James I.N. Introduction to Circulating Atmospheres

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5.2

The structure

transient eddies

119

If u'

> v'

, the eddies will be zonally elongated; if, on the other hand,

> i/

, they will be meridionally elongated.

Zonal means of t/

, i/

and Z'

are shown in Fig. 5.3. Taking typical values

for the midlatitude upper troposphere, v

280

, u

250

and

= 4 x 10

. Thus the eddies are slightly elongated meridionally. The

typical wavenumber is around

10~

; this corresponds to zonal wave-

number 5 at

40 °

of latitude. When filtering is applied, this picture changes.

The high frequency eddies are distinctly meridionally elongated and their

scale is somewhat smaller.

So far, we have considered eddies with their longer axes orientated either

east-west or north-south. There is no reason why they should not be orient-

ated at some intermediate angle. The orientation is related to the poleward

flux of momentum carried by the wave. Qualitatively, this can be seen

by considering Fig. 5.4, which shows a schematic view of an eddy which

tilts from south-west to north-east. Along section AB, both the u and v

components of eddy velocity are large and positive; the eddy is carrying

westerly momentum northwards. Along section CD, easterly momentum

is being carried southwards. The net effect of both these sections is to

transport westerly momentum northwards. The shorter sections BC and DA

are characterized by weaker winds and by southward transport of westerly

momentum. In the limit of extremely elongated tilted eddies, these two

sections would make a negligible contribution to the momentum transport

by the eddy. Averaging over the entire wavelength, it can be seen that there

is net poleward momentum transport, that is, that

v!v?

is positive for such a

tilted eddy. By the same arguments, an eddy tilted in the opposite direction

would transport westerly momentum southwards. There would be no net

transport of momentum by an untilted or exactly circular eddy, since the

contributions from the various segments would exactly cancel in those cases.

The poleward momentum flux is closely related to the orientation of the

eddies. Figure 5.5 shows latitude-pressure cross sections of the poleward

momentum fluxes for the DJF and JJA seasons. The eddy momentum

flux is largest in midlatitudes near the tropopause. The momentum flux is

poleward at lower latitudes, but tends to be equatorward at higher latitudes,

suggesting that the eddies are orientated in opposite directions to north and

south of the main cyclone belt. Such a configuration is especially clear in the

southern hemisphere, but is also important in the northern hemisphere. This

distribution of momentum flux implies, using the arguments of Section 4.4,

that an indirect Eulerian mean circulation is associated with the observed

eddy momentum fluxes.

The typical angle of tilt of the eddies can be calculated if the momentum

120

Transient disturbances in the midlatitudes

(a)

£3 -

-90 -60

-30 0 30 60

Latitude

(b)

~~i—

a t i t ud e

—]—

Fig. 5.3. Showing (a) zonal mean height variance [Z

] and (b) the variance of

velocities [u

] (solid) and [v

] (dashed).

flux and the velocity variances are known. Let us calculate the velocity

components in a frame of reference, denoted by

(5c,

y), which is rotated

through some angle

relative to the basic

(x,

y) frame of reference. Using the

usual formulae for rotation of coordinates, the transformed eddy velocities

are

—

cos (p

—

sin

cp,

(5.8a)

= u' sin

+ v

cos

cp.

(5.8b)

5.2 The structure of transient eddies

121

ttfl

(d)

^5-

-30

Latitude

-30

Lat itude

Fig. 5.3 (cont). (c) and (d) as (a) and (b) respectively, but for the high frequency

eddies. Based upon six years of ECMWF data at

kPa, DJF season.

Then the momentum flux in the rotated frame is:

= - \u

— v

sin2cp + u

cos2cp.

(5.9)

When cp is chosen so that the eddy has no_tilt in the rotated frame of

reference, the transformed momentum flux u

= 0. Hence the angle of tilt

will be given by:

(5.10)

122

Transient disturbances in the midlatitudes

Fig. 5.4. Schematic illustration of a tilted eddy.

This is also satisfied for

TT/2,

indicating that, of course, the eddy has

both a major and a minor axis.

The unfiltered transients have v

~ w'

, so that

°.

The high pass

filtered eddies have wV typically of

in the upper troposphere. This

implies that they must have a tilt of about 26°. However, the properties of

the eddies do vary from place to place. In Fig. 5.6, the anisotropy of the

eddies and the orientation of their major axes are shown. The 'anisotropy'

will be defined in Section 7.4 (Eq. (7.15) et

seq);

it is simply a dimensionless

number ranging from 0 for perfectly circular eddies to 1 for infinitely

elongated eddies. The unfiltered eddies are most anisotropic in the tropics,

a manifestation of low frequency tropical disturbances, the theory of which

will be discussed in Section 7.1. The high frequency eddies are meridionally

elongated in the midlatitudes, in the so-called 'storm track' latitudes.

We now turn to an examination of

the

poleward eddy heat transport due to

the transients. This is related to the vertical structure of the eddies. Figure 5.7

shows cross sections of the transient eddy temperature fluxes. The fluxes are

poleward in both hemispheres, and are largest in the lower troposphere at

midlatitudes. The transient temperature flux is unimportant throughout the

tropics, where the heat transport is mainly accomplished by the Hadley cell

and by ascent in small scale convective systems. The maximum zonal mean

temperature flux in the troposphere is around 15Km s"

, with somewhat

5.2 The structure of transient eddies

123

LATITUDE

Fig. 5.5. Latitude-pressure cross sections of the poleward transient eddy momentum

-2

fluxes observed in (a) the DJF and (b) the JJA seasons. Contour interval 5 m

negative values shaded. Based on six years of ECMWF data.

larger values in the winter stratosphere. The divergence of the temperature

flux gives an estimate of the heating or cooling due to transient eddies; this

is typically 0.5 K

day""

equivalent to

Wm"

. The seasonal cycle in the

temperature flux is marked in the northern hemisphere, but less so in the

southern.

124 Transient disturbances in the midlatitudes

LATITUDE

-90S

-60S

-30S

0 M

6'ON

' '

" "9'0N

LATITUDE

Fig. 5.6. Latitude-pressure sections showing the eddy anisotropy and the orientation

of the mean major axes of the eddies. Contours show the eddy anisotropy, contour

interval 0.1. Shading indicates values between 0.2 and 0.4. Vectors indicate the

orientation of the major axes of the eddies, vertical vectors indicating a north-south

orientation, (a) Unfiltered transient eddies, (b) High frequency transient eddies.

Based on six years of ECMWF data, DJF season.

5.2 The structure of transient eddies

125

-30S 0

LATITUDE

101

-90S

-60S

-30S

LATITUDE

30N

60N

90N

Fig. 5.7. Latitude-pressure sections showing the poleward transient eddy temper-

ature flux, [i/T

] for (a) DJF and (b) JJA. Contour interval 2 K m s"

, negative

values shaded. Based on six years of ECMWF data.

Now imagine an idealized geostrophically and hydrostatically balanced

disturbance. The geometry of the situation envisaged is shown in Fig. 5.8.

Three surfaces at pressure p

—

Ap, p and p + Ap are shown. The geostrophic

streamfunction is related to the geopotential height by

(5.11)

126

Transient disturbances

in the

midlatitudes

so that the wind components are

The temperature on the pressure surface is related to the thickness of a layer

by the hydrostatic relation; it follows that the temperature on level 0 is:

_PogZ[-Z'_

( 13)

R 2Ap '

1JJ

Suppose that the geopotential height varies sinusoidally in x. Furthermore,

suppose that the amplitude is constant with height, but that there may be a

change of phase with height. We may write

Zt = Z

+ A sin{kx + id), i =

-1,0,1,

(5.14)

where A is the amplitude of the wave. A sign convention has been chosen so

that positive S means the wave is tilting to the west with height. Then the

eddy temperature field at level 0 is

= J^-Acos(kx)

sin((5).

(5.15)

No fluctuation of temperature with x occurs unless there is a phase tilt. The

poleward wind is:

t/ =

Acos

(

516

)

Note that

the

temperature

and

poleward velocity waves

are in

phase. Hence

the poleward temperature

flux

is:

Inspection

of Fig. 5.8

makes this result intuitively obvious.

The

westward

vertical phase tilt means that thickness,

and

hence temperature,

is a

maximum

where

the

poleward wind

is a

maximum. Thus there must

be a net

poleward

temperature flux.

If the

phase tilt were reversed,

the

wave would transport

heat equatorwards. Careful attention

to the

sign convention

for

reveals

that these statements

are

equally true

in the

southern hemisphere.

An example

of the use of Eq.

(5.17)

is to

calculate

the

vertical phase tilt

typical transient from

the

and v'T

fields.

collection

circulation

statistics reveals that

70 kPa

and

45 °N,

the

high frequency height variance

35 m,

the

meridional wind variance

20m

and the

poleward

temperature flux is4Kms~

. From

Eq.

(5.6),

the

zonal wavelength

found

1070 km. Substituting values into

Eq.

(5.17),

it is

found that

the

phase

5.2 The structure of transient eddies

121

Fig. 5.8. Schematic illustration of a wave-like disturbance with westward phase tilt

with height.

difference between the 90kPa level and the 50kPa level is about 12° of

longitude.

As well as transporting heat polewards, the transient eddies also transport

heat vertically. Indeed, the demands of thermal wind balance ensure that

there is a close linkage between poleward and vertical temperature fluxes.

Figure 5.9 demonstrates the way in which this must happen. If an eddy is

characterized by warm advection at one longitude, and cold advection at

another, temperature gradients will develop in the zonal direction. This in

turn means that to maintain thermal wind balance, the upper level velocity

field must change. Such accelerations can be generated by ageostrophic

circulations in the height-longitude plane. The resulting flow is correlated

with the temperature field in such a way as to lead to a net upward flux of

temperature. We will show in the next section that such vertical temperature

fluxes are required if transient eddies are to develop spontaneously in a zonal

flow.

As discussed in Section 2.4, modern analysis and initialization methods

lead to a fairly reliable estimate of the unobservable vertical velocity field.

This can be used to calculate the vertical temperature flux associated with

the transient eddies. A cross section of the zonal mean transient vertical

temperature flux, [co'T'], for the DJF season, is shown in Fig. 5.10. The

most striking feature of this plot is the way in which the distribution of the

flux echoes that of the poleward temperature flux, in just the way that the

128

Transient disturbances in the midlatitudes

constant surface

warm

advection

cold

advection

Fig. 5.9. Schematic longitude-height section through

wave which is transporting

heat polewards.

thermal wind balance

is to

be maintained, the warm and cold

advections associated with the eddy lead

vertical circulations which transport

heat upwards.

arguments

the preceding paragraph would suggest.

The

flux

almost

uniformly upwards, with only small regions

very weak downward flux

away from the main storm track regions. Put another way, the temperature

flux vector has

rather constant gradient, pointing polewards and upwards.

The typical gradient is around 1 in 1000.

5.3 Atmospheric energetics

In order to gain some insight into the processes which generate the transient

eddies observed in the midlatitude atmosphere, we will present an approach

to large scale atmospheric energetics. We have already discussed in Chapter 3

some general thermodynamic arguments which showed that

net thermally

direct circulation is required

generate kinetic energy in the atmosphere.

In this section, we will present

discussion based on the quasi-geostrophic

dynamics of the large scale flow. This will, of course, confirm the deduction

from thermodynamics (which we would expect, since our equations include

the basic laws of thermodynamics) but will additionally give insight into the

types

dynamical structures needed

release the energy which thermo-

dynamics shows is available. The approach is

traditional one, pioneered

by Lorenz in the 1950s. The mathematical details will be expounded only

briefly since they are available in many other text books.

The kinetic energy of a unit mass of the atmosphere is simply (u

+ v

)/2

(we will ignore the very tiny increment

kinetic energy associated with

vertical motions). Then the total atmospheric kinetic energy JT is obtained