James I.N. Introduction to Circulating Atmospheres

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7.3 Midlatitude storm zones and jets

229

(d)

Fig. 7.11

(cont.).

Based on six years of ECMWF data.

230 Three-dimensional aspects of the global circulation

wave saturated and entered its decay phase, these temperature fluxes became

relatively small, but 'barotropic decay' involving strong conversions from

eddy to zonal kinetic energy became important. These conversions require

large poleward temperature fluxes, with convergence of zonal momentum

into the latitude of the main tropospheric

jet.

The reader will realize that this

description of the variations of eddy fluxes and variances in time through

a baroclinic lifecycle is very similar to the variations of the eddy fluxes

and variances in space as one passes along any one of the three major

tropospheric storm zones. For the short Atlantic storm zone, the disposi-

tion of cyclone tracks, with cyclogenesis concentrated near the western end

of the zone, and cyclolysis towards the eastern end, makes this interpreta-

tion straightforward. The length of the storm zone represents the distance

across which a typical North Atlantic depression develops and decays. The

much longer southern hemisphere storm zone must be interpreted rather

differently. Cyclones tend to spiral polewards across this storm zone; there

is a greater preponderance of developing cyclones in the section through

the Atlantic and western Indian Oceans and a preponderance of decaying

systems at the eastern end of the zone.

The concentration of

active,

growing eddies in the storm track regions sug-

gests that one might expect these regions to be more baroclinically unstable.

Linear theory leads to some possible measures of baroclinic instability. For

example, the growth rate of the most unstable baroclinic mode is, according

to Eady's theory:

(see Eq. (5.52)) where U is the basic state zonal wind field. Figure (7.12)

shows a plot of this quantity, with dU/dz taken as d | v \/dz

evaluated

between 70kPa and 85kPa during the northern hemisphere winter. Maxima

in the growth rate are seen near the start of the Atlantic and Pacific storm

tracks.

7.4 Interactions between transient and steady eddies

The aim of this section is to seek an explanation of why the midlatitude

cyclone belt is broken into discrete storm tracks. In fact, much of the discus-

sion will concern consistency relationships between the various circulation

statistics; it is much more difficult to make firm statements about causal

relationships. This requires more complex analyses of data, or, increasingly,

numerical experimentation.

7.4 Interactions between transient and steady eddies 231

Fig. 7.12. Contours of the Eady baroclinic growth rate at

77.5

kPa, as defined by

Eq. (7.11), for the DJF season. Regions which are beneath the ground surface have

been blacked out. Contour interval

0.1

day"

, values greater than

0.5

day"

shaded.

Based on six years of ECMWF data, (courtesy P.J. Valdes.)

We saw in the preceding section how the northern hemisphere storm tracks

are associated with the major tropospheric jet streams. This relationship is

indeed consistent with the distribution of cyclogenetic regions around a jet

stream, and is not merely a matter of chance association. At the level of

quasi-geostrophic theory, the time mean zonal momentum equations can be

written:

= fVa

(7.12a)

•

Vv = —

(7.12b)

These equations may be thought of as describing the variations of the

ageostrophic part of the wind as one follows the time mean track of fluid

parcels. For the purposes of the present argument, variations of / are

unimportant compared to variations of the wind speed. The jet streams

have a predominantly zonal orientation, are generally more or less straight

and are most intense near the tropopause. Consider Eq. (7.12a). This rela-

232

Three-dimensional aspects

the global circulation

tionship suggests that there is likely to be poleward ageostrophic flow in

the entrance to jets, and equatorward ageostrophic flow in the exit regions.

The ageostrophic wind will be small away from the jet. Figure 7.13 shows

the wind speed and ageostrophic wind vectors for the upper level northern

hemisphere troposphere in DJF. It confirms this general pattern rather nicely.

This ageostrophic wind field is divergent on the equatorward flank of the

jet entrance and on the poleward flank of the jet exit. Continuity requires

that there must be ascent at midtropospheric levels below these divergence

regions. (The high static stability of the levels above the jet core means that

the vertical velocities induced in the stratosphere will be much smaller.) Thus

meridional circulations will be set up in the jet entrance and exit regions,

with midtropospheric ascent on the equatorward flank of the jet entrances

and the poleward flank of the jet exit. In both these regions, there will

be generation of cyclonic vorticity by vortex stretching at low tropospheric

levels,

and hence a tendency to cyclogenesis. The Eady model of baroclinic

instability, discussed in Section 5, suggests that the growth rate of the most

unstable baroclinic disturbances is proportional to the Coriolis parameter

/ (see Eq. (5.52)). So the cyclogenetic effect is likely to be particularly

pronounced at higher latitudes, that is, on the poleward jet exit.

These arguments will be familiar to synoptic meteorologists. They are

an application to the time mean flow of 'development theories' which were

elaborated in the pre-numerical weather forecasting era to identify regions

where generation or deepening of cyclone systems was likely to occur.

They suggest that the meridional circulation associated with the major

tropospheric jet streams will favour baroclinic instability in the jet exits,

where the observed storm tracks are located. Whether this effect alone is

capable of generating the observed storm tracks is another matter. Indeed,

one could just as well argue that the existence of a region of enhanced

baroclinic instability would reduce the vertical wind shear, and hence lead

to a jet exit at upper levels.

Let us now consider the properties of the transient eddies themselves. We

will treat the upper tropospheric flow as approximately two-dimensional and

nondivergent. The correlations between the two fluctuating components of

the wind can be written in tensor notation as:

which can be written as the sum of a diagonal and a symmetric tensor:

M N

(K ON

(

7.4 Interactions between transient and steady eddies

233

5.0

Fig. 7.13. Isotachs of the mean wind at 25kPa for the DJF period, northern

hemisphere, together with vectors of the ageostrophic wind at the same level.

Contour interval

ms"

with values greater than

ms"

shaded. The sample

vector represents 5ms"

. Based on six years of ECMWF data.

where K = {u

+ v'

)/2

M = (u

- v'

)/2 and N = u'v'. The first tensor

measures the kinetic energy of the transient eddies. The second also has an

immediate physical interpretation: it yields information about the shape of

eddies which can be derived using the formulae of Section 5.2. We will refer

to this second tensor as the 'anisotropy' tensor. The tilt of the eddies is given

by Eq. (5.10). If the velocities are referred to axes rotated through the angle:

1 N

(7.15)

that is, to axes parallel to the major and minor axes of the eddies themselves,

then the anisotropy tensor is diagonalized. The component of eddy velocity

parallel to the major axis of the eddy is denoted u\ while that parallel to

the minor axis is v

. The mean departure of the eddies from circularity is

measured by these diagonal elements, which have magnitude M = (M

)

1//2

The quantity M/K may be thought of as a dimensionless measure

of the eddy anisotropy, varying from 0 when the eddies are exactly circular,

234 Three-dimensional aspects

of the

global circulation

and tending towards 1 when the eddies are so elongated that v

tends to

zero.

The properties of the velocity correlation tensor can be summarized in a

single plot, in which contours of K are plotted on top of vectors showing

the direction of the principal axis of extension of the mean transients, and

whose length is proportional to M. Figure 7.14 compares the results for

the high frequency transients with those for the low frequency transients for

the northern hemisphere winter. The high frequency eddies are strongest in

the two storm track regions, over the Pacific and Atlantic Oceans. They are

extended in the meridional direction, but towards the end and to the south

of the storm tracks, they develop stronger tilts. The low frequency eddies are

mainly zonally elongated; there is a tendency for the largest values to be in

the jet exit regions. We will return to the topic of low frequency variability

in Chapter 8.

A primary concern of all these studies is the way in which the mean

flow v is modified by the eddies and their transports. This becomes more

complicated in the case of a flow which varies in the zonal as well as in

the meridional direction. A possible approach would start with Eqs. (7.12a,

b),

ignoring friction, with the winds are expressed as a sum of mean and

transient eddy parts:

•

Vu +

@*/2)

+ (W)

= fv

, (7.16a)

•

W + (n't/)* + (v*)

= -fu

. (7.16b)

A given transient eddy forcing might be balanced either by accelerations of

the mean flow (the first terms on the left hand side of these equations) or

by an ageostrophic flow. Indeed, the largest eddy term in these equations

proves to be

)

and this will principally be balanced by fu

, not by

variations of the mean flow. The ageostrophic wind can be removed from

the balance by combining Eqs. (7.12a, b) to give a vorticity equation. The

changes of the mean vorticity would be balanced by the divergence of the

eddy vorticity fluxes. This is not a very practical solution, since such vorticity

flux divergences involve taking very high derivatives of a field which is likely

to be noisy anyway. The components of the eddy anisotropy tensor provide

a way round these difficulties.

The time mean vorticity equation without friction can be written:

v-VC + V-(VC

)=0. (7.17)

After a certain amount of manipulation using £' = v

—

, the transient

7.4 Interactions between transient and steady eddies

235

25.0

(b)

25.0

Fig. 7.14. The velocity correlation tensor for the DJF season at the 25kPa level: (a)

high frequency eddies; (b) low frequency eddies. Contours show X, contour interval

50m

s , while the vectors are parallel to the major axis of A and with length

proportional to M; the sample vector has length 25m

. Based on six years of

ECMWF data.

236

Three-dimensional aspects

the global circulation

eddy vorticity flux divergence can be written:

7? = (-M

+ N

-M

- N

), (7.18)

so that:

"" = ~2M

+ N

- Nyy.

(7.19)

Now it is seen from plots of the variances and covariances of u! and v'

presented in the last section that | N |<C| M |, and, furthermore, that the

maxima in the high frequency eddy statistics tend to be elongated in the

zonal direction. From this, it follows that | N

|<C| N

|. Neglecting the N

term in Eq. (7.19) enables the eddy vorticity flux divergence to be rewritten:

V-^C

)-—(V-E), (7.20a)

where the 'vector' E is defined as

E = (-2M, -AT) = (1/2 - u'

-u'v').

(7.20b)

Figure 7.15 illustrates the pattern of mean flow forcing associated with the

distribution of E. A local region where E converges will be characterized by

the forcing of anticyclonic vorticity to the north of the convergence region

and cyclonic vorticity forcing to the south. The signs are reversed for a region

of divergence. The net effect is that convergence is associated with easterly

acceleration of the mean flow and divergence with westerly acceleration. The

components of E are reasonably smooth, so plotting vectors of E gives a

fairly clear picture of its pattern of convergence or divergence.

The orientation of the E vector is closely related to the orientation of the

eddies themselves. The direction which the E vector makes with the x-axis

is tan"

(Af/2M), while the angle made by the major or minor axis of the

eddies is \ ta.n~

(N/M). When | N |<| M |, these angles are equal, and, in

fact, the difference between them is only 5% when N is as large as M/2.

One further interpretation of the components of the E-vector will be

discussed before turning to an examination of the observed data. They can

be related to the group velocity of the transient disturbances. The arguments

are essentially those already developed in Section 6.5, especially Eq. (6.65).

The group velocity of Rossby waves in a flow which varies in both the x-

and y-directions can be written:

/2)2

S (7.21a)

7.4 Interactions between transient and steady eddies

237

divergence

anticyclonic

convergence

Fig. 7.15. The forcing of the mean flow by convergence and divergence of the

E-vector.

(7.21b)

-*y " • (fe

+ /

)

Then, assuming that the scale of variations of the time mean flow is large

compared to the scale of individual disturbances, and assuming that the

eddies can be represented by sinusoidal disturbances, we may write

M (l

-k

) N 2kl

It then follows that:

)

= {-Ml

-Ml

(7.22)

(7.23)

Choose local coordinates in which the x-axis is parallel to contours of the

time mean absolute vorticity, £. Generally, the £ contours will only make a

small angle 9 with the latitude circles. The transformed M and N are given

(M,

JV)

= (M cos 26 + N sin

26,

-M sin 26 + N cos 26) (7.24)

(see Eqs. (5.8a, b) and (5.9)). In these rotated coordinates,

(7.25)

238 Three-dimensional aspects of the global circulation

Fig. 7.16. The directions of the eddy axes, the E-vector and the group velocity

relative to the mean flow for some typical eddies.

That is, the angle between the group velocity relative to the mean flow and

the

contours is approximately twice the angle tan~

(AT/2M) between the £

contours and the E-vectors. Figure 7.16 illustrates the relationship between

some typical eddies and the directions of the E-vector and the relative group

velocity vector. We conclude that, knowing the E-vector, we can summarize

information about the typical shape of the eddies, the interaction between

the eddies and the mean flow and the direction of the group velocity vector.

The latter is perhaps the least satisfactory application of the E-vector, but it

is still of qualitative value. Other, more exact diagnostics, have been devised,

but they are generally much more difficult to calculate.

Figure 7.17 shows an example of the E-vector, for the high frequency

eddies in the northern hemisphere winter. The fields have been averaged

in the vertical. The vectors are orientated more or less from west to east,

with largest magnitude downstream and slightly poleward of the main jet

core.

At the end of the storm tracks, the vectors develop a more meridional

component, particularly to the south. These features should be compared

with the maps of the various transient velocity variances and covariances

and with the maps of the mean major axes of the eddies shown in Fig. 7.14.

In terms of the forcing of the mean flow, the eddies appear to be reinforcing

the existing jet streams. The E-vector shows divergence, implying westerly

acceleration, in the region of the jet cores. The maximum divergence is

in fact rather poleward of the jet axis, so it is clear that other processes,

manifested by time mean ageostrophic circulations, must also act to generate

the observed

jets.

In the eastern part of the ocean basins there is convergence

of E, or easterly acceleration. These regions are where the westerly jet is

weaker. Indeed, these regions are characterized by blocking episodes, when

the westerly flow can reverse for periods of several days to some weeks. The