James I.N. Introduction to Circulating Atmospheres

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9.1 The seasonal cycle of the stratospheric circulation

309

-60

-30 0 30

Latitude

60 90

-90

Latitude

Fig. 9.4 (cont.). (c) Net heating, including long wave cooling, January; (d) net

heating, March. Contour interval

Kday"

. In (a) and (b), shading indicates values

in excess of

K day"

, and in (c) and (d) shading indicates net heating. (After Gille

and Lyjak 1986.)

310

The stratosphere

.001

-30S 0

LATITUDE

Fig. 9.5. Cross sections showing the zonal mean zonal wind

[u]

in the stratosphere,

based on the climatology of Fleming et. al. (1990): (a) January; (b) July. The contour

interval is

ms"

, with easterlies indicated by shading. In this, and succeeding

similar plots, the vertical coordinate is 1000 \og(p/pR). It may be noted that the tick

marks are therefore placed roughly every 14.7 km.

9.1 The seasonal cycle of the stratospheric circulation

(a)

311

.001

-60S

-30S 0 30N

LATITUDE

60N

9DN

.001

-30S 0

LATITUDE

Fig. 9.6. A comparison of radiatively determined and observed zonal mean strato-

spheric temperatures for July: (a) radiative equilibrium, kindly supplied by K. P.

Shine, and based on an extension of the calculations in Shine (1987); (b) Observed,

from the climatology of Fleming et al (1990). Contour interval 10 K, with shading

indicating temperatures less than — 60 °C.

312 The stratosphere

of the winter circulation. Easterlies persist until the following autumn. But

when the warming comes earlier in the winter, or is less strong, a gradual re-

laxation towards the normal winter flow takes place, essentially on radiative

timescales.

Figure 9.7 shows a typical sequence of fields during a stratospheric warm-

ing event. Although the actual reversal of the winds is indeed very abrupt,

anticyclones build up outside the polar vortex for some time before the

warming.

The dramatic changes during a warming, which take place on a timescale

of only a few days, point clearly to a dynamic origin for the event. Propaga-

tion of disturbances from lower levels seems to be implicated, and there

is evidence linking warmings to major anomalies (such as blocking) in the

troposphere. The transition in the southern hemisphere stratosphere is less

abrupt, and the polar night jet is less disturbed. This presumably reflects the

somewhat weaker steady long waves in the southern hemisphere troposphere.

The less disturbed state of the southern polar vortex is important in leading

to the extremely large depletions of ozone observed in recent years above

Antarctica. The lack of major disturbances means that air is trapped within

the polar vortex for long periods and thus destructive catalytic reactions

have time to occur.

The theory of upward propagating Rossby waves, outlined in Chapter 6,

accounts in broad terms for many of the observations recounted in this

section. At the same time, eddies in the winter stratosphere carry important

fluxes of heat and momentum; these modify the zonal mean state. As our

study of the Eliassen-Palm flux in Section 6.4 revealed, propagation of waves

and their interaction with the mean flow are related matters. We return to

the discussion of this in the next section.

9.2 Wave propagation and mean flow interactions

The requirements for the vertical propagation of Rossby waves rather neatly

account for the absence of waves in the summer stratosphere and the long

wave disturbances of the polar night jet observed in the winter. Figure 6.19

shows that steady Rossby waves will be evanescent in the vertical when the

zonal wind is easterly. Thus, we would expect the mean flow of the summer

stratosphere to be almost undisturbed by propagating waves. Indeed, it is

highly axisymmetric. Incidentally, it is not greatly disturbed by transient

eddies either. The theory of Section 6.4 is easily modified to the case of

transient waves, with a phase speed c. The result is simply to replace U by

—

cm Eq. (6.43). The phase speed of tropospheric Rossby waves is gener-

9.2

Wave propagation and mean

flow

interactions 313

ally fairly small compared to the magnitude of the summer stratospheric

easterlies, so that there is little possibility for transients to propagate into

the summer stratosphere. Vertical propagation is possible in westerly flow,

such as would be expected in the winter stratosphere. The longest waves

can propagate most readily, while shorter waves will be evanescent. The

filtering effect of the stable stratosphere in westerly conditions, illustrated by

Fig. 9.1, is accounted for in a straightforward way. Of course, the theory of

Section 6.4 is highly idealized, and its assumption of constant U and N is

hardly realistic. The Eliassen-Palm formalism provides a way of extending

the theory to more general conditions. In this section, we will concentrate

upon the winter situation in order to evaluate the transport properties of the

stratospheric planetary waves.

The temperature flux carried by a vertically propagating Rossby wave was

derived in Eq. (6.53) for the simplest situation of vertical propagation in a

uniform zonal wind. In addition to the temperature flux carried directly by

the eddies, there will be an indirect eddy transport of temperature by the

eddy induced mean meridional circulation. Together, these effects determine

the total heat transport resulting from the eddies. The same ideas will be

required to discuss the transport of trace constituents, such as ozone, by

the winter stratospheric circulation, an aspect to which we will return in the

next section. The eddy and meridional mean components of the temperature

flux tend to cancel out, a result which is qualitatively clear from the work

of Section 4.4. There we showed that a maximum of the poleward eddy

temperature flux in the midlatitudes would induce a thermally indirect mean

meridional circulation. That is, the induced circulation will tend to transport

heat downwards and equatorwards, while the eddies achieve the opposite.

To make the argument more quantitative, we will restate the analysis of

Section 4.4, here using pseudo-height z' as the vertical coordinate since we

are concerned with the stratosphere.

The Eulerian mean equations in local Cartesian coordinates were given as

Eqs.

(4.27) and (4.28), together with the thermal wind equation, Eq. (4.29).

In terms of the pseudo-height, they may be written:

^-/[•>]-~[«*i'*] +

i],

(9.1)

dt g dy

314 The stratosphere

Fig. 9.7. Sequence of synoptic fields at

kPa during a stratospheric warming event.

Solid contours show geopotential height, contour interval

500

with values greater

than

km indicated by shading. Dashed contours show temperature, contour

interval 5K, with values between

210

K and

220

K shaded. The outer latitude circle

°N.

(a) 23 December 1981; (b) 26 December 1981.

9.2 Wave propagation and mean flow interactions

315

(d)

Fig. 9.7 (cont.). (c) 29 December 1981; and (d) 2 January 1981. (Plots kindly supplied

by A. O'Neill.)

together with the thermal wind balance:

(9.3)

316

The

stratosphere

and continuity:

A useful transformation of these equations is obtained by defining a residual

wind such that:

The residual meridional wind still satisfies the continuity equation, and so it

defines a residual meridional circulation. Substituting into the zonal mean

momentum and thermodynamic equations, we have:

^/M

[I],

(9.6)

dt g

These equations are called the 'transformed Eulerian mean' (TEM) equations.

Note that the eddy term only appears explicitly in the momentum equation,

where the combined effects of the temperature and momentum fluxes are

contained in the V

•

F term. The thermodynamic equation reveals that

in steady flows, the residual meridional circulation is purely a response to

gradients of

heating.

The Eulerian mean circulation induced by the poleward

temperature fluxes has been removed from the residual circulation by the

transformations, Eqs. (9.5a, b).

In conditions of steady, frictionless and adiabatic flow, the TEM equations

establish an important result, called the Eliassen-Palm theorem. Consider

the special case of frictionless, adiabatic and steady flow. Then the TEM

equations reduce to:

-f[v]

— V-F,

(9.8a)

[w]

= 0.

(9.8b)

From

Eq.

(9.8b) together with continuity,

conclude that

[w]

= [v]

= 0,

and consequently that

•

F = 0. (9.9)

9.2

Wave propagation

and

mean flow interactions

317

This equation does not say that the eddy fluxes are zero. It does say that

the variation of

[v*9*]

in the vertical and of

[u*v*]

in the horizontal must

be such that the Eliassen-Palm flux has zero divergence. This is the case for

the simple vertically propagating Rossby wave discussed in Chapter 6. We

showed there that the poleward temperature flux associated with this wave

is substantial. However, its vertical variation is such that:

^7(p>*n)=0.

(9.10)

That is, Eq. (9.9) is satisfied if

[u*v*]

= 0. The physical interpretation of this

result is that although there is a poleward temperature flux associated with

the Rossby wave, it is exactly balanced by the equatorward temperature flux

carried by the induced Eulerian mean circulation.

We have considered other situations where the Eliassen-Palm theorem

does not apply. The temperature fluxes of a growing Eady mode, discussed

in Section 5.4 do not satisfy V

•

0, even though the Eady model

frictionless and adiabatic. Strictly, there is infinite divergence of F in an

infinitesimal layer near the lower boundary, and corresponding convergence

near the upper boundary. The reason is that the Eady wave is exponentially

amplifying, so we are not dealing with a steady flow. Although the induced

meridional circulations offset the poleward temperature fluxes carried by

the Eady mode, they do not exactly compensate for it, and so there

net poleward temperature transport. In the tropospheric global circulation

itself,

similar considerations apply. The Ferrel cell is thermally indirect and

so tends to transport heat equatorward. But the flow is highly unsteady,

and, furthermore, friction acts near the ground. Thus there is certainly net

poleward temperature flux associated with midlatitude tropospheric eddies.

Our arguments so far have been largely based upon the linear theory of

vertically propagating Rossby waves. The essence of a Rossby wave is

small meridional displacement of

fluid element. The element conserves

its potential vorticity as it moves into a region where the ambient potential

vorticity is different. The resulting circulations tend to return the Rossby

wave to its 'home' latitude, providing the restoring tendency which results in

wave motion. Note that this displacement is a totally

reversible

displacement.

However, as the Rossby wave propagates higher, its amplitude may vary,

and can lead to large, irreversible distortions of potential vorticity contours

or 'breaking'. At least two mechanisms for breaking are possible.

The first will operate even when

[u]

is constant; according to Eq. (6.45),

the amplitude will increase exponentially with height as the wave propagates

to levels where the density is lower. The wave action density is conserved

318 The stratosphere

as the wave propagates to higher levels, but the mean particle displacement

increases. This will mean that above some level, the linear assumptions of

the theory will break down. Typically, this will happen when the meridional

displacement of a fluid element becomes comparable with the wavelength

of the waves. In this situation, contours of potential vorticity become

highly distorted, and the potential vorticity is pulled out into long filaments.

Such long, thin filaments are barotropically unstable and might be expected

to break up rapidly into discrete vortices. The instability means that the

distortion of the potential vorticity contours rapidly becomes irreversible.

Various dissipative processes can act on these small scale fluctuations to

smooth out the potential vorticity field. These processes are not simple to

describe in analytical terms, and the rapid reduction of scales makes them

difficult to represent in a numerical model. Consequently, the details of the

saturation process are not fully understood. The overall effect, though, is to

mix the potential vorticity throughout the region occupied by the saturated

Rossby wave, giving a broad region in the midlatitudes where potential

vorticity gradients are small.

The second breaking mechanism may be associated either with meridional

or vertical propagation, and will occur when the zonal wind changes with

height. In particular, if

[u]

changes sign, a 'critical line', such as we discussed

in Chapter 6, will be associated with the locus of points where

[u]

= 0. As

a Rossby wave approaches a critical line, its amplitude will increase and its

group velocity will decrease. The assumptions of linearity will break down,

and irreversible shredding and mixing of the potential vorticity field will

occur; the Rossby wave will break in much the same way as for the increase

of amplitude with decreasing density.

There is an analogy between this saturation and breakdown of vertically

propagating Rossby waves, and the well-known behaviour of surface gravity

waves on deep water as they approach a shelving beach. As the waves

propagate into regions of shallower water, they become steeper. Eventually,

they become so distorted that they 'break', that is, the reversible displace-

ments of water associated with wave motion become irreversible, and a rapid

shrinking of the scales of motion takes place. This analogy has led various

authors to speak of the 'breaking' of planetary scale Rossby waves in the

middle stratosphere, and to refer to the region of small potential vorticity

gradient as the 'stratospheric surf zone'.

Figure 9.8 shows a map of the Ertel potential vorticity (see Eq. (1.79))

on the

850

K potential temperature surface during the northern hemisphere

winter. The vortex circumscribed by the polar night jet shows up as a

strong maximum in the potential vorticity field, distorted by the upward