James I.N. Introduction to Circulating Atmospheres

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4.3 More realistic models of the Hadley circulation 99

Fig.

4.11.

Schematic illustration of the effects of including moisture in the Held-Hou

Hadley cell model: (a) dry case; (b) some incoming energy evaporates water; (c) all

incoming radiation evaporates water.

100 The zonal mean meridional circulation

scale of the ITCZ is no more than 100km or so; mesoscale processes, which

lie beyond the scope of this book, are involved in determining this scale.

This reasoning suggests that the basic mechanism in the Held-Hou model,

namely, angular momentum mixing and thermal wind balance, gives a

general form of Hadley circulation which is insensitive to the details of

how the heat enters the system. The actual location of the ITCZ and its

strength will depend upon details of the tropical boundary layer, and the

fluxes of moisture out of the surface; these processes need to be represented

in GCMs and weather prediction models. Much of the flux of moisture into

the boundary layer is controlled by the sea surface temperature. Because of

the large thermal capacity of the ocean, this has a smaller seasonal cycle

than the land, and so the asymmetry of the circulation about the equator

is somewhat reduced by the moist processes. Figure 4.1 shows that it is

nevertheless quite substantial. We will return for a further discussion of

convection and heating processes on the large scale tropical circulation in

Chapter 7. Meanwhile, we will move to higher latitudes, and consider the

meridional circulation beyond the confines of the Hadley cell.

4.4 Zonal mean circulation in midlatitudes

Our study of the Hadley circulation of subtropical latitudes was simplified

by the observation that eddy fluxes are small equatorward of

30 °

of latitude.

So the circulation can be thought of as approximately axisymmetric at low

latitudes. But in the midlatitudes, the eddies are large and the fluxes of heat

and momentum that they carry are a major part of the global circulation. In

describing the zonal mean circulation at these latitudes, we must recognize

the role of the eddies explicitly. We then face a conceptual difficulty. We

mentally partition the flow into eddies and a zonal mean part which we

try to examine separately. But this distinction is artificial. The eddies

induce changes in the mean flow which in turn affect the distribution and

vigour of the eddies. In recent global circulation studies, a good deal of

effort has been devoted to isolating the effects of eddies from the effects

of other, genuinely axisymmetric, processes. In this section, we will discuss

a traditional approach to the effect of eddies on the mean flow. We will

suppose that the fluxes of heat and momentum associated with the eddies

are prescribed (for example, by reference to observations) so that we can

consider the response of the zonal mean flow to these forcings.

Our starting point is the quasi-geostrophic equation set, described in

Section 1.7. The quasi-geostrophic vorticity equation, using the pressure as

4.4

Zonal mean circulation

midlatitudes

101

vertical

coordinate,

may

written:

f4«)

|(»|.)

A=/|

,4.22,

where g

is the relative geostrophic vorticity v

—

, and

F = &\

—

schematic form of the friction term, with

and #"

the x- and ^-components

of acceleration due to friction respectively. Note that we will ignore the effects

of the Earth's curvature, save through the

(3v

term,

this stage. If we average

with respect

to the

zonal direction, we may write:

|g||

|;[^]

(4.23)

or, from continuity,

liW + fytt +

jjim-^

(4.24)

But

since

-[«]„,

we may

write:

} = 0.

(4.25)

Integrating with respect

to y,

and

determining

the

constant

integration

from

the

condition that d[u]/dt

= 0 in the

absence

friction, Coriolis

acceleration

dynamical fluxes, we have

quasi-geostrophic version

the

zonal mean momentum equation:

[u]+-^[vu]-f[v]

= [^

(4.26)

The momentum flux

[uv]

may be written

[u] [v]

[u*v*];

scaling arguments

such

those

Section

1.7

show that u*

[u]

whereas v*

[v],

that

the mean momentum flux may

neglected compared

the eddy

flux.

The

final form

the zonal momentum equation that we will use

therefore:

[u]t=M-[u*v*]

[*i\- (4.27)

where

the

subscripts

t and y

indicate differentiation. Alternatively, this

equation could have been derived

taking

the

zonal mean

Eq. (1.65a)

and using quasi-geostrophic scaling arguments.

The

first term

the

right

hand side represents changing

[u]

due

angular momentum conservation

by axisymmetric meridional motion;

the

second represents accelerations due

to eddy fluxes;

and the

last,

course, represents friction.

The thermodynamic equation

given

Eq. (1.71);

zonal averaging,

102

The

zonal mean meridional circulation

it reduces to

[0]t

= j[a>] + [2]. (4.28)

Once again,

[v]

has been neglected compared to

so that the poleward

eddy potential temperature flux dominates over the vertical eddy potential

flux or mean poleward flux. The vertical advection of the basic stratification

dominates the mean advection.

Equations (4.28) and (4.27) are linked through the zonal mean form of

the thermal wind equation. This is conveniently written in the form:

f[u]

h[0]

(4.29)

The

meridional circulation which is required to maintain a state of ther-

mal

wind balance despite the unbalancing tendencies of eddy temperature

and

momentum fluxes, friction and heating can be calculated from the set

Eqs.

(4.27),

(4.28) and

(4.29).

It is convenient to define a meridional

streamfunction xp

such that

where the sign convention has been chosen so that a maximum of

corres-

ponds to a direct circulation, with ascent at low latitudes (small y) and

descent (large y) at higher latitudes in the northern hemisphere. Then

multiply the momentum equation by / and differentiate with respect to p,

and multiply the thermodynamic equation by h before differentiating with

respect to y; subtract the two equations to eliminate the derivatives by

forming the balance condition:

This elliptic equation relates the mean meridional circulation to a number

of source terms. It can be solved if suitable boundary conditions on xp

are specified. We will apply

[v]

= 0 at bounding values of y (which may

be taken to be the equator and pole) and [co] = 0 at p = 0 and p = p

that is,

= constant (which may without loss of generality be taken to be

zero) along the meridional and vertical boundaries. The elliptic operator

means that a maximum in the source terms on the right of Eq. (4.31) is

associated with a minimum in the meridional streamfunction, that is, with

an indirect circulation in the northern hemisphere. Figure 4.12 illustrates

this.

Furthermore, the meridional streamfunction will reflect the larger

4.4 Zonal mean circulation in midlatitudes

103

Fig. 4.12. Schematic illustration of the solution of Eq. (4.31).

scale features of the source terms while smoothing out their smaller scale

structures.

Equation (4.31) is a diagnostic relationship. It states what meridional

circulations must take place in order to maintain thermal wind balance

when nonzero source terms are present. It is analogous to the co-equation

discussed in Section 1.7 and, indeed, may be derived from a zonal average

of the co-equation which is then integrated with respect to y.

Consider a simple application of this diagnostic relationship. Suppose

that friction and eddy fluxes may be ignored, leaving only the source term

associated with heating. The heating is taken to be positive for small y and

negative for large y

with a maximum of

—[£]

in midlatitudes. Hence, the

source term -(h/s

)[£]

will be negative throughout the midlatitudes. xp

will therefore have an associated maximum value in midlatitudes, implying

a thermally direct circulation in which air rises where the heating is large

and descends where it is small, as illustrated in Fig. 4.13. The interpretation

of this result needs care, however. The buoyancy terms have been scaled out

of the quasi-geostrophic set, so it is not a simple matter of warm air rising.

Rather, the heating generates horizontal pressure gradients which result in

meridional acceleration of the air. Continuity then implies ascent at low

latitudes and descent at high latitudes.

This meridional circulation accomplishes two things. First, the ascent at

low latitudes results in upward advection of potentially colder air, tending

to offset the temperature rises due to heating. A converse argument holds

at high latitudes. Thus, the temperature gradient increases by less than we

might expect if the atmosphere were unable to circulate. Second, the Coriolis

forces acting on the poleward moving air at upper levels impart a westerly

acceleration to the flow, while at low levels an easterly acceleration occurs.

In this way, the wind shear is increased so that a thermal wind balance with

the evolving temperature field is maintained.

104

The zonal mean meridional circulation

P,,

Heating

Cooling

7/777/77777777777777777/7/777/y

Fig. 4.13. The meridional circulation generated in response to a gradient of heating.

Fig. 4.14. The meridional circulation induced by surface friction.

As a second example, let us explore the effect of friction. In the midlatit-

udes,

surface winds are observed to be westerly. We expect that friction will

impart a strong easterly acceleration to the flow at low levels (large p) but

will be less effective at higher levels. Hence, the source term will be positive

and, this time, an indirect circulation will be forced. Figure 4.14 illustrates the

argument. In physical terms, a flow is induced from low to high latitudes at

low levels. The Coriolis force acting on this poleward flow produces westerly

acceleration, offsetting the easterly acceleration which results directly from

the friction. At the same time, descent causes warming in the tropics, while

ascent cools the higher latitudes. In this way, the temperature field is brought

back into thermal wind balance with the increased vertical wind shear at the

top of the boundary layer.

The friction induced circulation illustrates an important principle con-

cerning the role of boundary layer friction on the global circulation. To

the south of the jet core, where the vorticity associated with the zonal wind

is negative, the friction induces descent. To the north of the jet core, the

relative vorticity is positive and friction induces ascent. Such boundary layer

pumping is important in leading to the rapid spin up of the flow in the

4.4 Zonal mean circulation in midlatitudes 105

upper troposphere. For simple forms of the friction, analytic relationships

between the vertical pumping velocity at the top of the boundary layer and

the interior vorticity can be derived. For example, when the friction term is

of the form, Ku

, K being a constant 'eddy viscosity' coefficient, the vertical

pumping velocity is

1/2

. (4.32)

The resulting vortex compression or extension will dissipate relative vorticity

on a time scale (2H

/fK)

, H being a typical height scale (i.e., the tropo-

sphere depth). Such a boundary layer is called an 'Ekman layer'. It occurs

readily in laboratory experiments with rotating tanks of fluid in which the

boundary layer flow is laminar, but is a rather crude model of the turbulent

atmospheric boundary layer. However, even with more realistic parametriza-

tions of the boundary layer friction, a qualitatively similar pumping still

takes place. The Ekman boundary layer model provides a rationale for

the Rayleigh friction in the simple global circulation models which was

introduced rather arbitrarily in Section 2.4.

Now let us consider the role of the eddy flux terms in Eq. (4.31). The

poleward eddy temperature fluxes are largest at lower levels in the midlatit-

udes (see Section 5.2 for some examples). In the region of maximum flux, it

follows that

[v*0*]

must be negative. The temperature flux will therefore

contribute a positive source term to the right hand side of the circulation

equation, Eq. (4.31), and will force an indirect circulation. The result, shown

schematically in Fig. 4.15, is easily appreciated if the reader has followed

the argument given above for the effect of a gradient of heating. The eddies

tend to shift heat from low latitudes to higher latitudes. There will therefore

be a poleward gradient of heating associated with the eddies. An indirect

circulation is therefore required to reduce the upper level westerlies relative

to the lower level winds and so maintain thermal wind balance. At the same

time,

the vertical motions will offset the heating by the eddies.

The pattern of eddy momentum fluxes is more complex. The transient

momentum fluxes for the northern hemisphere winter season are small in the

lower troposphere, but increase with height, with maximum values near the

tropopause. At this level, there is a general convergence towards latitudes

around

°N,

with poleward flux to the south and equatorward flux at more

northerly latitudes. Thus

[u*v*]

is generally positive in midlatitudes, leading

to a positive source term in the circulation equation and hence to an indirect

circulation. Thus, the tendency of the eddy fluxes to accelerate the westerly

flow at midlatitudes near the tropopause is opposed by equatorward flow at

106 The zonal mean meridional circulation

Fig. 4.15. The meridional circulation induced by poleward eddy temperature fluxes.

upper levels, and poleward flow at low levels. The effect is to reduce the

vertical shear and make the flow more barotropic. The same effect becomes

extremely important in the decay stages of the lifecycle of a midlatitude

depression (see Section 5.5) when it leads to a massive build up of barotropic

kinetic energy.

From these examples, a general rule of thumb emerges. In the quasi-

geostrophic framework, any forcing of the zonal mean flow induces a meridi-

onal circulation which offsets the effect of that forcing. This is true whether

the forcing is a direct result of friction or heating, or due indirectly to eddy

transports. At the same time, the zonal accelerations or temperature changes

induced by the circulation are such as to restore thermal wind balance.

For a typical midlatitude location, we conclude that the typical observed

patterns of eddy fluxes and friction will all induce indirect circulations. The

diabatic heating will induce direct circulation. There is the possibility that

there could be a good deal of cancellation between the various source terms.

Let us estimate the typical meridional winds induced by the midlatitude

eddies. Ignoring all effects save those of the poleward temperature flux, and

assuming that \p

and d(f

)/dp are comparable, then we have, from

Eq. (4.31):

JLfoV].

(4.33)

Midlatitude winter values of

[v*0*]

have a maximum of around ^Kms"

while s

/h = —d6

/dp is typically around 5 x 10~

K Pa"

. The typical

maximum value of

is therefore 2 x 10

Pams~

. The poleward component

of the zonal mean wind is estimated to be

[v]

~ xp/Ap, (4.34)

where Ap is the typical pressure difference between the ground and the

4.5

Lagrangian view

of the

meridional circulation

107

midtroposphere. The poleward wind induced by the midlatitude eddy tem-

perature flux is therefore around 0.3

ms""

Similar reasoning can be applied

to the circulation induced by heating; in this case

V^-

\2L\

(4.35)

where L is typical horizontal distance. Typical values of \2\ in the tropo-

sphere are around 1.5 K day"

and so

[v]

can again be estimated; a reasonable

value is 0.3 ms"

. Thus the thermally direct and indirect components of the

midlatitude circulation tend to cancel out. All this proves is that a much

more careful calculation is needed if the direction of circulation is to be

estimated. Careful numerical solutions of Eq. (4.31) reveal that, indeed, the

observed Ferrel circulation is driven by the midlatitude eddy temperature

and momentum fluxes.

4.5

Lagrangian view

of the

meridional circulation

The discussion in the preceding section was couched in traditional Eulerian

terms.

That is, fluid properties were measured above some fixed point on

the Earth's surface, and the zonal means were calculated around each latit-

ude circle. In this way, we were able to distinguish between the tropical,

thermally direct, Hadley circulation, and the indirect Ferrel circulation of

midlatitudes. But we should be aware that this analysis may give quite a

different result from averaging in a Lagrangian sense, that is, by tracing the

motions of individual fluid elements. It turns out that this distinction is very

acute in the midlatitudes; indeed, the Ferrel circulation may be regarded

as an artifice of Eulerian averaging, and is a misleading description of the

mean circulation if (for example) we wish to discuss the advection of tracers

around the atmosphere.

Historically, this distinction first showed up in attempts to account for

the observed distribution of ozone in the lower stratosphere. The formation

of stratospheric ozone is a result of photolysis of ordinary diatomic oxygen

molecules by the ultraviolet components of sunlight (see Section 9.3). This

process is most effective at heights of around

km in the upper troposphere

and will proceed most rapidly where there is plenty of sunlight, that is, in

the tropics and in the summer hemisphere. Yet the maximum concentrations

of ozone are observed in the lower stratosphere, near the pole and in the

early spring. Clearly, advection must deposit the ozone in these regions. A

thermally direct circulation (now known as the 'Brewer-Dobson' circulation)

with descent near the winter pole, was postulated to account for the spring

108 The zonal mean meridional circulation

maximum. Yet the winter stratosphere is characterized by a disturbed

westerly jet at around

°N (the 'polar night jet'). The arguments of

the preceding section would suggest a thermally indirect Ferrel type of

zonal mean circulation, with ascent near the pole. There is of course no

contradiction between these views. The ozone is virtually a passive tracer in

the lower stratosphere and its distribution indeed indicates the Lagrangian

circulation of fluid parcels. The Ferrel circulation is an artifice based on

Eulerian averaging. In Eulerian terms, we must say that the poleward eddy

fluxes of ozone more than compensate for the equatorward, mean meridional

transports.

To explore the Lagrangian circulation of the atmosphere, let us consider

an idealized midlatitude jet such as that shown in Fig. 4.16. In the upper

troposphere, fluid moves through the waves from west to east. The waves

transport heat polewards, so there must be northward advection of warm

air and equatorward advection of colder air. To deduce the Lagrangian

trajectory of fluid particles in the meridional plane, we must consider the

vertical velocity field associated with the wave. Application of the co-

equation, Eq. (1.76), shows that there will be ascent associated with the

ridges of the wave. This upward advection of potentially colder air partially

offsets the heating due to the poleward advection of warm air. Similarly,

descent is associated with the troughs. Consider a parcel of air initially at

point A in the ridge. At this point, it will experience rising motion. As it

moves further east, it leaves the ridge, and starts to move equatorwards. At

the same time, it moves into a region of weaker ascent. At point B, the ascent

is zero. As it moves further into the trough, the parcel begins to descend,

with maximum descent at point C. Finally, it moves back through point D

to its original latitude and level. The projection of this motion onto the

meridional plane is shown in Fig. 4.16(b). The parcel describes a clockwise,

elliptical orbit in the y-p plane.

itself,

this motion implies no mass transport. Individual parcels of air

describe small elliptical orbits but will return to their initial pressure and

latitude at the end of each wave period. To deduce the mass circulation,

it is necessary to consider higher order effects than are included in this

simple model. The eddies are observed to have maximum amplitude in the

midlatitudes in the vicinity of the tropopause; at other levels and latitudes

they die away, as indicated in Fig. 4.17. The diagrams in Section 5.1 show

some observations of the eddy kinetic energy, a convenient measure of the

eddy activity. Now consider a parcel of air to the south of the maximum

of the eddy activity. As it orbits in the meridional plane, it moves from

a region of slightly lower eddy activity to a region of slightly higher eddy