James I.N. Introduction to Circulating Atmospheres

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5.5 Baroclinic lifecycles and high frequency transients

159

30N

60N

LATITUDE

30N

60N 90N

LATITUDE

Fig. 5.26 (cont). Contour intervals: in (a) and (b), the contour interval is lOKms

for [v*T*] and 50m

for [uV]; in (c) the contour interval is 2Kms"

for [v*T*]

for

[u*v*].

Negative values indicated by shading.

and

marked changes from one stage of the lifecycle to the next. The patterns

associated with the initial linear normal mode have already been given, in

Fig. 5.24. The corresponding plots for the mature and decaying stages are

shown in Fig. 5.26. At first, the fluxes are concentrated near or below the

steering level. As the mode grows, the intensity of the fluxes increases, but

the pattern of distribution remains close to that of the linear stage. As

the disturbance approaches its maximum amplitude, the momentum fluxes

become stronger relative to the temperature fluxes, and their largest values

are found at upper tropospheric levels. In the decay phase, the temperature

fluxes become quite weak, but the momentum fluxes are strong and concen-

trated near the tropopause, which acts more or less as an upper boundary

to the eddy part of the flow. The fluxes averaged through the lifecycle are

shown in Fig. 5.26(c). They are strikingly similar to the temperature and

momentum fluxes observed in the winter troposphere, with large temperat-

ure fluxes at low levels and large momentum fluxes at upper levels. These

results suggest that the observed transients may be regarded as a random

superposition of lifecycle events, occurring at different times and at different

longitudes throughout the midlatitudes. Further insight into these processes

is afforded by relating the pattern of eddy fluxes to the propagation of

wave-like disturbances out of the baroclinic regions. This will be treated in

Section 6.6.

160 Transient disturbances in the midlatitudes

The zonal flow at the end of the lifecycle is nearly stable. The eddy

energy is decaying, although for some zonal flows, a generally weaker

'secondary development' may follow. A numerical eigenvalue calculation

shows that, indeed, the final state is at most only weakly unstable. Such

unstable modes as there are have narrow and contorted structures, so that

there is little possibility of the remaining wavelike disturbances projecting

on to and exciting these new normal modes. Yet, according to the general

criteria for baroclinic instability developed in the preceding section, the flow

should still be unstable. It possesses available potential energy, which has

been depleted only marginally by the first baroclinic lifecycle. There are

substantial contrasts of potential vorticity gradient and surface temperature

gradient, suggesting the possibility of further instability. The resolution of

this apparent paradox lies in the form of the final zonal flow. Figure 5.27

shows the difference in the zonal flow between the beginning and end of the

lifecycle event. It is nearly barotropic, that is, it is independent of height,

which is consistent with the small reduction in AZ. A westerly acceleration

has taken place near the latitude where the baroclinic waves were centred,

and there has been easterly acceleration to the south and north. Now

suppose the surface wind is subtracted from all levels in the model. This

barotropic change will leave the wind field still in thermal wind balance

with the temperature field, and in fact will make only small changes to the

potential vorticity gradients. This is because the term [u]

makes only a

small contribution to the potential vorticity gradient in the midlatitudes.

The modified zonal flow is not very different from the starting flow at the

start of the lifecycle, and eigenvalue analysis shows that it is very nearly

unstable. In other words, the baroclinic lifecycles stabilize the zonal flow not

so much by depleting the zonal available potential energy as by introducing

a horizontally sheared barotropic component to the flow. This component

severely restricts the possible balanced structures which eddies can take, and

thereby reduces their instability.

This last result illustrates the limitations of the frictionless adiabatic life-

cycle as a model of global circulations. Extremely large surface winds are

associated with the final state, with maximum speeds of up to 40ms"

. In

the real atmosphere, friction would moderate these winds quickly. At the

same time, the introduction of heating would tend to restore any changes

made to the initial zonal mean temperature fields. Such a model is really a

simple global circulation model, such as has been introduced in Section 2.4

and used for illustration elsewhere in this book. Figure 5.28 shows a low

level field of temperature from such a model. A number of cyclonic centres,

in different stages of development, can be seen around the globe. The zonal

5.6 Problems

161

100

30N

60N

LATITUDE

90N

Fig. 5.27. The difference between the final and initial zonal

flow

during the baroclinic

lifecycle shown in Figs. 5.25 and 5.26. Shading indicates regions where easterly

acceleration has taken place. Contour interval

ms"

mean temperature and momentum fluxes are quite similar both to those

observed and to those resulting from the time average of the lifecycle. No

single wavelength dominates, and just as for the observed atmosphere, the

levels of eddy energy are continually fluctuating. Experiments with various

geometries reveal that these fluctuations are a result of interactions between

the different wavelengths of eddies present, as well as between the unstable

eddies and the zonal mean flow. But the source of eddy energy on all scales

is clearly the result of baroclinic instability in the midlatitudes. The lower

frequency transients exhibited by such experiments will be considered further

in Chapter 7.

5.6 Problems

5.1 Given that the global average zonal wind < u > is 15ms"

, estimate

the typical atmospheric kinetic energy in Jkg"

. If the friction spins up

the atmosphere on a timescale of five days, work out the typical dissipation

of kinetic energy in Wm~

. Compare your value with the global mean

insolation.

5.2 Use Fig. 5.7(a) to estimate the divergence of the transient eddy heat

flux in DJF, northern hemisphere in the subtropics and the midlatitudes.

162

Transient disturbances

in the

midlatitudes

Fig.

5.28.

Temperature

kPa

at an

arbitrary instant

in a run of a

simple global

circulation model

of the

type described

Section

2.4.

Contour interval

The

quasi-horizontal exchange

parcels

warm

and

cold

air at

different longitudes

clearly seen,

and

demonstrates

the

baroclinic instability mechanism

work.

Hence calculate the heating and cooling rates due to transient eddies in

Wm~

5.3 Perform a similar calculation for the eddy zonal momentum flux at

°N in DJF (see Fig. 5.5(a)) to estimate the acceleration of the midlatitude

flow due to transient eddies. Estimate the 'spin-up time' x

required in order

that friction should balance the eddy convergence of momentum flux.

5.4 High frequency transients in the upper troposphere at 45 °N are

characterized by ifi = 40m

= 80m

3600

and

20m

Estimate the typical horizontal scale of eddies, the ratio of their

major and minor axes and their tilt.

5.5 A typical geopotential height variance at 70kPa and 45 °N is

30 m,

typical meridional wind variance is

ms"

and a typical poleward tempera-

ture flux is 6Kms~

. Estimate the horizontal separation between the trough

at 90kPa and at 50kPa.

5.6 The Brunt-Vaisala frequency is about 10~

in the troposphere.

The available potential energy is observed to be 4 x 10

Jm~

. Estimate the

5.6 Problems 163

pole-equator temperature difference, and compare your result with values

taken from cross sections of the observed (potential) temperature structure.

Using values of AE, KE and KZ in Fig. 5.15, estimate representative values

of the typical eddy temperature fluctuation, the typical eddy velocity and the

typical zonal wind speed in the annual mean.

5.7 Using the cross sections of eddy temperature and momentum fluxes

given as Figs. 5.5 and 5.7, together with the sections of

[u]

and [9] given in

Fig. 4.2, estimate the typical magnitudes of the two terms in the eddy energy

equation.

Wave propagation and steady eddies

6.1 Observations of steady eddies

Despite the eddy-zonal flow partitioning which we have employed in pre-

ceding chapters, the seasonal mean flow is very far from being zonally

symmetric. Such departures from symmetry are important in accounting for

regional variations of climate. They also modify the global patterns of heat

and momentum transport, especially in the northern hemisphere winter. In

this chapter, we will discuss some observations of the steady wave pattern,

and show how rather simple theories based on linear wave propagation can

account for some of the gross features of these observations.

The steady waves are most pronounced in the northern hemisphere win-

ter, and have their largest amplitudes in the upper troposphere. In some

circumstances, they also become very important at high levels in the winter

stratosphere, a point that we will return to in Chapter 9. Figure 6.1 shows

the winter mean geopotential height field at 25kPa in both hemispheres.

The characteristic features of the northern hemisphere picture are the pro-

nounced troughs over Canada and Japan, with ridges over the eastern side

of the two ocean basins. One's subjective impression is of a predominantly

zonal wavenumber 2 pattern. This general pattern is very persistent and

can be seen in individual seasons with only relatively small variations. The

corresponding picture for the southern hemisphere looks, at first sight, much

more axisymmetric. Closer examination reveals that the main vortex is sig-

nificantly displaced from the pole, while a characteristic three trough pattern

can be seen around the periphery of Antarctica.

The character of the non-zonal disturbances to the upper level flow is

more obvious if the zonal mean is subtracted from diagnostics such as those

shown in Fig.

6.1.

Such zonal anomalies are shown in Figs. 6.2 and 6.3. These

diagrams show the zonal anomaly of the time mean streamfunction

\p*

164

6.1

Observations

steady eddies

165

25kPa rather than the geopotential height anomaly. This is to emphasize

the important zonal anomalies of the flow in the tropics and subtropics as

well as at high latitudes. As we shall see in later sections, the influence of

the tropics on disturbances in midlatitudes is now believed to be substantial.

The presence of waves in the northern hemisphere winter, Fig. 6.2(a), is

clear; at high latitudes, wavenumbers 2 and 3 dominate, while a strong

wavenumber 1 signature is seen at lower latitudes. Many of the eddies show

a systematic phase tilt, with their major axes orientated from south-west to

north-east. Such phase tilts indicate that a significant poleward momentum

flux must be associated with the steady eddies. The steady eddies are also

of substantial amplitude, though with rather longer zonal wavelength, in the

southern hemisphere winter, shown in Fig. 6.3(b). However, little systematic

phase tilt can be discerned in the southern hemisphere.

Steady waves are also seen in the summer season, though with sharply

reduced amplitudes. The major feature in the northern hemisphere summer,

Fig.

6.3(a), is an anticyclonic circulation centred over south western Asia;

this is associated with the upper level Asian monsoon circulation and is more

pronounced at higher levels such as lOkPa. Apart from this, steady eddies

are rather weak and have a smaller scale than those in the winter season.

Similar remarks also apply to the southern hemisphere summer, Fig. 6.2(b);

it is dominated by a rather weak zonal wavenumber 1 pattern.

A notable feature of all these diagrams is the absence of much flow across

the equator. The steady disturbances fill their respective hemispheres, but by

and large do not spill over into the opposite hemisphere. The exception is the

Asian monsoon circulation, which does involve significant cross equatorial

flow (especially at low levels). The reasons for this apparent insulation of

the hemispheres from one another will be matters upon which our theories

of steady waves will be required to comment.

There is considerable continuity between the upper and lower level steady

waves.

Rather than present diagrams such as Figs. 6.2 and 6.3 for many levels

throughout the troposphere, Fig. 6.4 shows representative longitude-pressure

cross sections. Attention is restricted to the winter seasons. In the northern

hemisphere winter, the dominant zonal wavenumber 2 can be discerned.

More striking is the systematic phase tilt of the waves, with the upper level

troughs and ridges well to the west of their low level counterparts. As we saw

in Chapter 5, such a tilt is inevitably associated with a poleward transport

of temperature by the eddies. In fact, the steady eddies make an important

contribution to the poleward transport of heat in the northern hemisphere

winter. Although these sections do not resolve the stratosphere adequately,

the rapid increase of the steady eddy amplitudes at the uppermost levels is

166 Wave propagation and steady eddies

(a)

(b)

Fig. 6.1. The time mean geopotential height of the 25kPa surface, based upon

ECMWF data for six seasons: (a) Northern hemisphere, DJF. (b) Southern hemi-

sphere, JJA. Contour interval

100

6.1 Observations of steady eddies

167

(a)

Fig. 6.2. The time mean eddies of the

kPa flow, shown by the zonal anomaly of

the mean streamfunction,

\p*

for the DJF season. Contour interval 5 x 10

negative values dashed. Based upon six years of ECMWF data, (a) Northern

hemisphere, (b) Southern hemisphere.

(a)

Fig. 6.3. As Fig 6.2, but for the JJA season, (a) Northern hemisphere, (b) Southern

hemisphere.

significant. In the southern hemisphere winter, note the somewhat smaller

amplitude and longer wavelength of the disturbances. More importantly, the

vertical phase tilt is small; there is little systematic poleward temperature

flux by the steady eddies in the southern hemisphere.

The total effects of the steady eddies for the transport of heat and mo-

168

Wave propagation and steady eddies

120

180

LONGITUDE

Fig. 6.4. Longitude-pressure sections of the zonal anomaly of geopotential height

Z*,

based on six years of ECMWF data. Contour interval 50m, negative values

shaded, (a) For DJF at

°N.

(b) For JJA at

°S.

mentum are summarized by latitude-pressure sections of

the

steady poleward

temperature flux

[v*

T*] and momentum flux

[u*

v*]

shown in Figs. 6.5 and

6.6 respectively. The fluxes are significant only for the northern hemisphere

winter, when a poleward temperature flux extends throughout the depth of

the midlatitude troposphere. The maximum value of more than 12 K m s"

is comparable to the transient eddy temperature flux. At the same time,

there are large values of the momentum fluxes in the upper troposphere,

with convergence around

°N.

Once again, the distribution and magnitude

is comparable to the

fluxes

carried by the transients. In contrast, the

fluxes

the summer season and for the southern hemisphere are a good deal smaller

than the transient fluxes.

The persistence of these steady disturbances throughout an entire season,

and their repeatability from one year to another, strongly suggests that they