James I.N. Introduction to Circulating Atmospheres

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8.7 Chaos and ultra

low

frequency variability

299

LATITUDE

Fig. 8.22. Structure of the first EOF of

[u]

for the long integration of the simple

global circulation model. This EOF accounted for 31% of the total variability of

the time series, and an even larger fraction of its very low frequency behaviour.

low order equation set would have to possess. As a minimum, one variable

would be needed to represent the baroclinicity of

the

zonal

flow,

and a further

two variables would be needed to describe the (complex) amplitude of an

unstable, propagating wave. These are the main ingredients of the Lorenz set.

There is a feedback, such that strong developments of the waves weaken the

baroclinicity which makes them unstable. The extra element which is needed

in the Lorenz set to make its solutions chaotic is the interaction with a forced

stationary wave, represented in the last term in Eq. (8.19b). If it is strong

enough, this continuous forcing can disrupt any balance between the waves

and the zonal flow, so that the system is always unsteady. There is no forced

stationary wave in the simple global circulation model. Instead, we might

appeal to interactions with a more slowly propagating long wave component

to the flow, or to wave breaking, which would involve interactions with a

harmonic of the unstable wave. Clearly, the possibilities rapidly multiply

as extra degrees of freedom are introduced. One way of placing bounds

on the sort of equation sets which would be needed to describe the long

term behaviour of the global circulation model would be to determine the

fractal dimension of its chaotic solutions. That, at least, would tell us how

many variables would have to be retained in a low order approximation to

it.

300 Low frequency variability of the circulation

This turns out to be a very difficult task. The number of data points

required to determine the fractal dimension of a solution becomes very large

as the fractal dimension increases. Determinations of fractal dimensions

greater than about 7 or 8 are probably unreliable. Attempts to determine

a fractal dimension for the kind of integrations shown in Fig. 8.21 are not

encouraging. Their dimension is probably well in excess of 8, unless some

severe time smoothing is introduced. The most optimistic view is that the

simple global circulation model embeds some simpler model with dimension

perhaps 10 or 20; clearly, extracting such a model from integrations of the

full model would be a very difficult task. The more pessimistic possibility is

that the simple global circulation may not be reducible to simpler terms. In

that case, it would be a waste of time seeking any 'explanations' of its ultralow

frequency behaviour. More radically, we would be left with the concern that

a higher resolution model would exhibit qualitatively different behaviour,

since it is a set of many more equations. The use of global circulation

models to study the behaviour of the atmosphere relies on the implicit

assumption that, given enough resolution, the large scale, low frequency

evolution of the flow will be independent of the model. At present, it is an

open question whether this assumption is correct.

8.8 Problems

8.1 From a long numerical integration of the Lorenz set, Eq. (8.19a-c), the

following data were obtained:

U =

1.0606,

A = -0.0560, B = 0.3326,

1.4129,

= 0.7612, & = 0.9319,

UA = 0.0138, UB = 0.2685, AB = -0.0693.

Determine EOFs for the Lorenz system, and say how much variance each

EOF accounts for. Plot your EOFs on the U-B and A-B planes, and relate

them to Figs. 8.18 and 8.19.

8.2 Use Eqs. (6.9) and (6.10) to plot graphs of the mountain drag ver-

sus flow speed for a zonal wavenumber 2 variation of orographic height,

amplitude

1000

drag timescale

days,

channel width

5000

km.

You may

assume that U

ms"

and that the channel is centred at

°N.

Hence

determine whether multiple equilibria are possible according to the Charney-

Devore theory. If they are, say what the mean flow is for each equilibrium

state.

8.8 Problems 301

8.3 Suppose an isolated tropical heating anomaly were suddenly turned

on, generating zonal wavenumber 3 Rossby waves at

°N.

Given a typical

DJF flow, estimate the poleward group velocity of these Rossby waves, and

hence deduce how long it will be before the anomaly starts to perturb the

flow at

°N.

Repeat the calculation for zonal wavenumber 1.

The stratosphere

9.1 The seasonal cycle of the stratospheric circulation

Up until this point, we have concentrated almost exclusively upon the tropo-

sphere, which is characterized by a relatively weak stratification, with a

temperature lapse rate of around 6-7 K km"

. At the tropopause, the lapse

rate becomes close to zero; the lower stratosphere is nearly isothermal. The

corresponding change in stratification, as measured by the Brunt-Vaisala

frequency, is by a factor of around two, from values of

10~

in the

troposphere to values of

10~

in the lower stratosphere. In the upper

stratosphere, from heights of

km to around

km,

the temperature actu-

ally increases with height. The transition to stably stratified conditions is

called the tropopause, which is extremely sharp in the tropics and midlatit-

udes.

It is rather more gradual in polar latitudes, especially in winter when

there is no incoming sunlight. The abrupt increase of stratification at the

tropopause means that the stratosphere is dynamically very different from

the underlying troposphere. Baroclinic instability is virtually suppressed

and disturbances are mainly forced from below. The stratification acts

as a filter, removing the smaller scale disturbances and allowing only the

longest waves to propagate out of the troposphere to great heights in the

stratosphere. Shorter wavelength disturbances are thereby trapped in the

troposphere, which behaves as a waveguide, the upper boundary of which is

the tropopause.

Figure 9.1 illustrates this filtering action. Fields have been taken from a set

of ECM WF analyses to show the synoptic pattern of

flow

for the same winter

analysis time but at different levels, ranging from the upper troposphere up

to the middle stratosphere. At

kPa,

a number of sharp troughs associated

with surface depression systems can be seen, as well as an anticyclone to

the west of Ireland and a number of other disturbances. At higher levels,

302

9.1 The seasonal cycle of the stratospheric circulation

303

300

100

Fig.

9.1.

The streamfunction for 22 January 1987 at various levels over the northern

hemisphere, (a) 30kPa (about

9 km)

(b) lOkPa (about

17 km)

26 km)

and (d) lkPa (about

34 km)

. Contour interval 10

a steady transition is noted; by the 3kPa level (at about 24km above the

Earth's surface) the only features to be seen are the tight vortex displaced

from the pole towards northern Europe and a weak anticyclone over the

northern Pacific. A Fourier analysis of the streamfunction field at this upper

level reveals that it is composed almost entirely of zonal wavenumbers 1-3.

In the summer season, a similar, but even more dramatic, filtering of

wave like disturbances is seen. Plots of streamfunction for 22 July 1986 are

shown in Fig. 9.2. At

kPa,

the rather weak midlatitude westerlies are very

disturbed, with large amplitude, transient systems particularly evident over

the two ocean basins. By lOkPa, the cyclonic vortex has nearly vanished,

with just a vestigial feature over North America. Instead, the field at

this level is dominated by the massive anticyclone over the Middle East

304

The stratosphere

300

100

Fig. 9.2. Similar to Fig. 9.1, but showing the streamfunction for 22 July 1986. (a)

30kPa, (b) lOkPa, (c) 3kPa and (d) lkPa. Contour interval 5 x 10

and central Asia associated with the Asian monsoon. This anticyclonic

character becomes accentuated at

kPa, while at

kPa an almost precisely

axisymmetric anticyclone is centred over the north pole and fills the entire

summer hemisphere.

The theory of Section 6.4 gives a good account of the changing character

of the flow with height shown in Figs. 9.1 and 9.2, and we will return to the

application of this theory to the stratosphere in Section 9.2. But, first, it is

necessary to give some account of the zonal mean flow in the stratosphere.

This is primarily driven by radiative effects, but is strongly modified by

dynamical heat transports. It has a characteristic seasonal cycle.

Approximately 1% of the sunlight incident on the atmosphere, mostly

at ultraviolet wavelengths, is absorbed in the stratosphere. But because

of the very low density of the air, substantial heating rates can result.

9.1 The seasonal cycle of the stratospheric circulation 305

Figure 9.3 shows the daily mean flux of solar radiation incident on the top

of the atmosphere as a function of latitude and time of year. The maximum

insolation occurs at the summer pole, close to the time of the summer solstice.

Around the winter solstice, high polar latitudes receive no sunlight at all.

The most important absorber of ultraviolet radiation in the stratosphere is

ozone, the triatomic form of oxygen. Ozone has a maximum partial pressure

at around

km and a maximum mixing ratio at around

km.

It is an

extremely effective absorber of ultraviolet radiation with wavelengths of less

than

300

nm.

Thus the maximum heating rates, of up to

K day"

, occur

near the top of the ozone layer, at around

km.

It is this pattern of heating

which effectively determines the gross features of the vertical temperature

profile of the atmosphere, with stable stratification from the tropopause

to the stratopause at about

km,

and rather less stable stratification in

the mesosphere. The atmosphere develops strong temperature gradients in

response to this heating, until long wave cooling is sufficient to balance the

heating. Cooling is principally due to carbon dioxide, although both water

vapour and ozone itself have important infrared emission bands which make

large contributions to the cooling. Fig. 9.4 shows heating rates due to

absorption of solar radiation by ozone, and the net heating rates due to all

radiative processes. At the solstices, the heating follows the pattern we might

expect from the distribution of the solar radiation, with heating throughout

the summer hemisphere and intense cooling at high winter latitudes. At the

equinox, the pattern of heating is roughly symmetric about the equator, with

heating in the tropics and cooling at high latitudes in both hemispheres.

Although ozone has such a profound effect on the temperature structure

of the stratosphere, its total concentration is extremely small, amounting to

no more than 1 molecule in 10

at its most concentrated. The total volume

of ozone in a column is sometimes stated in terms of the depth of a column

of gas if the total ozone were isolated and compressed into a thin layer

100

kPa and

273

K. Typical values range from 2.6 to

4.5

mm,

depending

upon latitude and season. In contrast, the rest of the atmosphere measured

in this way would have a depth of

km.

The situation is complicated by the

fact that ozone is a highly variable constituent. The factors influencing its

concentration will be discussed in more detail in Section 9.3.

In the winter, the heating rate is zero at high latitudes, since there is

no incident sunlight. The heating rate builds up as tropical latitudes are

approached. The result is an intense gradient of temperature near the

poleward limit of sunlight at 66° of latitude, with very low temperatures

near the pole. In thermal wind balance with this temperature gradient, the

zonal winds increase with height at high latitudes, and are quite intense

306

The stratosphere

Incident solar radiation

270 330

Day of year

Fig. 9.3. The daily mean flux of solar radiation incident upon the top of the

atmosphere as a function of latitude and time of year. Contour interval 50Wm~

Shading denotes the polar winter regions of perpetual darkness.

above

kPa.

This high latitude jet stream or 'polar night jet' is generally far

from axisymmetric, at least in the northern hemisphere; as seen in Fig. 9.1,

it is distorted and displaced away from the pole.

Figure 9.5 shows latitude-height cross sections of the zonal mean zonal

wind in the stratosphere. In the summer hemisphere, the tropospheric

westerly jet dies away quickly in the lower stratosphere, and the wind regime

becomes predominantly easterly. In the winter hemisphere, the tropospheric

westerly jet also becomes weaker above the tropopause, while the polar

night jet rapidly strengthens, attaining values of around

ms"

at the

stratopause. Winds in the tropics are more difficult to estimate. They cannot

be inferred from thermal wind balance, and direct measurements of the

stratospheric wind field are difficult to obtain above the usual ceiling for

radiosonde ascents.

Although we have just discussed the gross features of the stratospheric

zonal mean temperature and wind field in terms of the radiative equilibrium

distributions of temperature, it must be appreciated that in some respects the

stratosphere is very far from a state of radiative equilibrium. Figure 9.6 shows

the results of a computation of the radiatively determined stratospheric tem-

perature field, obtained by integrating a model which included the seasonal

cycle of solar radiation but ignored dynamical heat transports, compared

9.1

The seasonal cycle

the stratospheric circulation

307

with that observed. The observed winter pole is considerably warmer than

radiative equilibrium, implying that

the

atmospheric circulation

transporting

heat into high latitudes. More strikingly, observations reveal a local maxi-

mum of temperature in the midlatitudes, with a cold equator and maximum

temperature near

°N.

The associated reversal of the usual temperature

gradient is responsible for the easterly shear above the tropospheric jet and,

consequently, for the characteristic jet maximum near the tropopause. The

existence of this cold stratospheric equator is clear evidence for a very active

circulation in the stratosphere which transports heat polewards.

The 'Charney-Drazin' theory introduced in Section 6.4 predicts the ob-

served filtering effect of the stratosphere on the eddies. We saw there that

in westerly flow, only long wavelength Rossby waves can propagate vertic-

ally. Shorter wavelengths are evanescent and will die away quickly with

height. The stronger the westerly flow, the longer the wavelengths of vertic-

ally propagating Rossby waves. The rather strong westerlies associated with

the polar night jet will prevent the upward propagation of smaller wave-

length disturbances which are seen to become attenuated rapidly above the

tropopause. At the highest levels, only wavenumbers 1, 2 and 3 are present

with significant amplitude. If anything, the amplitude of these wavelengths

seems to increase with height. This result is consistent with the density effect

predicted in our Charney-Drazin theory.

In midsummer, the incident sunlight is a maximum at the pole. The relat-

ively low angle of the sun is more than compensated by the long day at high

latitudes. In the troposphere, the scattering and absorption of sunlight along

the long atmospheric path of the beam means that insolation is still larger

in the summer tropics. But in the stratosphere, scattering and absorption

are less important. The temperature maximum is at the summer pole, and a

poleward temperature gradient is characteristic of the summer stratosphere.

Thermal wind balance suggests easterly shear within the stratosphere, and

indeed by the

kPa level, easterlies are established throughout the summer

hemisphere.

The transitions between the typical summer and winter flows are of

considerable interest. The autumn transition is generally gentle and accom-

plished on radiative timescales (typically 5-20 days, depending upon height).

As the solar heating declines at the pole around the equinox, cooling takes

place and a small cyclonic vortex starts to form over the pole. The easterlies

characteristic of the summer season persist in the midlatitudes. The polar

vortex strengthens and expands until a highly disturbed polar night jet is

established, while the easterlies retreat into the tropical latitudes. The trans-

ition is generally complete by the end of November. The spring transition

308

The stratosphere

(a)

-60 -30 0 30

Latitude

60 90

-90 -60

-30 0

Latitude

Fig. 9.4. Cross sections showing the heating rates in the stratosphere: (a) heating

due to ozone, 21 December; (b) heating due to ozone, 21 March.

is much more dramatic. The westerlies break down rapidly, sometimes on

timescales of only a few days. Anticyclonic circulations disrupt the polar

vortex and displace it from the pole. At the same time, the temperature rises

substantially over the polar regions. Such 'stratospheric sudden warmings'

are among the most dramatic synoptic events in the neutral atmosphere.

The sudden change of circulation establishes a summerlike easterly regime.

When the warming occurs late in the winter, it can mark the final breakdown