James I.N. Introduction to Circulating Atmospheres

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10.4 The

atmospheric circulation

the

giant

planets

369

wavenumber kp where:

gx 1/2

j) , (10.25)

U being a characteristic eddy velocity, the nonlinear and /} terms become

equal; eddies begin to propagate as coherent Rossby wave packets. Now the

frequency of Rossby waves can be written (see Eq. (6.13)) in the form:

—

cos a, (10.26)

where a is the angle between the wavenumber vector and the x-axis, while

the frequency of waves in the turbulence regime is simply

= Uk. (10.27)

Once energy reaches the wavenumber kp, further cascade to low frequencies

is blocked by the Rossby wave propagation. Hence kp is sometimes called

the 'Rhines blocking wavenumber'. The cascade can proceed only if the total

wavenumber remains fixed, while the wavenumber vector rotates towards

the meridional direction. That is, eddies become elongated in the zonal

direction and contract in the meridional direction. The end result is a set

of alternating easterly and westerly currents, whose characteristic width is

Lp = n/kp. Figure 10.10 illustrates this process schematically.

Numerical experiments confirm these rather heuristic arguments. We

conclude that the end result of arbitrary forcing of eddies at some large

wavenumber kf on a rapidly rotating planet will be a set of zonal jets

whose width Lp depends upon the rotation rate and the level of eddy kinetic

energy. Figure 10.11 shows the result for a Jovian parameter setting. Friction

induced meridional circulations could then account for the banded clouds

associated with the jets.

In the model used to produce Fig.

10.11,

the forcing was simply a pre-

scribed mechanical forcing. The model serves to demonstrate that if isotropic

kinetic energy is generated at some large wavenumber kf, two-dimensional

turbulence with a /?-effect will eventually generate a number of zonally ori-

entated jets. The question for the gas giants then concerns the nature of the

eddies that force the large scale flow. Since some hydrodynamical instability

of the flow must be present to generate such eddies, one would anticipate

some interaction between the jets and the eddies. For example, baroclinic

instability might be concentrated in the jet regions, where the vertical shears

are large, and would be weaker in the horizontally sheared, but more weakly

baroclinic, regions either side of the jets.

Two major theories exist for the eddies on Jupiter and similar planets. The

370

Planetary atmospheres and other

fluid

systems

Turbulence

Rossby waves

Zonal wavenumber k

Fig. 10.10. Schematic illustration of the evolution of geostrophic turbulence on a

/?-plane towards a pattern of parallel zonal jets.

first is that baroclinic instability is responsible. The temperature contrast

between the poles and the equator on Jupiter is very small. Presumably,

variations of the solar input across the planet are compensated by variations

in the heat flux from the interior of

the

planet. Larger horizontal temperature

gradients might be present within the individual jets. Even then, it is not

clear that baroclinic instability will be present, other than in a very weak

form. The Eady instability relies on the presence of rigid boundaries at top

and bottom where

= 0. The sudden change of static stability in the Earth's

tropopause approximates this boundary condition for deep baroclinic waves.

The Charney model requires a boundary with w = 0 at the bottom, and a

critical level above. In a planet with no solid boundary, and with an adiabatic

lapse rate in the deep interior, these situations are not encountered. There is

a weak tropopause on Jupiter, and this might sustain Charney-type modes

of instability in regions of easterly vertical shear, but it is difficult to see how

baroclinic instability could be strong in regions of strong westerly shear.

The other possibility for baroclinic eddies is 'internal' baroclinic instability,

the necessary condition for which is that the potential vorticity gradient, [q]

should change sign at some level in the atmosphere. While such a condition

can be envisaged, it is difficult to see how it could be sustained over large

10.4 The atmospheric circulation of the giant planets 371

DAY 4.6 DAY 23

Fig.

10.11.

Evolution of the streamfunction in a rapidly rotating thin shell of

barotropic fluid in which small scale eddies are randomly excited. (From Williams,

1978.)

372 Planetary atmospheres and other fluid systems

parts of the planet. Such internal instability is unusual on the Earth or in

laboratory systems; its finite amplitude evolution has not been studied very

thoroughly. Such studies as are documented suggest that the eddies would

generally be weaker than those in the Charney or Eady idealizations.

Vertical convection, driven as a result of the destabilizing effect of the

internal heat flux, might be an alternative source of eddy kinetic energy. On

Earth, small scale convection is capable of organizing larger scale flows, in

cloud cluster systems, or tropical cyclones. Similar processes are thought to

be important in generating the shallow mesoscale systems known as 'polar

lows'

at high latitudes. The predominantly anticyclonic circulation and high

cloud associated with the spots observed on Jupiter and Saturn suggest an

analogy with the anticyclonic upper tropospheric outflow from convectively

driven tropical cyclones in the Earth's atmosphere. However, terrestrial

tropical cyclones depend crucially upon large convergence of latent heat

fluxes in the planetary boundary layer. It is not clear whether analogous

systems could be formed in the absence of a rigid boundary.

The crucial test to distinguish between these possibilities is to estimate the

Richardson number, Eq. (10.6), for Jupiter or Saturn. If baroclinic instability

dominates, Ri should be large. If the flow is driven by vertical convection,

Ri will be very small. However, since both 9

and 9

are quite small, rather

accurate determinations of 9

and 0

(or, equivalently, u

) are needed to

establish Ri accurately. Present estimates of 6, which are based on infrared

and microwave radiance data, have too coarse a vertical resolution and do

not extend sufficiently far beneath the upper cloud layers to give useful

estimates of Ri. A major goal of future space missions will be to obtain

more accurate vertical soundings of winds and temperatures.

The discussion in this section has concentrated upon the possibility that

the observed circulation of Jupiter is confined to the optically thin, meteoro-

logically active layers of the planet. It is seen as an extreme development

of the sort of circulation which we have studied on Earth. There is another

possibility, however. In a planet which is fluid throughout its bulk, the

motions we observe might simply be the surface expression of circulations

extending throughout the deep bulk of the planet, driven by the internal

heat flux. Studies of Jupiter's magnetic field indicate that at deep levels, the

hydrogen which dominates its composition becomes electrically conducting,

so that dynamo effects generate strong magnetic fields. Determining the

circulation then becomes a complicated problem in thermally driven magne-

tohydrodynamics. A description of the circulation in these terms would be

highly speculative and grounded on poorly understood physics.

Before concluding this section, it is worth making a few remarks about

10.5 Large scale ocean circulation 373

Uranus. This is a very similar planet to Neptune, except that its internal

heat flux is not more than 10% of the intercepted solar heat flux, and indeed

is probably almost totally absent. Its other peculiarity is its very large axial

tilt, which means that, averaged through its year, maximum insolation is

received at the poles and minimum insolation at the equator. The depth of

the atmosphere and the weak solar constant mean that the radiative timescale

is very long, and so seasonal effects will be negligible, at least until very high

reaches of the atmosphere are considered. Thus, the meridional temperature

gradients will be reversed from those of the terrestrial planets. Images from

the Voyager probe indicate a very inactive atmosphere, with few clouds and

therefore weak vertical motions. This suggests that the destabilizing effect

of the internal heat fluxes is a most important feature of the other gas

giants. Very large zonal winds were determined from the few clouds that

could be observed; a profile is shown in Fig. 10.12. The zonal winds are not

consistent with a radiative equilibrium temperature variation. This would

have a warm pole and cold equator, implying that the winds would become

increasingly retrograde with height. Instead, strongly prograde winds were

observed at the cloud top levels in the summer hemisphere, especially at high

latitudes. The temperature distribution, too, is very far from radiative equil-

ibrium. Voyager 2 recorded the largest temperatures at the equator. Despite

its bland appearance, it seems that the atmospheric structure of Uranus is

profoundly modified by dynamical transports of heat and momentum.

10.5 Large scale ocean circulation

The oceans form a planetary scale fluid system whose dynamics has much

in common with the dynamics of the atmosphere. Away from the equa-

tor, the small flow speeds mean that the Rossby number is very small, so

that quasi-geostrophic flow dynamics are appropriate. In the tropics, var-

ious equatorially trapped waves such as those discussed in Section 7.1 are

important, though with different characteristic space and timescales than

those of their atmospheric counterparts. However, although the dynamics

of individual oceanic systems are essentially the same as in the correspond-

ing atmospheric systems, the global scale circulation of the ocean is very

different. This is due to two major factors:

(i) The distribution of heating is totally unlike that of the atmosphere,

(ii) The flow is mainly confined to nearly closed ocean basins, with boundaries on

their eastern and western sides.

We consider each of these factors in turn.

374

Planetary atmospheres and other fluid systems

60-

30-

3 0

Lati

-30-

-60-

" ^ ^

-100

100 200

Zonal wind (m s~

)

Fig. 10.12. As Fig. 10.9, but for Uranus.

The sunlight which enters the ocean is entirely absorbed in its uppermost

layers. The typical vertical temperature structure which results consists of

a thin layer of warm, highly stratified surface layers overlying the much

colder, nearly isothermal, abyssal ocean. The highly stratified layer is called

the thermocline; it is generally not more than 1.5km deep. The abyssal

layers of the ocean have temperatures between 2 and 4°C, and may be

3-5 km deep. Thus, at least for the bulk of the oceans, there is no possibility

of releasing potential energy through upward heat fluxes, and so the ocean

circulation cannot be driven thermally. The oceans act as a heat pump,

rather than as a heat engine. The source of mechanical energy is provided

by the surface winds of the atmosphere, which apply a surface stress to the

ocean. At the same time, the fluxes of heat and water vapour from the ocean

surface provide a most important source of thermal energy for atmospheric

circulation, especially in the tropics.

The exception to this is at high latitudes, where strong winter cooling pro-

duces surface water which may be denser than the cold abyssal waters. If this

occurs, thermally driven convection can result, and new bottom water can be

10.5 Large scale ocean circulation

375

0.002

Wind Forcing

0.007

Dissipation

0.009

Fig.

10.13.

Energetics

of the

ocean circulation. Wind forcing

by the

atmospheric

circulation provides

additional input

to the

kinetic energy. Units

as Fig. 5.15.

(Based

data

Peixoto

Oort, 1992.)

formed. Two regions have been identified where such generation of bottom

water occurs. One is in the north Atlantic, and the other is in the Weddell

Sea. The downward convection of cold water in these regions must be

compensated by slow upwelling elsewhere, a circulation called the 'thermo-

haline circulation'. The thermohaline circulation is important in exchanging

water between the thermocline and the abyssal ocean. These exchanges are

important in considering the global budgets of dissolved constituents such

as salt and carbon dioxide. The thermohaline circulation is thermally direct,

and so converts available potential energy into kinetic energy. However,

measurements reveal that it makes a rather small contribution to the kinetic

energy budget of the oceans. The energetics of the oceanic circulation are

illustrated in Fig.

10.13.

Available potential energy and kinetic energy are

given in the same units as were used for the atmospheric energy diagram

Fig.

5.15. The kinetic energy of the oceans is much smaller than that of the

atmosphere. A small amount of mechanical energy is generated thermally,

but over 75% of the observed kinetic energy comes from wind forcing.

The confinement of the flow by continental boundaries gives rise to impor-

tant asymmetries in the circulation in each individual ocean basin. Consider

the flow in a barotropic ocean which is set into motion by wind stresses at its

surface. The flow is described by means of the barotropic vorticity equation

d£ __ k

•

V x

(10.28)

where x

represents the stress resulting from the winds, p

is the density of

seawater and H is the depth of the ocean. If, as we expect, the currents

are weak, the advection terms will be small and so our equation will be

376 Planetary atmospheres and other fluid systems

dominated over most of the basin by a balance between /to and the curl of

the wind stress, k

• V

x x

li. Thus, where there is forcing of anticyclonic

vorticity there will be an equatorward drift of water, while there will be

a poleward drift where cyclonic forcing operates. This so-called 'Sverdrup

balance' is illustrated in Fig. 10.14, which shows the circulation induced in

a square ocean basin by a wind field which has a midlatitude maximum of

westerly

flow.

That

is,

the vorticity forcing is cyclonic in the north of the basin

and anticyclonic in the south. In a narrow region near the western boundary,

however, Sverdrup balance cannot operate. The boundary condition of

= 0

at meridional boundaries means that there must be a meridional flow which

balances the Sverdrup drift at the western boundary. This generates vorticity

which reinforces the wind stress forcing, and so the western boundary current

becomes narrower and stronger.

In the example of Fig. 10.14, an artificial diffusion term was added to

Eq. (10.28). This prevented the western boundary current becoming in-

finitely narrow. In the real ocean, the western boundary current becomes

hydrodynamically unstable, and so eddy transports prevent it from collapsing

entirely. Considerable current oceanographic research is presently devoted to

exploring the nature and role of ocean eddies. The narrow western boundary

currents are clear in any map of mean surface currents in the oceans, and

are very important in transporting warm water poleward in the subtropics,

and cold water equatorward in the high latitude parts of the ocean basins.

It is perhaps no coincidence that the northern hemisphere storm tracks start

just where these tropical and polar waters meet, forming strong meridional

gradients of sea surface temperature. But cause and effect are difficult to

separate; the location of the atmospheric jet itself determines the ocean

circulation and hence the latitude of the largest sea surface temperature

gradients.

Steep orography can lead to analogues to the western boundary currents

in an atmospheric context. The low level jet over east Africa, which is an

important component of the Asian monsoon circulation described in Sec-

tion 7.1 is one such example. Mars is characterized by very high orography,

with plateaux and depressions more than a Rossby radius across, and with

a vertical scale in excess of a pressure scale height. It has been suggested

that boundary currents might be an important component of the circulations

here also.

10.6 Laboratory systems

Fluid can be forced into motion in the laboratory in a way which is closely

analogous to the development of global circulation. Such systems are amen-

10.6 Laboratory systems

377

1.0

0.0

Ctf

0.6

•

I—*

•i-H

0.2

0.0

0.2

0.4 0.6

Zonal distance x

0.8

1.0

Fig. 10.14. An example of the circulation of a barotropic ocean confined to a square

ocean basin, based upon

numerical integration of Eq. (10.28).

able to experiment and suggest ways in which circulation might be modified

under changes of imposed parameter. They enable certain primary dynamical

mechanisms

be isolated. On the other hand, there are some respects

in which they introduce elements which have no analogue

in a

planetary

atmosphere.

The basic apparatus consists

fluid

in an

annular container. This

mounted on

rotating platform, and subjected to

horizontal temperature

gradient.

typical configuration

shown

Fig. 10.15. Different upper

boundary conditions can be imposed. The fluid may be in contact with

solid lid, so that the top and bottom boundary conditions are symmetric, or

378

Planetary atmospheres and other fluid systems

camera

illumination

-a-

working fluid

-b-

Fig. 10.15. The configuration of the basic 'annulus' experiment. The notation used

in the text is defined.

else

may be in contact with air. This latter stress-free boundary condition

is more closely analogous to atmospheric configurations and will be assumed

in what follows. The fluid is commonly water, but other substances can be

used

order to change such parameters as the ratio

thermal diffusion

to viscosity (the 'Prandtl' number). A variety of methods of diagnosing the

resulting flow can

used. These include mounting temperature sensors

in the working fluid,

following the motion

neutrally buoyant plastic

beads.

The equations governing the flow

the annulus system are much the

same as those which govern the atmospheric circulation. The equation of

state is written:

«(T

)),

(10.29)

where

is the expansion coefficient. In fact, for water,

quadratic term in

— To)

is also significant. Since the arguments of this section would not be

altered by including such a complication, it will be ignored. The fluctuations

of density are quite small, and so the Boussinesq approximation

valid.

The other difference from the atmosphere is that the Coriolis parameter is