Salby M.L. Fundamentals of Atmospheric Physics

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500 15 The General

Circulation

O(E ='l " ) "\

...........

I; ,~;iz;--~;;cz;

+ O~)(z)

............ .~. ............. ', ....

O(E =a') ~\

O(z)= g (~ / ~'

j. / /

+ E 2 U ~ /

~// / J. 1

/ )

:; iiI?I ....................

/ (a)

CI (b)

/ _ ~ ..

.. __j

Too

Too- 81"

(d) To(~~

.......................

0 r 90 ~

Figure 15.4 Zonally symmetric circulation in a Boussinesq spherical atmosphere of depth H

that is driven thermally by imposing different temperatures along its rigid lower boundary and

upper free surface. Motion is valid in the limit of fast rotation and small friction, which makes the

Ekman number E

small. The solution accurate to

O(E89

): ~ = v -(~ + E89 -(1), is characterized by

a strong zonal jet, with meridional motion confined to shallow boundary layers along the bottom

and top. (a) Zonal motion. (b) Meridional motion. (c) Streamfield in a meridional cross section.

(d) Meridional profile of imposed temperature. Adapted from Charney (1973) with permission of

Kluwer Academic Publishers.

15.2 Heat Transfer in an Axisymmetric Circulation

501

At O(E), the zonal momentum equation reduces to

--f"v -(2) -- K~

a~/(o)

o~Z 2

= 0. (15.36)

Then continuity implies

o~(2)

--" 0~

o~z

so ~(2) is independent of height in the interior and equal

m(2)(H). Air

expelled from the upper boundary layer drives a gentle

O(E)

subsidence in

the interior, which must be absorbed in another boundary layer at the bottom.

The lower boundary is a rigid surface, so motion there has the form of a

standard Ekman layer. Introducing the stretching transformation

z = E~ ~: (15.37)

inside the lower boundary layer magnifies sharp changes to make viscous

terms comparable to others in the momentum balance. Then, at

O(E89

the

momentum equations reduce to the Ekman balance

-f-O~ 1) = K ~

a~:2 , (15.38.1)

f-U(a 1) --

K ~

Osc2 (15.38.2)

with the geostrophic contribution again automatically satisfied by ~1). Requir-

ing the total solution accurate to

O(E89

to satisfy (15.21.6) and to reduce to

the inviscid behavior outside the boundary layer yields

-'l" U(a 1) -- 0 V(a 1)

= 0 ~: - 0 (15.38.3)

~1) ,~, 0

V(a 1) 0 (15.38.4)

The boundary value problem (15.38) has solution

~1)(~:) _ _~(gl)e-r~ cos y~:,

~(al)(~ :) = ~(~l)e-r~: sin y~:

(15.39.1)

(15.39.2)

(Figs. 15.4a and b), which is analogous to the Ekman structure in Fig. 13.3.

Because ~ vanishes in the interior, meridional mass transfer inside the

lower boundary layer must be exactly compensated by meridional mass transfer

502 15

The General Circulation

inside the upper boundary layer. This requires ~Cal) to satisfy the condition

fo fo

E 9(al)ds r - -E 9(al)ds r

KR 3T o

= EaHf---5 d--~"

(15.40.1)

Thermal Wind (15.24) then implies

K(ga)

= R dT o

(15.40.2)

aHyf a~

for the

O(E89 )

geostrophic motion, which completes the solution to this order.

Figure 15.4 illustrates the motion for the profile T0(~b ) = Too- 6T sin4 ~b

(Fig. 15.4d). This temperature structure may be thought to result from the dis-

tribution of radiative heating (Fig. 1.29c) and is sufficiently flat at low latitude

for the solution to remain valid up to the equator. The motion characterizes

a zonally symmetric Hadley cell. Air heated near the equator rises, gradually

spirals poleward in a strong zonal jet along the upper surface, sinks in the ex-

tratropics, and then returns along the lower boundary. Because motion in the

interior is zonal and parallel to isotherms, meridional heat flux is concentrated

inside the boundary layers, which have thickness

O(E 1)

and v =

O(E89

f~ increases, E --+ 0. The zonally symmetric circulation then becomes increas-

ingly geostrophic, with meridional motion confined to shrinking Ekman layers

along the walls in which poleward heat transfer collapses like E. Hence, ro-

tation acts to choke the poleward transfer of heat by the zonally symmetric

circulation.

In the atmosphere, the zonal-mean meridional circulation (Fig. 15.5) qual-

itatively resembles the idealized Hadley cellmbut only in the tropics. Rising

motion is found near the equator in the summer hemisphere, coincident with

the Inter Tropical Convergence Zone (ITCZ) and organized latent heat re-

lease (Fig. 9.38). This is compensated almost entirely by sinking motion in the

winter hemisphere. Strong meridional motion appears near the tropopause

and surface, but only equatorward of 30 ~ Like subsidence, that ageostrophic

motion is found chiefly in the winter hemisphere, where the meridional tem-

perature gradient is steepest (15.40).

The observed Hadley cell can, in fact, be understood from considerations

of angular momentum conservation like those discussed at the beginning of

this chapter (see, e.g., James, 1993). Air moving poleward in the upper tro-

posphere accelerates via the Coriolis force to produce strong westerlies (see

Fig. 15.8a later in this chapter). This acceleration is observed, however, only

out to the poleward edge of the Hadley cell, which happens to coincide with

the subtropical jet (compare Fig. 1.8). At higher latitudes, the thermally di-

rect Hadley cell in Figure 15.5 is replaced by a thermally indirect

Ferrell cell,

15.3 Heat

Transfer in a Laboratory

Analogue

503

200 -

E 400-

UJ 0 -2

2 -2

n-U J00

600L.

800

looo ~

-80-70-60-50-40-30-20-10 0 10 20 30 40 50 60 70 80

LATITUDE (Degs)

Figure

15.5 Zonal-mean meridional circulation observed during December-February.

Adapted from Peixoto and Oort (1992).

in which air sinks at low latitude and rises at high latitude. 3 Contrary to the

Hadley cell, which is driven thermally by heating at low latitude, the Ferrell

cell is driven mechanically by zonally asymmetric eddies that develop along

the poleward flank of the subtropical jet and dominate the extratropical cir-

culation.

15.3 Heat Transfer in a Laboratory Analogue

Insight into why the observed meridional circulation differs from the zonally

symmetric model is provided by laboratory simulations. A rotating cylindrical

annulus (Fig. 15.6) is driven thermally by imposing high temperature along its

outer surface, which represents the equator, and low temperature along its in-

ner surface, which represents the pole. Figure 15.7 shows the motion observed

in the rotating frame, as a function of its rotation rate 1~. At slow rotation,

the motion is zonally symmetric and has the form of a thermally direct Hadley

cell. Fluid along the surface gradually spirals inward in a zonal jet, converges,

and then sinks along the interior wall. As ~ increases, fluid moving radially

experiences an intensified Coriolis force that deflects the motion increasingly

parallel to isotherms and spins up a strong zonally symmetric jet in the inte-

rior. Heat transfer then becomes concentrated inside the Ekman layers along

the walls, diminishing like E.

3This thermally indirect motion is actually an artifact of the Eulerian mean along latitude

circles, disappearing in other representations (see, e.g., Holton, 1992). For this reason, the Ferrell

cell is not the principal mechanism for meridional transport in extratropical regions.

504 15 The General Circulation

<3=

q>O

Figure 15.6 Schematic of a rotating cylindri-

cal annulus that is driven thermally by imposing

high temperature along its outer surface and low

temperature along its inner surface.

~!iiiiii~ ....... ~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiili ....

d e

i ~ ........

~"~"~i ~:!i!ili i i~i~'~' ......

ii ~':~ ..:,i ~' .....

Figure 15.7 Surface flow pattern in the rotating cylindrical annulus for fixed temperature

contrast and as a function of increasing rotation rate. Adapted from Hide (1966).

15.3

Heat Transfer in a Laboratory Analogue

505

Beyond a critical rotation rate, zonally asymmetric waves appear. These

waves develop from a steep temperature gradient that forms in the interior

when radial heat transfer by the zonally symmetric circulation is choked by

rotation. The zonally symmetric stratification is then strongly baroclinic, pos-

sessing available potential energy. By thermal wind balance (12.11), the ra-

dial temperature gradient is accompanied by strong vertical shear, which ren-

ders the zonally symmetric flow

baroclinically unstable

(Chapter 16). Unsteady

disturbances then amplify by extracting available potential energy from the

zonal-mean state and converting it to eddy kinetic energy. Baroclinic waves ac-

complish this by transferring heat radially in

sloping convection.

Warm inward-

moving fluid overrides heavier fluid and ascends, whereas cool outward-moving

fluid undercuts lighter fluid and descends (compare Fig. 15.3). The result is

net heat transfer inward. This asymmetric heat transfer occupies much of the

interior, so it is far more efficient than the shallow

O(E)

heat transfer of the

zonally symmetric circulation that has been confined along the walls by rota-

tion. Although more complex, sloping convection likewise lowers the overall

center of gravity to release available potential energy.

The wavelength of the dominant baroclinic disturbance decreases with in-

creasing rotation rate (Fig. 15.7b-e). In certain ranges of [l, the annulus cir-

culation resembles observed flows in the troposphere. The pentagonal struc-

ture in Fig. 15.7e is similar to patterns observed in the Southern Hemisphere

(Fig. 2.10). Because its storm track is almost zonally symmetric, cyclones of-

ten appear uniformly spaced about the pole. At sufficiently fast rotation, the

dominant wavelength becomes small enough to make the wave itself unstable. 4

Wave motion then breaks down into isolated vortices that transfer heat effi-

ciently by rolling up and eventually mixing fluid across the annulus. Behavior

similar to that in Fig. 15.7 occurs if the imposed temperature contrast ex-

ceeds a critical value, which likewise renders the zonally symmetric circulation

baroclinically unstable.

In the troposphere, nonuniform heating (Fig. 1.29c) continually makes

the zonal-mean stratification baroclinic, producing available potential energy.

Baroclinicity is strongest at midlatitudes, between regions of radiative heat-

ing and cooling. Unstable eddies develop on the strong vertical shear of the

zonal-mean jet, where they transfer heat poleward through sloping convection.

Compared to the troposphere, heating in the stratosphere is fairly uniform.

Therefore, baroclinicity remains weak, as does the potential energy available

to generate baroclinic eddies.

By transferring heat poleward, baroclinic motions along with the Hadley

circulation make the general circulation of the troposphere behave as a heat

engine (Fig. 6.7). Air is heated at high temperature while it is near the equator

4Vertical shear of the wave reinforces that of the zonal-mean flow, which makes shear strong

enough locally to render the motion baroclinically unstable. A parallel exists in the Northern

Hemisphere, where planetary waves reinforce zonal-mean shear in the North Pacific and North

Atlantic storm tracks to provide conditions favorable for the development of baroclinic eddies.

506

15 The General Circulation

and surface, whereas it is cooled at low temperature after being displaced

upward and poleward. The second law then implies that an individual air

parcel performs net work when executing a thermodynamic cycle. Kinetic

energy produced in this fashion maintains the circulation against frictional

dissipation. About half of the organized kinetic energy of the troposphere

is dissipated inside the planetary boundary layer through turbulent mixing

(Sec. 13.1.2). The remainder is dissipated in the interior through turbulence

generated by dynamical instability (Sec. 14.7) and convection.

15.4 Tropical Circulations

At low latitude the Coriolis force is weak, which allows the circulation to

possess a greater contribution from divergence and be thermally direct. The

relatively flat temperature distribution (Fig. 1.7) implies little available po-

tential energy to drive large-scale motion. Instead, latent heat release inside

organized convection provides the primary source of energy for the tropical

circulation, which in turn is derived from evaporative cooling of the oceans.

Precipitation inside the ITCZ (Fig. 9.38) exceeds evaporation directly be-

neath it. Thus, moisture must be imported into convection from surrounding

areas. Much of this converges in the easterly equatorward

trade winds

near

the surface (Fig. 15.8b). The trades comprise the lower branch of the zonal-

mean Hadley cell in Fig. 15.5 and develop when equatorward-moving air is

deflected westward by the Coriolis force. Air subsiding from the upper branch

of the Hadley cell dries the troposphere and stabilizes it through mechanisms

discussed in Sec. 7.4.4. By inhibiting convection, subsidence in the Hadley

circulation maintains the deserts common at subtropical latitudes.

In addition to the zonal-mean Hadley cell, the tropics contain two other

classes of circulation. These are also thermally direct, but zonally asymmetric.

Monsoon circulations are established by horizontal gradients of surface tem-

perature, which vary annually. During summer, subtropical landmasses such

as India and northern Australia become warmer than neighboring maritime

regions. 5 During winter, the horizontal temperature gradient reverses, with

continents becoming colder than the surrounding maritime regions.

Surface heating during summer favors enhanced convection over land,

which is fueled by moist inflow at low levels. Latent heat release then warms

a deep layer, which leads to increased thickness between 1000 and 100 mb by

the hypsometric relationship (Fig. 1.30). Isobaric surfaces that have been de-

pressed in the lower troposphere introduce a surface low over land, which re-

inforces convection through frictional convergence inside the boundary layer.

5 Higher temperatures over land result from the comparatively small heat capacity of soil and

its immobility, wherein only diffusion is available to transfer heat away from the surface. By

comparison, water has a much greater heat capacity and can efficiently transfer heat away from

the surface through fluid motion. Both effects make seasonal swings in ocean temperature much

smaller than corresponding swings over continents.

0 ,--

(s6ec])

qC]NIIIV-I

~ ~=

,JD ~ ~D

~ o

~88Z~

*-~ (D

1=~ ,-~ ""

~.~

~ ~

= .

"~ .~ ~ ,.

~~ ~o

508 15 The General Circulation

The influx of water vapor then increases

0 e

to make the lower troposphere

potentially unstable (Fig. 7.12). Convergence, which extends well above the

boundary layer, also reduces the moist static stability through mechanisms

described in Sec. 7.4.3. These changes provide conditions favorable to deep

convection by allowing air to penetrate more easily to the level of free con-

vection (LFC). Low-level air that converges into the convective region spins

up cyclonically to conserve potential vorticity (e.g., over northern Australia in

Fig. 15.8b). Convected to the tropopause, that air then diverges from the con-

vective region and spins up anticyclonically (e.g., also over tropical Africa and

South America in Fig. 15.8a).

The onshore circulation during the summer monsoon is analogous to a

seabreeze circulation, which is likewise generated by horizontal gradients of

heating. Both may be understood in terms of solenoidal production of horizon-

tal vorticity, which occurs under baroclinic stratification (Fig. 15.9). Nonuni-

form heating introduces a horizontal variation of temperature that drives

isochoric surfaces (a = const) out of coincidence with isobaric surfaces. A

solenoidal production term proportional to

V p x Va

then appears in the

L ~

Z ~

p = const.

-- -- o~ = const.

- ~ ~

_. ....._ _..._.

~ ~ ~

~_....."

------ ......,.......... ....

Warm

Figure 15.9 Schematic of a thermally direct monsoon circulation that is established through

solenoidal production of horizontal vorticity under baroclinic stratification. Heating over land

expands the vertical spacing of isobaric surfaces (solid lines), whereas it draws down isochoric

surfaces (dashed lines) of larger a from aloft. The reverse occurs from cooling over ocean.

The nonuniform distribution of heating drives isochoric surfaces out of coincidence with isobaric

surfaces to produce baroclinic stratification. Solenoidal production of horizontal verticity Vp x Va

then spins up an onshore circulation.

15.4 Tropical Circulations

509

budget of horizontal vorticity, analogous to the one in the budget of vertical

vorticity (12.36). This spins up horizontal vorticity as an onshore circulation,

with rising motion over the heated region and sinking motion over the cooled

region.

During winter, the monsoon circulation is reversed. Land, colder than

neighboring water, then cools the overlying air to produce subsidence. This

process inhibits convection by drying the environment with air from aloft,

which in turn reduces 0e and stabilizes the stratification. The compressed

thickness between 1000 and 100 mb forms a surface high (Fig. 1.30). Frictional

divergence at low levels then drives anticyclonic motion (e.g., as appears over

the south Indian ocean in Fig. 15.8b) and subsidence, which strengthens the

moist static stability of the lower troposphere (Sec. 7.4.3).

The other class of circulation important in the tropics is the Walker cell,

which is depicted in Fig. 1.30. A zonally asymmetric overturning in the equato-

rial plane, the Walker circulation is characterized by rising motion over regions

of heating and sinking motion to the east and west. The Pacific Walker circu-

lation is forced by latent heating in the western Pacific. Low surface pressure

in the western Pacific supports deep convection, in the same way it does mon-

soons. Strong surface easterlies along the equator fuel convection with mois-

ture that has been absorbed by air during its traversal of the western Pacific,

where sea surface temperature (SST) approaching 30~ provides an abundant

source of water vapor (Fig. 5.1).

Latent heating that drives zonal overturning in the Walker cell also drives

meridional overturning, which is likewise zonally asymmetric. Figure 15.10a

shows surface motion forced by a steady heating anomaly (shaded area) over

the equator. Surface easterlies along the equator that converge into the heating

resemble the Pacific Walker cell and are mirrored in the upper troposphere

by westerlies (Fig. 15.10b). Flanking ascending motion inside the heating are

two cyclonic gyres in the subtropics, which reinforce inflow along the equator

to the west. Those gyres represent a Rossby wave mode that is trapped about

the equator (see, e.g., Gill, 1982). They are evident in the 200-mb streamfield

(Fig. 15.8b), along with zonal flow to the east that comprises the Walker cell.

The Pacific Walker circulation changes markedly on interseasonal

timescales.

El Niho,

which occurs with a frequency of 3 to 5 years, is

marked by warming of SST in the eastern and central Pacific (Fig. 15.11). 6

The equatorial Pacific is especially sensitive to temperature perturbations

because SST exceeds 25~ there. The exponential dependence in the Clausius-

Clapeyron equation (4.39) then implies that small variations of SST produce

large changes in evaporation and latent heat transfer. When the tempera-

ture anomaly in Fig. 15.11 is introduced, the warmest SST and convection

shift eastward from their normal position (Fig. 9.38) to the central Pacific.

6The name refers to "the child" because warm waters off the coast of Peru, which signal its

onset, appear around Christmas.