James I.N. Introduction to Circulating Atmospheres

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10.1 Major influences on planetary circulations 349

Fig. 10.1 (cont.). (c) Jupiter, also in visible light.

gaseous until very great depths. There may be a small rocky core near

the centres of these bodies. But the exchange of momentum between the

meteorologically active layers of the atmospheres and the deep interior is

likely to be small, and therefore the spin down time for circulation will be

much longer than for a terrestrial planet. Paradoxically, this does not imply

that atmospheric circulation will necessarily be more vigorous. Consideration

of energetics shows that the generation of kinetic energy by upward heat

fluxes must ultimately balance the frictional dissipation of kinetic energy. If

the friction were reduced, then the generation of kinetic energy, and hence

the poleward and upward temperature fluxes, must also be reduced. The

result would be a meteorologically inactive atmosphere, with the velocity

vectors predominantly parallel to the temperature contours.

Significant orography on the solid surface of a planet will modify atmo-

spheric circulation. Mars has considerably higher orography over wide areas

than Earth, and so the contribution of steady, forced eddies to global heat

transport might be larger. The most extreme example is provided by the

Earth's oceans. The ubiquitous presence of longitudinal barriers dominates

the circulation of most of the oceans.

In the following sections, we will consider groups of known planetary

atmospheres, and attempt to relate these principles to what is known about

their atmospheric circulations. We will also discuss briefly the circulations

in related systems such as the oceans and rotating fluid systems in the

laboratory.

350 Planetary atmospheres and other fluid systems

10.2

Terrestrial circulations

Terrestrial global circulation is characterized by an essentially axially sym-

metric 'Hadley regime' at low latitudes, and a zonally disturbed 'quasi-

geostrophic' regime at higher latitudes. The two regimes interact strongly,

and forced steady waves, particularly in the northern hemisphere winter,

introduce further complexity. These matters have all been dealt with in

various degrees of detail in the earlier chapters of this book. The aim of the

present section is to investigate the range of conditions for which this broad

description remains helpful. In particular, the quasi-geostrophic regime will

be discussed in this section, while the Hadley regimes will be considered

further in the next section.

Heat transport in the terrestrial quasi-geostrophic regime occurs princip-

ally in baroclinic disturbances which exhibit a characteristic westward phase

tilt with height. The Eady theory of Section 5.4 sets out a linear instability

theory which gives a reasonable prediction of the scale and structure of

disturbances observed in the Earth's midlatitudes. Recall that the zonal

wavenumber of the most unstable wave is

1.61K

(10.2)

where K~

{

= L

, the 'Rossby radius of deformation', given by

= — (10.3)

The Rossby radius is the characteristic length scale of disturbances in the

midlatitudes. A first necessary condition for the existence of such disturb-

ances is that the planet should be large enough to accommodate the unstable

wavelengths, that is, that

< a. (10.4)

For the Earth, L

/a is around 0.12, so that it falls comfortably within the

range in which baroclinic instability is possible. For Mars, the same ratio

is about 0.33, suggesting that baroclinic waves are still likely, but that they

will have considerably larger scale relative to the planetary radius.

The growth rate of the most unstable wave was given in Eq. (5.52). It can

be rewritten

a = 0.31i?r

1/2

/, (10.5)

where Ri is the dimensionless Richardson number, defined as

/(u

)

best

thought of as a dimensionless measure of static stability. If thermal wind

10.2

Terrestrial circulations

351

balance holds, Ri can be written entirely in terms of the potential temperature

gradients:

The quasi-geostrophic analysis which led to Eq (5.6) is valid only for large

that is, for rapid rotation and strong stratification, but for relatively weak

horizontal temperature gradients. The Earth's midlatitudes are characterized

by Ri « 50, so that they fall within this range. But on the equatorward flank

of the subtropical jet, the local values of Ri are much smaller, generally

less than 10. The theory can be modified for smaller Ri. But as Ri

approaches unity or less, the dynamical regime becomes entirely different,

being dominated by 'symmetric instabilities', with zero zonal wavenumber,

but a limited meridional scale. Thus, the static stability of an atmosphere is

revealed as a crucial property in determining the character of its circulation.

The difficulty we have is that the horizontal and vertical gradients of

potential temperature are not externally determined characteristics of an

atmosphere. Radiative heat sources and sinks acting alone might serve to

determine radiative equilibrium values of the temperature gradients, but the

circulation of the atmosphere itself will modify, possibly profoundly, the

temperature distribution. Apart from the Earth, it is generally not straight-

forward to determine the actual values of Ri for planetary atmospheres with

any great degree of certainty. This is because it is difficult to measure the

stratification sufficiently accurately. Indeed, for the gas giants, the lapse

rate below the cloud tops is unknown, so that it remains an open question

whether the circulations on these planets are quasi-geostrophic, baroclinic in

character, or whether they belong to a totally different class.

To progress further, we need to parametrize the temperature transport of

quasi-geostrophic circulations in terms of the imposed temperature gradi-

ents and other quantities. Despite many efforts, no very satisfactory way of

doing this is possible. Indeed, this is one of the reasons why global circula-

tion studies are now dominated by GCM modelling experiments, which

endeavour to model the lifecycle of each individual baroclinic disturbance

explicitly. The use of complex numerical models is also perhaps the best way

forward in theoretical explorations of the dynamical regimes of planetary

atmosphere flows. But for the purposes of this exposition, we will explore

a limited attempt to parametrize baroclinic heat transports and hence to

derive an analytical model of the structure of planetary atmospheres for a

range of external parameters.

The analysis is based on the Eady linear model of baroclinic instability.

352 Planetary atmospheres and other fluid systems

Expressions were derived in Section 5.4 for the horizontal and vertical

temperature fluxes, given as Eqs. (5.54) and (5.56) respectively. For the

purposes of the present order of magnitude estimates, the wavenumber of

the most unstable wave will be substituted in these equations, and the fluxes

will be evaluated at a midlevel and midlatitude point. For these particular

choices, the temperature fluxes reduce to

, (10.7)

(108)

Here, *P is the streamfunction amplitude. A typical value of

*¥

must be

estimated so that temperature fluxes can be calculated from Eqs. (10.7) and

(10.8).

Some relationship between *F and the mean state of the atmosphere

must be assumed; such a relationship is called a 'closure hypothesis'. A

number of such closure hypotheses are possible, all equally defensible; the

closure assumption made in this section is that the waves reach a typical

amplitude such that

* = -^ » At/. (10.9)

This implies that the streamlines make angles of around

45 °

with the un-

disturbed zonal background flow when growth ceases. Using Eq. (10.9), in

conjunction with the form of

the

normal mode solution for the most unstable

wave, given by Eqs. (5.44) and (5.46), one obtains estimates for the total

temperature flux:

pel'

(io.il)

These expressions relate the expected typical poleward and vertical temperat-

ure

fluxes

to the imposed horizontal and vertical temperature gradients. Were

they accurate, much current effort in global circulation modelling would be

wasted; we could simply parametrize the baroclinic systems of midlatitudes.

As a result, climate and global circulation studies would be much more

straightforward. In reality, these estimates are extremely crude; at best,

they provide merely an approximate upper bound on the actual temperature

fluxes. Our purpose in this chapter is not to obtain very accurate predictions

10.2

Terrestrial circulations

353

of the actual magnitudes of the fluxes in particular circumstances, but, rather,

to suggest the parametric dependence

temperature fluxes

planetary

properties such as the rotation rate.

The temperature fluxes associated with the baroclinic waves will modify

both the horizontal and vertical temperature gradients. These effects will be

offset by radiative processes which, acting in isolation, will set up

state of

radiative equilibrium. Let us write the thermodynamic equation as follows:

f *„.«.,_ «

[wV

]

JMHW.

,10.12,

oy oz

The smaller, opposing temperature fluxes carried by the induced zonal mean

circulation have been ignored. Differentiating with respect

to y or to z

then yields equations for the poleward and vertical potential temperature

gradients. According

Eady theory, [v*0*]

= 0;

the cross derivative

[w*0*]z);

will also be neglected. The second derivatives of the temperature

fluxes may be approximated by finite difference formulae such that

[v*0*]

[v*6*]/a\

[w*6*]

[w*9*]/H

(10.13)

The first of these expressions assumes that there is a single zone of baroclinic

waves between the equator and pole; in the case which we will consider in

Section 10.4 when there may be be several parallel storm tracks,

rather

smaller scale than a would be needed in this formula. The equations for the

temperature gradients may be written

+ O

J^-V

= ^—-,

(10.14)

V^^B

(10.15)

where, for convenience,

and

are defined as

-[0]

[0]

(10.16)

so that

is positive for the usual situation when the poles are colder than

the equator; B

and V

are the radiative equilibrium values of B and

The steady state climatology

the planet

found by setting the time

derivatives

zero

Eqs. (10.14)

and

(10.15); the problem reduces

solving

pair of simultaneous algebraic equations in

and

By way of

example, suppose that the equilibrium atmosphere is neutrally stratified so

354

Planetary atmospheres

and

other fluid systems

that V

= 0. Then the horizontal potential temperature gradient is given by

and the vertical potential temperature gradient is

The structure of the atmosphere is determined by the value of a single

dimensionless parameter y where

£T

2/3

(5 B

i =

-&TT

(1(U9)

and 5 is the geometrical aspect ratio, H/a. Figure 10.2 illustrates the

solution as y is varied. For very large y, baroclinic instability is so effective

that it destroys virtually all the horizontal potential temperature gradients

and leaves the vertical profile very close to neutral. When y tends towards

zero,

the atmosphere is close to radiative equilibrium. For intermediate

the horizontal temperature gradients are reduced and the atmosphere is

stabilized. The maximum stability is achieved for y = 2. For the Earth, y

has the value 5.0, while for Mars, y is 1.3. Both are fairly close to the regime

where the eddies achieve maximum stabilization of the atmosphere. This

simple calculation suggests that horizontal temperature gradients will be

closer to radiative equilibrium for Mars than for Earth, though the vertical

profile for both atmospheres will be significantly stabilized by the eddies.

The baroclinic disturbances of the Earth's atmosphere are highly irregu-

lar. Individual systems have a wide range of intensities, phase speed and

lifetime. The Earth's tropospheric flow is therefore inherently unpredictable.

Small differences in a given initial state lead to very different evolutions

of individual baroclinic systems, and hence rapidly to very different global

flow fields. Surface observations suggest that the Martian atmosphere, while

essentially similar to that of the Earth in that heat transport by baroclinic

transients dominates the winter circulation, is much more predictable. Fig-

ure 10.3 shows the pressure record from the Viking landers. During the

summer, the atmosphere is very inactive meteorologically, with only small

fluctuations of pressure. A diurnal variation of the surface wind, related to

the local slope of the ground, is almost the only significant weather feature.

In autumn and winter, the pressure rises; this is due to the evaporation of

large amounts of carbon dioxide which condensed out of the atmosphere on

to the southern polar cap during the southern hemisphere winter. The flow

10.2 Terrestrial circulations

355

Fig. 10.2. Showing the horizontal and vertical temperature gradients predicted by

an analytical model of a quasi-geostrophic planetary atmosphere for various values

of the dimensionless parameter y. Selected values of y are indicated by circles. B

has been scaled by

and V by B

/S.

associated with the condensation of carbon dioxide at the winter pole and

its corresponding evaporation at the summer pole is an important element in

Martian global circulation. Simple calculations suggest that it could intensify

the winter jets by as much as

15-20

ms"

During the winter, fluctuations of

the surface pressure as baroclinic cyclone systems passed over the lander were

observed. These fluctuations were highly regular, with similar amplitudes

and periods, suggesting the regular passage of wavenumber 4 disturbances

of rather uniform amplitude over the landing site. A spectral analysis of the

pressure record shows a single sharp peak centred at a period of about four

Martian days.

The simple theory of this section is severely limited. It predicts the typical

length scale of baroclinic eddies, and suggests the degree of modification

of the radiative equilibrium temperature profile. It suggests how these

quantities might depend upon such imposed parameters as rotation rate and

radiative timescale. But it cannot be expected to give an accurate quantitative

prediction of the lapse rate or horizontal temperature gradient. Neither can

356

Planetary atmospheres and other fluid systems

seasons (northern hemisphere)

Sum.|Aut.|Wint.| Spr. | Sum.

Aut. |Wint.| Spr. | Sum.

Aut.|Wint.| Spr. | Sum. |

Year 1

T Year 2 Year 3 I

1500

1750

0.04

0.2

(b)

0.4 0.6 0.8

frequency (cycles/day)

1.0

Fig. 10.3. (a) Pressure versus time (in Martian rotation periods) for the Viking

landers on the Martian surface, (b) Spectral analysis of the winter pressure record,

showing the concentration into a single sharp peak.

it give any indication of the momentum fluxes by the eddies, which are not

predicted by Eady theory, or about the interaction between the midlatitude

quasi-geostrophic regime and the tropical Hadley regime. Investigating these

questions requires a global circulation model. Figure 10.4 shows the results

of a sequence of runs with a global circulation model with different rotation

rates.

The parametrizations of boundary layer processes, radiative heating

and cooling, and of convection and other moist processes were essentially

10.2 Terrestrial circulations 357

terrestrial. The diagram simply shows the zonal mean wind,

[u],

for each

run. The zero rotation rate run shows virtually no systematic zonal flow,

as one would expect. The low rotation rate runs have a strong jet at the

poleward edge of a Hadley cell which extends to very high latitudes. Around

the terrestrial rotation rates, the familiar pattern of a subtropical jet and a

weaker high latitude jet begins to appear. At higher rotation rates, the zonal

flow becomes quite weak, consistent with thermal wind balance. But several

midlatitude jets, each with its own associated maximum of eddy activity,

are present at midlatitudes. At the highest rotation rates, a strong westerly

equatorial jet dominates the flow. Such a pattern is more reminiscent of the

gas giants than the terrestrial planets, and suggests that the variations of

rotation rate might be the most important difference between terrestrial and

Jovian regimes. This is a contentious view which will be discussed further in

Section 10.4.

A sharp reminder of how misleading a parametrization of baroclinic

eddies in terms of Eady (or Charney) type baroclinic instability might be is

provided by a final example. Illustrated in Fig. 10.5 are two contrasting runs

with the simple global circulation model introduced in Section 2.5. Both

runs are for identical radiative equilibrium conditions, designed to simulate

the terrestrial winter circulation moderately faithfully, and for identical

planetary properties, with one exception. The difference is that the first run

has a realistic frictional spin-up time of about five days, while the second

has a very long spin up time of 1250 days. The effect on the eddies is

dramatic. In the low surface friction case, there are virtually no eddies, and

the temperature field is essentially the radiative equilibrium field. The high

friction case has active eddies, and a thermal structure which is consistent

with the simple model illustrated in Fig. 10.2. The surface winds in the high

friction case are small, but in the low friction run they are very large, and

the horizontal wind shears are large. It is this latter feature which causes

the Eady parametrization to break down. The Eady model includes no

horizontal variation of

[u];

its inclusion leads to an intractable, nonseparable

problem. Numerical solutions of the linear baroclinic instability problem

with large horizontal shears show that such shears reduce the growth rate of

unstable waves, primarily by restricting their structure so that they cannot

tap the available potential energy effectively.

The last result emphasizes the very important role that the upper and lower

boundaries play in the baroclinic instability process. The Eady and Charney

models both depend upon the presence of rigid boundaries to support

instability. Even for 'internal baroclinic instability' in which a change of sign

of the potential vorticity gradient at some level makes instability possible,

358

Planetary atmospheres and other fluid systems

MOIST (fl*)

ZONAL VELOCITY

(f)

ZONAL VELOCITY

Fig. 10.4. A sequence of pressure-latitude cross sections of the mean zonal wind

[u]

from numerical simulations of an Earth-like planetary atmosphere, in which the

rotation rate of the planet has been varied. (From Williams, 1988a.)

without some layer where friction can occur, horizontal shears will build up

during the growth of baroclinic waves to finite amplitude, and these shears

will eventually stabilize the fluid. The implications for a planet with a very

deep atmosphere, or for an atmosphere with no solid lower boundary, are

that baroclinic instability might not be a very important mechanism. This

is a consideration to which we will return in Section 10.4 when we consider

the circulation of the giant planets.