James I.N. Introduction to Circulating Atmospheres

Подождите немного. Документ загружается.

4.2

The

Held-Hou model

of the

Hadley circulation

A<?

(410)

assuming that T

does not vary with latitude. Furthermore, 9

— 6

y = 7, giving a second equation in Y and 6

-^-Y

(4.ii)

These two equations contain two unknown quantities Y and 6

. It is

straightforward to solve for either of them; the resulting formulae are:

(

4i2)

and

5A6^gH

Let 0

= 255 K and A6 = 40 K (a typical observed value). Then we find

Y = 2200 km and 8

- 6

= 0.8 K. Comparison with Fig. 4.1 shows that

the estimate of the Hadley cell width is at least roughly in agreement with

observations, albeit slightly on the small side.

The model also leads to a picture of the upper tropospheric zonal winds

associated with the Hadley circulation. For y < 7, the upper level zonal

wind is simply equal to U

Eq. (4.5). For y > 7, the temperature is equal to

the radiative equilibrium temperature, and the zonal wind can be calculated

by applying thermal wind balance to 6

; this wind may be denoted u

and,

when the small latitude approximation is applied, it is easily shown to be a

constant, equal to gHA6/aQd. There is a discontinuity of the zonal wind at

y = Y. This prediction has both realistic and unrealistic elements. It suggests

the existence of a subtropical jet at the poleward edge of the Hadley cell.

Indeed, Fig. 4.1 shows just such a jet. But the discontinuity of wind speed

beyond the edge of the Hadley cell is not observed, and indeed would be

violently unstable.

The difference between 9

, the actual equatorial temperature, and 0

£O

the radiative equilibrium equatorial temperature, can be used to estimate

the meridional flow speeds associated with the Hadley circulation. On the

equator, by symmetry, there must be a balance between vertical advection

and heating, so that

dz x

90 The zonal mean meridional circulation

Here, N is the Brunt-Vaisala frequency of the atmosphere. Taking x

15 days, and assuming N = lO^s"

, we find that w ~

0.24

mm s"

. From

continuity, a typical horizontal velocity in the subtropical part of the Hadley

cell will be

v ~ —w. (4.15)

With the present parameters, this works out to be

0.5

cm s"

. A comparison

with Fig. 4.1 shows that the observed meridional winds in the Hadley cell

are closer to lms"

. So we may say that our simple theory has provided

a reasonable estimate of the geometry of the Hadley cell, but a very poor

estimate of its strength.

However, the Held-Hou theory is perhaps better than these simple

estimates suggest. We have been considering the annual mean Hadley cell,

which is symmetric about the equator. But during the solstices, the distribu-

tion of solar heating is asymmetric about the equator, with a maximum of

heating in the summer hemisphere. The theory can fairly easily be general-

ized to this situation. The radiative equilibrium temperature distribution can

be written:

(0)

—£*- sin

(/>

sin

0 -

1), (4.16)

where A9

is the mean pole-equator temperature difference and A6

is the

temperature difference between the summer and winter poles. The ascending

motion will now not be at the equator. Consequently, angular momentum

conservation will lead to upper level easterlies over the equator and (by

thermal wind balance) to maximum 6

away from the equator at the latit-

ude of maximum ascent. Figure 4.6 illustrates the forms of

and 6

in this

asymmetric case. The lack of symmetry means that the maximum ascent

need no longer coincide with the latitude of maximum radiative equilibrium

temperature, nor with the streamline dividing the northern and southern

hemisphere cells. Consequently, we have a more complicated system with

four unknown parameters, determined from four matching conditions in the

manner of Eqs. (4.12) and (4.13).

The results of these asymmetric calculations are summarized in Fig. 4.7.

The cell rapidly becomes highly asymmetric as the radiative equilibrium

maximum is moved off the equator, with a small weak cell in the summer

hemisphere and a much stronger cell with ascent in the summer hemisphere

4.2 The Held-Hou model of the Hadley circulation

30°S

15°S

Fig. 4.6. The Held-Hou model for the case of a heating maximum away from the

equator. The latitudes </>

<j>

and

<j>

as well as the equatorial temperature 6

are

to be determined.

and descent in the winter hemisphere. The mass flux carried by the two cells

(proportional to the area between the 9

and 6

curves in Fig. 4.5) differs

by a very large factor for even modest asymmetry of the heating, and we

would expect the annual mean Hadley circulation to be dominated by the

two winter cells. If this is so, our estimate of Y will be increased somewhat,

but our estimates of w and v will be increased by an order of magnitude,

bringing them more in line with the observations.

An important principle is illustrated by these calculations. The strength

and character of the cells react in a highly nonlinear fashion to changing the

latitude of the heating maximum. Consequently, the annual mean Hadley

circulation is very different from the response to the annual mean forcing

which we would predict. In various forms, this problem of 'nonlinear

averaging' is a central difficulty in parametrizing many diabatic forcing

processes in the atmosphere. It means that a complex general circulation

model has to be run even if one is only interested in the annual and zonal

mean circulation.

The zonal mean meridional circulation

30°N-

15°N-

2° 4° 6° 8° 10° 12° 14°

15°S-

30°S-

(b)

1 .

-1 -

-2 -J

Winter Cell

8° 10° 12° 14°

Summer Ce

L L

Fig. 4.7. Results for the asymmetric Hadley cell, assuming the same parameters

as previously: (a) variation of 0

, (j)

and

(j)

as a function of 0

, the latitude of

maximum radiative equilibrium temperature; (b) variation of the mass flux carried

by the winter and summer cells as a function of 0

4.3 More realistic models of the Hadley circulation 93

4.3 More realistic models of the Hadley circulation

There were two glaring omissions in the discussions in the preceding sections.

The first was that no effects of friction in the upper layer of the atmosphere

were included. Since the overturning timescale of the Hadley cell predicted

by the model was several radiative timescales, even very weak dissipation

could modify the flow significantly. The second was that the effects of latent

heat release by condensation of water vapour were ignored. In fact, latent

heat release dominates heating in the tropics. This is not to say that using

such models is folly. Indeed, the aim of any scientific modelling is to separate

crucial from incidental mechanisms. Comprehensive complexity is no virtue

in modelling, but, rather, an admission of failure. The Held-Hou model is

remarkable, not for the effects it omits, but because even with such minimal

assumptions it reproduces so many observed features of the meridional

circulation and the zonal wind field. But in this section, it is necessary to

examine how much of an effect these various complicating factors might

have.

We will examine the effects of friction using a variant of the 'simple global

circulation model' introduced in Section 2.4. This variant is axisymmetric,

with all variations in the longitudinal direction suppressed. A slightly more

complicated form of friction is introduced, with the momentum equation

written:

Here, 5£

and J

represent the linear and nonlinear dynamical terms,

respectively, and K is a constant vertical diffusion coefficient. The diffusion

term on the right hand side of this equation is a crude parametrization of

the turbulent transports of momentum in the planetary boundary layer, and

K is sometimes called an 'eddy viscosity' coefficient. The effect of such a

diffusion term is to introduce a classical Ekman boundary layer into the

flow, in which the depth of the boundary layer is given by

D =

(—J .

(4.18)

Noting that the midlatitude planetary boundary layer is around lkm in

depth, it follows that a reasonable value for K is around 10m

. Such

an Ekman boundary layer induces secondary circulations which dissipate

vorticity in the fluid above the boundary layer by stretching or shrinking of

vortex tubes. Figure 4.8 illustrates this. The typical timescale for this 'spinup'

The zonal mean meridional circulation

cyclonic

ant icy don ic

Fig. 4.8. The dissipation of vorticity by the Ekman boundary layer.

process is

(4.19)

which is a few days for typical midlatitude conditions. In this way, if the

overturning timescale of the Hadley cell is long compared to the spinup

timescale, the strong meridional shears which develop in the vicinity of the

subtropical jet in the Held-Hou model will be moderated by the presence of

a midlatitude boundary layer.

The results of two integrations of the simple global circulation model with

this friction term are shown in Fig. 4.9. The first has 6

symmetric about

the equator, as in the simplest form of the Held-Hou model. There is a

weak Hadley circulation extending into the subtropics, with an associated

subtropical jet. The zonal wind speeds are much less than those predicted by

the frictionless model, and there is no discontinuity of wind associated with

the edge of the Hadley cell. Indeed, the Hadley circulation does not have

a sharp poleward boundary, although it does become very weak at high

latitudes. The dimensions of the Hadley cell are very much in line with the

predictions of the Held-Hou model. But the zonal winds generated by the

circulation are considerably weaker. The second integration is for solstitial

conditions, with the midlatitude temperature gradients around twice as large

in the winter hemisphere as in the summer hemisphere. The maximum of

is at 6°N. The asymmetry between the summer and winter Hadley cells is

marked, though not as extreme as in Fig. 4.7. The summer cell has only a

weak subtropical jet at its poleward edge, consistent with the dominant effect

4.3 More realistic models of the Hadley circulation 95

of friction on such a slowly overturning flow. The winter cell has a much

more vigorous circulation. The jet is correspondingly much stronger, though

not as strong as it would be if the

flow

conserved angular momentum exactly,

and it shows signs of very sharp shears on its poleward flank. As we will see

in later chapters, this flow would be violently unstable if the axisymmetric

assumption were relaxed. The resulting eddy fluxes of heat and momentum

would quickly modify the jet profile into a more stable form.

Now consider the effects of moisture. Satellite images show that the Hadley

cells in each hemisphere are separated by a band of cumulonimbus clouds.

This band forms at the line of convergence where the low level trade winds

originating in each hemisphere meet; it is sometimes called the intertropical

convergence zone (or TTCZ'). Figure 4.10 shows a schematic diagram of

the formation of the ITCZ and its relationship with the Hadley circulation.

Descending air at the poleward edge of the Hadley cell reaches the boundary

layer with an extremely low humidity. Air returns towards the equator in

the low level

flow,

picking up heat and moisture from the underlying surface

as it goes. When it meets the air from the opposite hemisphere, it is forced

to rise. As it rises, it quickly becomes saturated, so that condensation and

latent heat release take place. The release of latent heat is balanced by ascent,

and so the ITCZ is characterized by deep convection extending through the

depth of the troposphere. The principal effect of latent heat release is to

concentrate most of the ascending motion of the Hadley circulation into the

narrow ITCZ.

Calculating the nature of the Hadley circulation associated with this

moist model would appear to be extremely difficult, since it involves various

boundary layer exchanges of heat and moisture, as well as deep cumulus

convection in the ITCZ. As we saw in Section 2.4, parametrizing such

processes is a crucial but poorly resolved problem in global circulation

modelling. However, placing some general bounds on the width and strength

of the Hadley circulation is in fact more straightforward than might initially

be expected.

Let us continue to work with the Held-Hou two-layer formulation, with

the lower layer representing the moistening boundary layer, and the upper

layer representing the frictionless poleward flow in the middle and upper

troposphere. For simplicity, we return to the case of a Hadley circulation

which is symmetric about the equator. The zonal wind in the upper layer will,

as before, be determined by angular momentum conservation, and hence the

variation of potential temperature with latitude will be given by 0

, defined

by Eq. (4.9). The cooling due to the radiation of long wave radiation to

space can be related to

Equally, in the absence of fluid motion, the

The zonal mean meridional circulation

-90S -60S -30S 0 30N 60N 90N

LATITUDE

(b)

-60S -30S

LATITUDE

30N 60N 90N

Fig. 4.9. The Hadley circulations and zonal winds set up by a simple axisymmetric

global circulation model, (a) and (b) show a case for which 6

is symmetric about

the equator, and (c) and (d) for a case where the maximum of 6

is located at about

°N.

temperature would be 6

, the radiative equilibrium potential temperature

distribution, given by Eq. (4.4). The distribution of short wave radiative

heating is simply

;

the total heating is given by

H= / -^dy.

(4.20)

J0 T

4.3 More realistic models of the Hadley circulation

(c)

LATITUDE

Fig. 4.9 {cont). (b) and (d) show the mass streamfunctions with a contour interval

kg s"

, while (a) and (c) show zonal winds, contour interval

ms"

, heavy

shading denoting values in excess of

ms"

Now let us make the extreme assumption that all incoming solar radiation

is used to evaporate water into the boundary layer. This heat is realized as

latent heat of condensation in the ITCZ, where the total heat released will

be if, Eq. (4.20). As in Section 4.2, the width of the Hadley cell can be

determined by the requirement that the heating and cooling integrated over

The zonal mean meridional circulation

ITCZ

—

Tropopause — — — —

Trade winds Trade winds

\\\\\\\\\\

Fig. 4.10. Schematic illustration of the ITCZ and the Hadley circulation.

the width of the Hadley cell must balance, i.e., that

(4.21)

But we have met this equation before, as Eq. (4.10). It is identical to the zero

net heating condition

for

the dry Held-Hou model. The other condition,

namely 6

at y = 7, is

also identical

the two models.

this

case,

the inclusion of moisture leaves the width and mass flux of the Hadley

cell unchanged; the difference from the Held-Hou model is simply that the

ascent and heating is concentrated into

narrow region

the ITCZ while

there is descent and cooling throughout the rest of the cell. If

made a less

extreme assumption, that some of the incoming sunlight went to evaporate

water, while the remainder led to sensible heating, we would still obtain the

same result for the width and strength of the cell, but with some ascent over

the tropical part of the cell and

stronger concentration of the ascent at the

ITCZ. Figure 4.11 illustrates these possibilities.

To summarize, the effect of moisture is to introduce an asymmetry between

large scale ascent and descent

the Hadley circulation.

the dry case,

roughly equal areas of the cells are ascending and descending. As conden-

sation and latent heat release become more important, the vertical motions

become stronger and more concentrated, while the bulk of the cells are char-

acterized by descent. The total circulation of the cells remains unchanged,

however, because there is no direct change in the ultimate thermal forcing,

which

provided by the incident flux

solar radiation.

The

final

width

of the ITCZ depends upon the structure

the individual cumulus towers

which make

up; turbulent mixing between the strongly ascending towers

and the surrounding clear air determines the size

the cumulus elements

and hence the width

the ITCZ. Observations show that the meridional