James I.N. Introduction to Circulating Atmospheres

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7.1 Zonal variations in the tropics

209

(a)

~, ,.,,,

: :

20.0

(b)

• x xj "-i

—••'««,

-->-

i -•

•

;--

, , .

- .,) i '••;.

: -

,-± , xV^^-

-,-•'.

-wICl

'-. !

10.0

•^"••.... '

' r

^ *

Fig. 7.1. Vectors of the time mean horizontal wind, v, for the DJF season. Based

on six years of ECMWF data for the DJF season, (a) At 15kPa, near the tropical

tropopause: the sample vector represents

ms"

(b) At 85kPa, just above the

atmospheric boundary layer. The sample vector represents

ms"

is near the tropopause in the deep tropics. There, essentially zonal winds

are dominant in both seasons. This is one immediate reason why the eddies

do not contribute significantly to meridional transports in the tropics: the

meridional part of the eddy wind field is very small. However, the eddy zonal

wind is by no means small; there are a number of significant extrema of u

distributed around the globe. In the DJF season, there are strong easterlies

over Indonesia and strong westerlies over the eastern Pacific and over the

Atlantic. The zonal mean zonal wind therefore involves a good deal of

cancellation; in fact, at these levels, the mean wind is westerly (see Fig. 4.1).

In the JJA season, the zonal component again dominates, but the pattern of

maxima is quite different. There is a region of strong easterly flow stretching

from Africa in the west to Indonesia in the east. Elsewhere, there are weaker

easterlies, with very little trace of the patches of westerly flow seen in DJF.

Turning now to the low level wind fields, the same dominance of the zonal

over the meridional components of the wind is clear. In DJF, the wind is

mainly easterly, with regions of convergence more or less corresponding to

regions of divergence at 15kPa and vice versa. In JJA, a similar pattern is

found over most of the tropics. However, dramatically different fields occur

over the Indian Ocean. Here we see strong meridional flow, in a broad

210

Three-dimensional aspects of the global circulation

(a)

—*

(b)

* /

.»

••* / / • •

..\S

^si

i:;

-^.;....^,.^..^,H-.

- - r - -S'.«- ^

- ^

/ / / r--

•>->.»

• • -

•-.-.-•.' "r--:

20.0

4 >-

10.0

Fig. 7.2. As Fig. 7.1, but for the JJA season.

band parallel to the African coast, which links easterly flow in the southern

subtropics and westerly flow over India. This flow is part of the Asian

monsoon system, a planetary scale system which is sufficiently important to

merit separate discussion in the next section.

Comparing the flows at 85 and 15kPa, there is a strong tendency for low

level convergence to be associated with upper level divergence, and vice versa.

This is consistent with midlevel ascent and descent linking the two levels, and

suggests that the tropical response to localized heating consists of a set of

overturning circulations in the longitude-height plane. Such circulations are

sometimes called 'Walker circulations', and were first identified by comparing

the surface meteorological records at Darwin, north-east Australia and Tahiti

in the mid-Pacific.

In attempting to account for these observations, it has to be conceded that

elegant theories of tropical dynamics comparable to the quasi-geostrophic

and related formulations for midlatitudes do not exist. A group of theories,

based on a linearization around a resting atmosphere, represents the simplest

approach, and will be outlined in this section. But it is not straightforward to

relax the very restrictive assumptions of such a theory. The alternatives soon

involve rather elaborate numerical solutions of the full dynamical equations.

Scaling arguments show that temperature fluctuations will be very much

7.1 Zonal variations in the tropics 211

smaller in the tropics than in midlatitudes; in fact

A0 U

, .

A0 fUL, .„ . «

-5-

—77

(tropics), — « -—-(midlatitudes), (7.1)

u gh u gti

where if is a typical height scale and L is a typical horizontal scale; Ad/9

is around 10~

for the tropics and an order of magnitude larger in the mid-

latitudes. This means that horizontal temperature gradients in the tropics are

small so that horizontal advection cannot balance the large heating observed

in convecting regions. Instead, that heating must be balanced by vertical

advection, so that:

Strong ascent, of around 3cms

, is expected in the convecting regions,

where the heating may be as large as SKday"

. Throughout the rest

of the tropics, weak descent of around 0.3cms"

is needed to balance

radiative cooling. Continuity implies that these vertical motions will in turn

excite horizontal velocities. The character of the horizontal velocity field is

elucidated by considering the momentum equations.

The Coriolis force is zero on the equator, but varies rapidly with latitude.

In the dynamical equations, the Coriolis parameter can be approximated

by / = Py where P = 2Q/a. This approximation is sometimes called the

'equatorial P plane'. Making this approximation, the flow is linearized about

a motionless state in which atmospheric variables change only in the vertical

direction. The solution can then be separated into a height dependent part

and a part which depends upon the horizontal coordinates and time. The

horizontal part is governed by the linearized 'shallow water' equations, with

a suitable 'equivalent depth'. These equations may be written:

du dh' u

-p

= -

g 9

(7.3a)

Ot OX T

.3b)

^ = -hoV

• v

+ J - —. (7.3c)

Here, Eq. (7.3c) is the linearized continuity equation for a shallow layer

of incompressible fluid of depth ho; J represents a forcing which can be

related to the heating. Dissipation in the form of Rayleigh friction or

Newtonian cooling acting on perturbations has been included. The simplest

212 Three-dimensional aspects of the global circulation

interpretation of the equivalent depth ho is provided by considering an

incompressible atmosphere with constant Brunt-Vaisala frequency N and a

rigid lid at height H. Then the gravest vertical modes have pressure and

horizontal velocity perturbations which vary as cos(nz/H) and a perturbation

vertical velocity which varies as sin(nz/H). The equivalent depth is given

by:

^f-. (7.4)

Setting H to the depth of the tropical troposphere, around

km, a typical

equivalent depth is around 400

A more satisfactory way of obtaining the

same equations, but for a compressible atmosphere with variations of

JV,

etc.,

in the vertical, is to solve an eigenvalue problem for the vertical structure;

similar equivalent depths for the gravest vertical modes emerge. We will

restrict our attention to the gravest vertical modes which have a single

maximum of vertical velocity at midtropospheric levels, and zero vertical

velocity at top and bottom. This is justified because the vertical distribution

of tropical heating is expected to excite disturbances which project principally

on to such a structure.

Before considering the full forcing problem, it is worth considering some

of the wave motions which can take place in the tropical atmosphere. The

shallow water equations support gravity waves which propagate at speed

{gho}

But further large scale tropical modes are also possible, and these

dominate the large scale response to isolated heating maxima. The simplest

solutions to Eqs. (7.3a-c) are obtained when friction and heating are ignored,

and the meridional velocity is set to zero. The equations become:

du dh'

pyu = -g|£, (7.5b)

dh'

= -

5c)

This set is satisfied by solutions of the form:

u =

<*(y)f(x

- c

t), ti = a®(y)f(x - c

t). (7.6)

Substituting in, we find:

= ±(gh

)

1/2

, a -

/g)

1/2

(7.7)

7.1 Zonal variations in the tropics 213

and:

Equation (7.8) is simple to integrate, giving:

) (7.9)

The negative root for c

is not physically reasonable, since it would lead to

exponentially increasing zonal velocity perturbations away from the equator.

The choice of the positive root leads to nondispersive, eastward propagating

waves, which are confined to the region of the equator. The meridional scale

of these waves is (2c

//?)

1//2

or around

2000

km when the equivalent depth is

400 m. Their eastward phase speed is about

ms"

These trapped disturb-

ances are called 'equatorial Kelvin waves' and are an important component

of the response of the tropical atmosphere to localized thermal forcing.

A variety of further trapped equatorial wave solutions to Eqs. (7.3a-c)

can be found, with v

=£

0 in general. The general dispersion relationship is:

CoJ

OJ Co

where n is an integer. The meridional scale for all these trapped equatorial

waves is of order (CQ/2/?)

, but their frequencies, and hence their phase

speeds, vary substantially. The dispersion relationships are illustrated in

Fig. 7.3. The Kelvin wave dispersion relationship is also consistent with

Eq. (7.10) when n =

—1,

and this is also plotted on Fig. 7.3. For n > 1, there

exist high frequency equatorial gravity waves which propagate either to the

west or to the east. But there is also a group of low frequency 'planetary

waves' which propagate to the west; a typical phase speed is around 20ms"

for

400

These latter, along with the Kelvin wave, are an important

component of the tropical response to localized heating. The n = 0 mode

is known as the mixed Rossby-gravity wave. For large positive /c, these

waves are similar in structure to the eastward propagating gravity waves.

But for negative fe, they are slowly westward propagating modes, similar to

the planetary waves.

Now consider the flow forced by a localized heating maximum in terms of

these equatorially trapped waves. The problem is rather different from the

midlatitude problem discussed in Section 6.2. There, the response to forcing

was discussed in terms of Rossby waves which had zero phase speed with

respect to the forcing. In the equatorial case, the phase speeds of all the

waves shown in Fig. 7.3 are large compared to the typical tropospheric zonal

214

Three-dimensional aspects

the global circulation

-5

Zonal Wavenumber

Fig. 7.3. The dispersion relationship, Eq. (7.10),

for

equatorially trapped wavelike

solutions

to the

linear shallow water equations

on an

equatorial jS-plane.

The

frequency is given in units of Q and the wavenumber in units of a~

. The family of

curves

governed by the single parameter c

/(Qa) which took the value 0.134

for

this calculation.

flow speeds, and so

is pointless to seek wave solutions which are stationary

with respect

the forcing. Instead, solutions can

constructed

which

the zonal flow

taken

to be

zero,

but in

which

Kelvin wave propag-

ates eastward from

the

forcing region, while planetary waves propagate

westward. Steady solutions can

constructed

there

some dissipation

in the system,

that the

u, v

and

perturbations decay

on a

timescale

Then the disturbance decays away from the forcing region

on a

spatial

scale

L =

) where

dco/dk

the zonal component

the group

velocity of the waves. For the Kelvin waves, this scale is large. Taking

five

days,

we have

L =

26000 km. Hence the Kelvin wave emanating from

a single heating maximum would fill much

the tropics. The westward

extensions

the response would decay more rapidly;

typical

would

around 8000 km.

A typical solution

to the

forced problem

shown

Fig. 7.4.

As one

would expect, there

low level convergence into the heating region.

But

7.1 Zonal variations in the tropics

215

Fig. 7.4. The linear response to a localized heating maximum centred on the equator,

showing the low level wind vectors and contours of the pressure perturbation. The

upper level fields are, of course, simply the same but with the signs reversed. The

plot is based on a numerical solution of the linear shallow water equations; the

wind vectors have been arbitrarily scaled so the maximum is

ms"

this convergence is dominated by du/dx. The meridional winds are virtually

zero to the east of the heating region, consistent with the response there

being a Kelvin wave, and are generally small to the west of the heating.

Thus the major response consists of a zonally overturning 'Walker' cell, with

two cyclones on the north-west and south-west flanks of the heating region.

Any meridional temperature flux will be small, and will have opposite signs

at upper and lower levels. The important conclusion is that the localized

heating maxima noted in Fig. 3.8 will have little impact on the meridional

heat transport out of the tropics. Rather, the meridional heat tranports are

dominated by the mean heat transports. Localized heating will lead to zonal

overturning and associated zonal heat transports.

A similar solution can be constructed when the heating maximum is

centred away from the equator. The heating can be decomposed into a

part which is symmetric about the equator (as in Fig. 7.4) and a part which

is antisymmetric, and the response to each forcing superposed. Figure 7.5

illustrates such a solution. To the east of the forcing region, only the

symmetric part of the heating has any effect, and the solution is identical to

that of Fig. 7.3. To the west of the heating maximum, a large asymmetry in

the cyclones either side of the heating region develops, with the cyclone in the

same hemisphere as the heating maximum quickly dominating. This solution

has similarities to the JJA circulation in the Indian region, a point to which

216

Three-dimensional aspects of the global circulation

Fig. 7.5. As Fig. 7.4, but showing the surface pressure and surface wind vectors for

the case of a heating maximum located 10° to the north of the equator.

we will return in the next section. The meridional winds are stronger, with

the result that the zonal mean of the solution shows a significant Hadley

cell straddling the equator. The strongest ascent is in the vicinity of the

heating maximum, north of the equator, and the descent is largely to the

south of the equator in the 'winter' hemisphere. The results have the same

general pattern as the axisymmetric theory of Section 4.2, although, in this

linearized problem, the strong subtropical zonal winds associated with the

Hadley circulation are absent because there is no meridional advection of

angular momentum.

It is as well to conclude this section with a word of warning. The

vertical modal decomposition on which the theory of this section is based

involves linearizing around a state of zero motion. This is a very restrictive

assumption which, strictly, can only be justified for very light winds and

weak forcings. Such solutions do indeed have much the same qualitative

appearance as large scale disturbances observed in the tropical circulation.

But more elaborate nonlinear (and generally numerical) models have to be

used to establish the quantitative details of the response to large amplitude

forcing with general vertical structure. Although waves with structure like

the Kelvin and planetary waves discussed in this section are observed in the

troposphere, their phase speeds and detailed structure do not accord with

the simple theory. The effects of feedbacks between the large scale flow

and moist convection, and more realistic boundary layer processes, have

to be included. These matters are more properly left to a text on tropical

atmospheric dynamics for their fuller discussion.

7.2 Monsoon circulations 217

7.2 Monsoon circulations

One of

the

largest and most dramatic steady departures from zonal symmetry

in the tropics is the summer Asian monsoon circulation. The term 'monsoon'

strictly means any seasonal reversal in the circulation. However, the summer

monsoon over India and south-east Asia is such a dominant component of

the circulation, and is of such enormous human and economic importance

that it is often referred to simply as 'the monsoon'. The circulation is

primarily associated with changes in the distribution of heating between

summer and winter. But other effects, such as the feedbacks between large

scale circulation and the release of latent heat in cumulus scale convection,

and the influence of mountains also play an important role, making the

Asian monsoon a complex and still imperfectly understood component of

the JJA circulation.

The essential character of the monsoon is well illustrated by Fig. 7.2.

Throughout the JJA season, a strong anticyclone at 15kPa is centred near

India. The zonal extent of this anticyclone is truly enormous. Its influence

is clear over more than

90 °

of longitude, from North Africa to the western

parts of the Pacific Ocean. Its meridional extent is around

3 000

km.

To its

south, strong easterly winds are located near the equator; these make a very

significant contribution to the zonal mean easterlies during this season. The

low level winds, at

kPa for example, tend to exhibit a cyclonic circulation.

But the most prominent low level feature is the jet which crosses the equator

along the east African coast and, from there, across the Arabian Sea into

India. The heating was shown in Fig. 3.8(b); the intense maximum over

south-east Asia is one of the most prominent features of this plot. It is

largely associated with the seasonal precipitation in this part of the world

which lends the monsoon its significance for human activities.

The summer monsoon circulation is a thermally driven circulation which

arises primarily from the temperature differences between the hot contin-

ental areas in the summer northern hemisphere and the colder oceans in the

southern hemisphere. This is something of an oversimplification, though, as is

clear from the observations that comparable strong monsoonal circulations

do not feature over other subtropical continental areas such as northern

Australia or South America. There is a complex feedback between the

flow field and the heating, especially through the interaction between moist

convection and large scale flow, which is poorly understood but which is

undoubtedly involved in ensuring that the Asian monsoon is so much more

prominent than the monsoonal circulations over other continents. The special

orography of that particular area also modifies the circulation considerably.

218 Three-dimensional aspects of the global circulation

The result is that the monsoon is a particularly complex weather system. Its

quantitative description is beyond the scope of simple models and probably

requires the use of a full global circulation model to do it justice. As a

consequence, the brief discussion in this section will be mainly descriptive.

Consider, first, the situation shown in Fig. 7.6, in which a hot land mass

lies to the north of the equator, and cooler ocean lies to the south. The

asymmetric Hadley cell discussed in Section 4.2 provides a zonally symmetric

example of such an idealized monsoonal circulation. Low level cross equato-

rial flow is set up by the heating distribution. Under the combined influence

of pressure gradient forces, friction and Coriolis accelerations, air parcels

will have consistently negative relative vorticity, i.e. cyclonic in the southern

hemisphere and anticyclonic in the northern hemisphere. The convergence

of moist air at low levels in the heating region will lead to convection and

release of latent heat, thus giving a positive feedback which will reinforce

the monsoonal circulation. At the same time, the warm anomaly over the

continent implies, by thermal wind balance, increasing anticyclonic vorticity

with height, so that a strong upper anticyclone will overlie the low level 'heat

low'.

Now consider the effect on this circulation of a high mountain barrier

lying across the equator. Such a barrier is provided by the highlands of

east Africa, where the mean height of the continent rises to

1 — 2

km within

a short distance of the coast. The low level barrier provided by these

mountains means that any zonal component of the low level flow is blocked

at the equator. In an effect somewhat analogous to the formation of western

boundary currents in the ocean (see Section 10.5), the northward flow is

concentrated in a low level jet along the foot of the orography. The low

level east African jet is a prominent feature of Fig. 7.2(b). It dominates the

low level flux of mass across the equator in the JJA season. The strong low

level winds are also effective at evaporating water from the Arabian Sea,

strengthening the convection over the northern continent. This provides yet

another positive feedback which amplifies the monsoon circulation.

The Tibetan plateau and the Himalayan mountains also provide an import-

ant element in the observed monsoon, in the form of a total barrier to low

level meridional winds. The ascent and rainfall caused as the low level flow

encounters the mountain barrier both increases the strength of the heating

and confines it to the region south of the Tibetan plateau. At the same

time,

it has been suggested that the sensible heating, caused as sunlight is

absorbed by the elevated Tibetan plateau, provides a high level heat source

which strengthens the upper level anticyclone.

The qualitative arguments for the existence of the monsoonal circulation