstrate that no work is performed on the atmosphere as a whole.

15.7. Show that the pressure averaged over an isentropic surface ff is pre-

served under an adiabatic rearrangement of mass.

15.8. Discuss why uniformly heating an atmosphere increases its store of total

potential energy, but makes none available for conversion to kinetic

energy.

15.9. Show that the integration limit p - 0 in (15.15) vanishes.

15.10. Derive expressions (15.19) for available potential energy.

15.11. Consider the following thermal structure, representative of zonal-mean

stratification in the troposphere:

T(th, p)=

T(p)-

T'(th, p),

where

T(p)= T0-rHln --p- + P0 '

T,(ch p ) _ A T [sinch la ( P ) A__~~ ( p )

' To- To'

and an overbar denotes global average. For T O - 268 K, F - 4 Kkm -1,

H = 8 km, AT = 70 K, 4,0 - 45 ~ and P0- 1000 mb, (a) plot the ther-

mal structure and zonal wind as functions of latitude and ~ -

ln(po/p),

(b) calculate the available potential energy d of the atmosphere, and

as a function of ~.

15.12. For the atmosphere in Problem 15.11, determine the fraction of total

potential energy that is available for conversion to kinetic energy.

15.13. For the atmosphere in Problem 15.11, determine a characteristic eddy

velocity if thermal structure is driven adiabatically into barotropic strat-

ification and if the kinetic energy produced is concentrated between

4-30 ~ and +60 ~ latitude. What conclusion can you draw?

15.14. Show that E89 is the appropriate power of Ekman number in which

equations (15.21) should be expanded.

15.15. Verify that equations (15.21) at

O(E89

have the same form as (15.23).

15.16. For the spherical atmosphere treated in Sec. 15.2, (a) verify that (15.33)

describes the

O(E89 )

behavior inside the lower Ekman layer and (b) ver-

ify that (15.39) describes the

O(E89 )

behavior inside the upper Ekman

layer.

Problems

515

15.17. Show that the temperature profile

To(b) - Too - 6Tsin 4 ~b

15.18.

15.19.

15.20.

15.21.

15.22.

15.23.

15.24.

makes the solution in Sec. 15.2 valid up to the equator.

Observed zonal-mean flow in the troposphere is easterly at low lat-

itudes and westerly at middle and high latitudes (Fig. 1.8). Surface

drag must then represent a sink of easterly momentum at low latitudes,

which is equivalent to a source of westerly momentum there, and a

sink of westerly momentum at middle and high latitudes. On average,

the atmosphere's angular momentum remains constant, so the Preced -

ing source and sink of momentum must be compensated by a poleward

transfer of westerly momentum. (a) Demonstrate that transport of an-

gular momentum by the zonal-mean Hadley circulation produces a

momentum flux fi-~ of the correct sense to accomplish this transfer of

westerly momentum. (b) Not all of the momentum transfer required to

maintain equilibrium is accomplished by the zonal-mean Hadley cir-

culation. The remainder occurs through large-scale eddy transport. In

terms of horizontal phase structure and group velocity, describe the

meridional propagation of Rossby waves needed to accomplish the re-

maining momentum transfer between tropical and extratropical regions.

The length of day can vary through exchanges of momentum between

the atmosphere and solid earth. (a) Estimate the velocity fluctuation

over the equator corresponding to observed fluctuations in the length

of day of order 10 -3 s, if the earth has a mean density of order 5.0 x

103 kg m -3 and if the atmosphere responds through uniform changes of

angular velocity. (b) More generally, where would velocity fluctuations

most effectively introduce changes in the length of day?

Describe how sloping convection lowers the center of gravity of air to

release available potential energy.

Discuss how horizontal rearrangement of air by baroclinic eddies drives

thermal structure toward barotropic stratification.

Relate heat transfer inside convective and cloud-free regions to the

distribution of surface pressure in the tropics.

(a) Use observed distributions of precipitation rate (Fig. 9.38) and

total precipitable water vapor (Fig. 1.16) to calculate a characteristic

timescale for the column abundance of water vapor in the tropics.

(b) Discuss this timescale in relation to the efficiency of dehydration

inside individual convective cells, their fractional coverage, and the

efficiency with which water vapor is produced at the earth's surface.

Describe the disturbed Walker circulation and trade winds in the Pacific

anticipated during E1 Nifio, when convection is found near the dateline.

516

The General Circulation

15.25.

15.26.

15.27.

15.28.

The variation with latitude of planetary vorticity (i.e., /3) is difficult

to simulate directly in the laboratory. Use the vorticity budget for an

incompressible fluid to construct an analogue of /3 in terms of the

variation of fluid depth inside a rotating cylindrical annulus.

The height of convective towers in the tropics is controlled by the

convective available potential energy (CAPE) of surface air (Sec. 7.4.1),

which in turn is controlled by fluxes of sensible and latent heat from

the surface. A surge of extratropical air into the tropics produces the

temperature profile

T(z)= I T~ z < z~

Tr z >_ zr,

where

T o

= 290 K, F= 7.5

Kkm -1,

z T

12 km, and Tr = 200 K.

(a) Calculate the CAPE of surface air if, through contact with an iso-

lated landmass, it attains a temperature of 300 K and a mixing ratio of

25 g kg -~. (b) As in part (a) but if the air is dry. (c) Estimate the height

to which convection would develop under the conditions of parts (a)

and (b).

In terms of the first law, hydrostatic equilibrium, and the budget of hor-

izontal vorticity, (a) describe the development of stratification over a

continental and neighboring maritime region in the tropics under mon-

soonal conditions and (b) describe how the accompanying thermally

direct circulation spins up.

Many components of the global energy budget, like convection, cloud

albedo, and fluxes of sensible and latent heat, vary diurnally. Discuss

how systematic diurnal variations of such properties contribute to the

time-mean energetics illustrated in Fig. 1.27.

Chapter 16

Hydrodynamic Instability

Wave propagation is supported by a positive restoring force, which opposes

air displacements by driving parcels back toward their undisturbed positions

(Chapter 14). Under certain conditions, the sense of the restoring force is re-

versed. Air displacements are then reinforced by a negative restoring force,

one which accelerates parcels away from their undisturbed positions. Instabil-

ity was encountered earlier in connection with hydrostatic stratification (Chap-

ter 7). If distributions of temperature and moisture violate the conditions for

hydrostatic stability, small vertical displacements produce buoyancy forces that

accelerate parcels away from their undisturbed positions. Unlike the response

under stable stratification, this reaction leads to finite displacements of air.

Fully developed convection then drives the stratification toward neutral stabil-

ity by rearranging air.

Two classes of instability are possible.

Parcel instability

follows from re-

inforcement of air displacements by a negative restoring force, such as that

occurring in the development of convection.

Wave instability

occurs in the pres-

ence of a positive restoring force, but one that amplifies parcel oscillations

inside wave motions. Unstable waves amplify by extracting energy from the

mean circulation (e.g., from available potential energy associated with baro-

clinic stratification and vertical shear; Chapter 15). Strong zonal motion that

results from the nonuniform distribution of heating and geostrophic equilib-

rium makes this class of instability the one most relevant to the large-scale

circulation. Like parcel instability, it develops to neutralize instability in the

mean flow, which it achieves by rearranging air.

16.1 Inertial Instability

The simplest form of large-scale instability relates to the inertial oscillations

described in Sec. 12.4. Consider disturbances to a geostrophically balanced

zonal flow K on an f-plane. If the disturbances introduce no pressure pertur-

bation, the total motion is governed by the horizontal momentum balance

dt fv-O,

(16.1.1)

dt + f(u

- ~) = 0, (16.1.2)

517

518 16 Hydrodynamic Instability

where geostrophic equilibrium has been used to eliminate pressure in favor

of ~.

Since a parcel's motion satisfies

(16.1.1) implies

(16.2)

v- dt'

dt = f "

(16.3)

Integrating from the parcel's initial position Y0 to its displaced position Y0 + Y'

gives

u(yo + y') - -a(yo) = fy'.

To first order in parcel displacement, this can be expressed

d-~y,

u(Yo) + -~y - -u(yo) = fY'

OK) y,

(u - ~)ly0 -

- 7y = O.

Then incorporating (16.1.2) yields

d2y '

dK) y'

et---r + f f - Uy = o.

(16.4)

f+ st= f (16.5)

has sign opposite to the planetary vorticity f. Displacements then amplify

exponentially and the zonal flow is

inertially unstable.

These circumstances

make the specific restoring force

f(f + ()y'

negative, so the ensuing instability

is of the parcel type. Because f + s r usually has the same sign as f, the criterion

for inertial instability is tantamount to the absolute vorticity reversing sign

somewhere.

Inertial instability does not play a major role in the atmosphere. Extratrop-

ical motions tend to remain inertially stable, even locally in the presence of

synoptic and planetary wave disturbances. However, the criterion for inertial

instability is violated more easily near the equator, where f is small. Evidence

of inertial instability exists in the tropical stratosphere, where horizontal shear

If the mean flow is without shear, (16.4) reduces to a description of the

inertial oscillations considered previously. In the presence of shear, ' displace-

ments either oscillate, decay exponentially, or grow without bound. The system

possesses unstable solutions if the absolute vorticity of the mean flow

16.2

Shear Instability

519

flanking the strong zonal jets (Fig. 1.8) can violate the criterion for inertial

stability.

16.2 Shear Instability

More relevant to the large-scale circulation is instability associated directly

with shear. Shear instability is of the wave type, so it requires a more involved

analysis than that applying to an individual parcel. Like the treatment of wave

motions, describing shear instability requires the solution of partial differen-

tial equations that govern perturbation properties. Closed-form solutions can

be found only for very idealized profiles of zonal-mean flow K(y, z). How-

ever, an illuminating criterion for instability, due originally to Rayleigh, can be

developed under fairly general circumstances.

16.2.1 Necessary Conditions for Instability

Consider quasi-geostrophic motion on a beta plane in an atmosphere that

extends upward indefinitely and is bounded below and laterally at y - +L

by rigid walls. Disturbances to the zonal-mean flow K(y, z) are governed by

first-order conservation of potential vorticity (14.75)

DQ'

where

D/Dt - 3/3t + -~3/3x,

-- -]- 1)'/~ e --" 0 (16.6.1)

la (fff ,~g,')

(2'- v2q/+ (16.6.2)

po ~ ~P~ '

o~2~ 13 (f2 3~)

~ Oy 2 PO Oz -~ Po -~z

- 3y'

(16.6.3)

and z denotes log-pressure height. Requiring vertical motion to vanish at the

ground (which is treated as an isobaric surface) gives, via the thermodynamic

equation and thermal wind balance, the lower boundary condition

=0 z-0. (16.6.4)

Dt \ 3z J 3z 3x

Physically meaningful solutions must also have bounded column energy, which

provides the upper boundary condition

finite energy condition

z ~ oo. (16.6.5)