Planets - University of Leedssmt/planets.pdf · Terrestrial planets, with a rocky mantle and...

Planets

Chris Jones

Department of Applied Mathematics

University of Leeds, UK

Leeds STFC Summer School September 10th

1

Why study Planets?

A vast amount of new information has come from space exploration.

Much of this new data can be �tted into a theoretical framework, but there is

still much that is unexplained.

Planetary Science poses many interesting scienti�c challenges which relate to

our understanding of fundamental physics: e.g. Fluid Dynamics, Dynamo

Theory, High Pressure Physics.

Exoplanets have been discovered recently, and new planets are being found all

the time. Some of these are very di�erent to anything in the solar system,

others may have life.

We live on a planet! Studying planetary physics and planetary meteorolgy

helps us understand our planet, its climate and its weather.

STFC Summer School September 10th Leeds 1/60

2

Some current research areas

Each of these is a huge area of Science!

(i) Formation of the planets, in the light of recent discoveries

(ii) Planetary structure and the physics of their interiors

(iii) Planetary magnetic �elds and their creation

(iv) Dynamics of Atmospheres: planetary meteorology

(v) Exoplanet detection

(vi) Planetary dynamics: ring formation, N-body problem, tidal interactions

etc.

(vii) Planetary Chemistry (Geochemistry)


3

What are planets?

All bodies with radius > 1000 km, that are smaller than brown dwarfs.

Anything large enough to be still hot in the core, but not hot enough

for nuclear reactions. Natural to include the larger moons as well as the

conventional planets.

Fall into three categories:

Terrestrial planets, with a rocky mantle and (usually) an iron core. Mercury,

Venus, Earth, Mars, Moon, 4 Galilean satellites of Jupiter, Titan and Triton.

Gas Giants: Mostly hydrogen and helium, probably with a small solid core.

Jupiter, Saturn and most of the exoplanets, since large exoplanets easier to

observe.

Ice Giants: Uranus and Neptune, mostly water and ammonia. Pluto and

Charon.


4

How did the planets form?

Current view is that planets formed from an accretion disk at the same time

as the Sun. Star and planet formation all part of the same process.

This idea goes back to Kant and Laplace. It fell out of favour in the mid-

nineteenth century, because of the angular momentum problem. However, this

is not seen today as an insuperable di�culty.

(i) Planetary formation 4/60

5

Passing star theories?

Passing star theories were fashionable from 1850-1950. Never really solved the

angular momentum problem, also dynamical arguments suggest you can only

create planets close to the Sun this way.

Lithium is depleted by nuclear burning in the Sun, but not depleted in planets:

suggests that planets not made from fully formed solar material.

Recently, accretion disks observed around T-Tauri stars, which are stars being

born.


6

Star forming regions

Most information about solar system formation comes from observations of T

Tauri disks.

Left: a real T-Tauri disk with an axial jet. Right: artists impression of a

T-Tauri protoplanetary disk.


7

Angular momentum issue

Angular momentum problem: most of the mass of the solar system is in

the Sun, most of the angular momentum is in the planets. Total angular

momentum now much less than in the original protoplanetary nebula. Solar

wind removed this problem.

In an axisymmetric, nonmagnetic disk with negligible viscosity, angular

momentum is conserved, i.e. it stays constant in each �uid element.

If the initial condition is a large roughly uniformly rotating cloud, angular

momentum transfer and loss must have taken place to arrive at what is

observed.

Turbulence enhances viscosity? Dynamo action in disks? Gravitational coupling

from nonaxisymmetric parts of the disk? All active research areas.


8

Planet building

How do planetary systems form from the protoplanetary nebula?

Very active area of research, but still highly controversial. Many unresolved

problems.

Gravitational accretion or collisions of planetesimals?

Diversity of planets suggests collisions are important: e.g. Mercury has a large

core and small mantle: lost its mantle in a collision?

Moon has only very small iron core: result of impact between proto-Earth and

proto-Moon?

Theories in which near circular orbits are formed (as in solar system) have to

come to terms with very eccentric orbits of many exoplanets.


9

Three stages of Planet formation

Grain condensation and growth

The molecular low temperature (

10

Late stage collisions

As this process continues, all planetesimals on crossing orbits get merged,

leaving much larger bodies on non-crossing orbits.

However, self-gravitation perturbs the orbits over a 10Myr timescale, leading

to late stage large impacts.

The evidence of such large impacts can still be seen.


11

Gas giants and Ice giants

The major problem with all this is how gas giants form.

The protoplanetary nebulae of T-Tauri stars appear to lose their gas in less

than 1Myr, short compared to the time required to build rocky cores.

However, there must have been enough gas around for Jupiter and Saturn to

develop, so presumably these planets must have developed rather quickly, for

some unknown reason.

Situation for exoplanets of the 'hot Jupiter' class is even more di�cult, as

these orbit close to the parent star, where gas would be expelled very quickly.

Possibly these planets were formed at large distance, where the gas can hang

around, and migrated inwards at a later stage to where they are now. Migration

occurs due to planets interacting with gas.

Planetary migration an active area of celestial mechanics.


12

Di�erentiation of terrestrial planets

Whatever the detailed processes, the gravitational energy of formation,

GM2/R, must have turned into heat, and this is enough heat to ensure

that the Earth and other terrestrial planets started hot enough to melt the

rock.

The heaviest element present in large quantities, iron, made its way to the

centre of the planet, releasing more gravitational energy and ensuring the core

is hot. This process is known as di�erentiation, and can happen in a few Kyrs

only.

This forms the structure of terrestrial planets, with iron cores and rocky

mantles.


13

Radioactivity in the Core?

Radioactive elements are present in the mantle, and contribute a large part of

the heat coming out of the Earth's surface (44TW).

An interesting question is whether the di�erentiating iron took any radioactive

materials with it down to the core, like Uranium or radioactive Potassium

K40. Potassium is depleted in the mantle, but did it evaporate into space at

formation or end up in planetary cores?

If it didn't, then the core is gradually cooling down, at a current rate of around

1K every 10Myr, which is possible.

If it did take radioactive material down with it, then the core might be in

thermal equilibrium, with the heat �ux out of the core balancing the radioactive

input.

These di�erent scenarios have implications for solid inner core formation and

for dynamo theory.


14

Structure of terrestrial planets

Interior structure of Earth Interior structure of Ganymede

Earth's �uid outer core radius 3480 km, solid inner core radius 1220 km.

Earth's solid inner core is deduced from seismology: don't yet know whether

other planets have solid inner cores.

Why is there a solid core where the temperature is largest? Increasing pressure

raises melting point.

(ii) Interiors of Planets 14/60

15

Convection in the core

Outer core is probably convecting. Driven by thermal convection and

compositional convection.

Compositional convection occurs because the iron in the liquid outer core is

actually a combination of iron and lighter elements (Sulphur, Oxygen). When

almost pure iron solidi�es onto the inner core, it releases buoyant light material,

which rises and stirs the �uid.

The light material released may collect at the top below the CMB (stably

strati�ed `inverted ocean') or it may just mix.

If we assume no radioactivity in the core, then the rate of cooling is fast

enough that the inner core formed only 1Gyr ago. Much younger than Earth,

and much younger than geomagnetic �eld, so dynamos shouldn't depend solely

on compositional convection.

Temperature of core: CMB about 4000K, inner core 5,500K.


16

Mantle Convection

Large density jump at the Core-Mantle Boundary (CMB). Earth appears to be

the only planet currently have plate tectonics (mantle convection).

Although the mantle is a solid on short timescales (seismology) it can �ow

slowly on long-timescales. Shifts the plates around on 100Myr timescale.

Mantle convection transports the 44TW of heat generated in the interior to

the surface. Heat �ux at the CMB probably around 10TW. Mantle convection

is an active area of research. Main problem is how to deal with descending

plates, part �uid, part solid.

Plumes coming out of the Core-Mantle boundary, may go right through the

mantle and emerge at hotspots like Hawaii.


17

Plate tectonics on other planets? Venus

Venus might be expected to have plate tectonics, but the surface suggests not.

Venus surface does look quite recent, around 500Myr old however, with a lack

of cratering compare to the Moon.

Possibly Venus undergoes periodic resurfacing: heat from radioactivity builds

up in the interior because it can't escape by mantle convection.

This gradually builds up buoyant stresses, which eventually lead to a breakdown

and a rapid release of heat to the surface, melting it.

It then settles down and the cycle repeats.


18

Plate tectonics on other planets? Mars

Mars also doesn't seem to have mantle convection at present.

However, Mars has two very di�erent hemispheres

Left Topographic map with Tharsis region prominent: Right Magnetic �eld

of Mars, indicating the hemispheric structure is deep-seated. (Note longitude

plotted di�erently! Hellas basin (blue) is nonmagnetic.)


19

What happened on Mars?

It has been suggested that a dipolar m = 1 (spherical harmonics Pml ) mantleconvection may have occurred in the past, giving rise to this structure.

Alternatively, could be due to a giant impact.

Crustal magnetization is strong and global, so Mars must have had a strong

magnetic �eld when the Southern Uplands formed. The Hellas basin formed

about 500Myr after Mars formation, and is nonmagnetic.

Mars used to have a dynamo, but it switched o� about 350Myr after formation.

What caused it to fail?


20

Structure of the giant planets

The metallic hydrogen layer is caused by the high pressure. The matter is

squashed into a small space, and so the particle velocity goes up (exclusion

principle). In consequence, the hydrogen ionizes, and so becomes electrically

conducting. This allows currents to �ow and hence a dynamo.

The rocky core is actually entirely conjectural. It can't be seen in the gravity

�eld, and consistent models can be produced with no core. Its drawn in

because its hard to understand how Jupiter or Saturn formed without a core.


21

Saturn and the ice giants

Saturn's lower mass leads to lower pressures and hence a smaller metallic

hydrogen region.

Helium rain: the upper parts of gas giants have lower helium content than

solar ratio. Suggestion is that there is an inhomogenous region where a

helium/hydrogen phase transition occurs leading to helium droplets forming

and helium `rain'.

Ice Giants, Uranus and Neptune. These planets have a mantle believed to be

a water-ammonia ocean, which has an ionic electrical conductivity. This is

where the dynamo is believed to be.

However, ionic conductivity is much lower than from metallic hydrogen or

liquid iron. Further complication is that Uranus has very little heat �ux coming

from interior.


22

Planetary magnetic �elds: Terrestrial planets

Earth: mainly dipolar magnetic �eld. Inclination of dipole axis to rotation axis

currently 11.5◦. Field at Core-Mantle boundary, CMB, can be reconstructedfrom satellite and observatory data. Strength ∼ 8× 10−4Tesla = ∼ 8 Gauss.

Field at the CMB in year 2000, units 10−3 T. Note the intense �ux patchesnear Canada and Siberia. Patches under Africa are moving westward. Field at

poles surprisingly low.

(iii) Planetary magnetic �elds 22/60

23

Magnetic �elds of other terrestrial planets

Mercury: internal �eld of dipolar structure. Strength at CMB ∼ 1.4×10−6T,which is surprisingly small. Rotates only once every 57 days. Liquid iron core,

radius 1900 km.

Venus: no magnetic �eld.

Ganymede: approx dipolar, inclination ∼ 10◦. Strength at CMB ∼ 2.5 ×10−4T. Rotates once every 7 days. Core radius 480 km

Io: too close to Jupiter's magnetic �eld to tell whether it has its own magnetic

�eld. Other large moons don't have internal �elds at present.

Extinct Martian �eld is deduced from strong crustal magnetism, which suggests

Mars had a dipole �eld in the past. Lunar rocks also have remanent magnetism


24

Planetary magnetic �elds: Giant planets

Giant planets are all rotating rapidly.

Jupiter: strong magnetic �eld, basically dipolar (more higher harmonics than

Earth), inclined 10◦ to rotation axis, about 17 gauss at surface.

Saturn: very axisymmetric �eld, about 2.5 gauss at surface.

Dipole �eld of SaturnAurora of Saturn, produced by

particles moving along �eld lines


25

Dynamo theory for planetary magnetism

No permanent magnetism above the Curie point, 800◦C for iron. Earth'souter core between 4000K and 5500K.

Thermolelectric e�ects, �eld induced by solar wind, suggested for Mercury, too

small for Earth.

Fluid motion in the core produces dynamo action (Larmor, 1919)

∇×B = µ0j Pre-Maxwell equation

j = σ(E + u×B) Ohm's law in a �uid moving at velocity u

Currents are generated by motion across magnetic �eld lines.

∂B∂t

= −∇×E = ∇× (u×B)−∇× (j/σ), Faraday's law


26

Induction equation

So the magnetic �eld obeys the induction equation which is

∂B∂t

= ∇× (u×B) + η∇2B,

if the electrical conductivity is assumed constant.

The induction term must overcome the di�usion term. The magnetic di�usivity

η = 1/µ0σ, where σ is the electrical conductivity and µ0 = 4π × 10−7 isthe permeability.

Earth's core magnetic di�usivity η ∼ 2m2s−1

The ratio of induction to di�usion is approximately the Magnetic Reynolds

number Rm = U∗`/η. Here U∗ is the typical �uid velocity, ` the typicallength scale, either outer core radius or gap between inner/outer core.


27

It can be proved that for induction to overcome di�usion in spherical geometry,

Rm must be greater than π2, and in practice in the numerical dynamo

simulations Rm of around 50 is needed for dynamo action.

Earth's core velocity U∗ ∼ 4 × 10−4ms−1, and Earth's core size ` ∼3.5× 106m, giving Rm ∼ 700

U∗ from secular variation studies `Westward Drift'

Ohmic decay time `2/π2η ∼ 20, 000 years

How is the velocity at the CMB found from observing the geomagnetic �eld?


28

Core-�ow inversion

Ignore di�usion, (large Rm) and take radial component of induction equation,

∂B∂t

= −(u · ∇)B + (B · ∇)u,

and since ur = 0 at CMB, get

∂Br∂t

= −(u · ∇)Br. (1)

Since Br and ∂Br/∂t can be observed, we might hope to use this equation

to determine u. Unfortunately, there are two unknown components of u andonly one equation. Velocities along contours of constant Br give no signal.

Assume tangential geostrophy, that is near CMB

2ρΩr̂ cos θ × u = −∇p,

Reasonable, as buoyancy and Lorentz forces small close to CMB.


29

Taking the radial component of the curl of this force balance gives

∇ · cos θu = 0, (2)

and (1) and (2) are enough to reconstruct u.

20.0km/yr

20 km per year is 6× 10−4 metres/sec. Slower than a snail! Note westwarddrift in S. Atlantic, Indian Ocean. Paci�c has low secular variation. Waves or

�ow?


30

Energy Sources for Planetary Dynamos

• Dynamo energy source: Precession, Tidal interactions, Thermal Convection,Compositional Convection

Tides and precession derive their energy from the Earth's core rotation. Tides

distort the CMB, precession is caused by the torques on the Earth's equatorial

bulge. Earth's axis of rotation precesses once every 26,000 years.

Precessing systems lead to a core �ow which is unstable, and these instabilities

can drive motion, just as buoyancy instabilities can.

Successful simulations have been done, but only with Precession/rotation

ratios of 10−3. Stabilised even by very small viscosity, so not clear it works inEarth's core.


31

Thermal convection

Thermal convection only occurs if the heat �ux produced is greater than the

amount that can be carried by conduction. A plausible reason for some planets

like Venus not having a dynamo is that all the heat �ux in the core is carried

by conduction.

The adiabatic temperature gradient is

T−1ad

(dTdr

)ad

= −gα/cp,

Here α is the coe�cient of thermal expansion and cp the speci�c heat. The

heat �ux carried down this gradient by conduction is

Fad = −κρcp(dTdr

)ad

For convection need the actual heat �ux F > Fad


32

The geotherm

At bottom of mantle there is

a thermal boundary layer D′′.

In the outer core, geotherm is

close to adiabatic.

Heat �ux carried by

conduction only ∼ 0.3TW atthe ICB, rising to ∼ 6TW atthe CMB. Mostly because of

larger surface area.

If latent heat is dominant heat source, possible that core is superadiabatic

(convecting) near ICB and subadiabatic (stable) near CMB.


33

Wiedemann-Franz law

In a metallic core, thermal and electrical conductivities are proportional:

Wiedemann-Franz law κρcp = 0.02T/η

High electrical conductivity (low η) implies high thermal conductivity, making

Fad large.

Stevenson's paradox: high electrical conductivity is bad for dynamos, because

it makes Fad larger than F . Since the heat is conducted rather than convected,

nothing to stir the core!


34

Dipole moment

The dipole moment is M =4πr3sµ0

g1, where

g1 =((g01)

2 + (g11)2 + (h11)

2)2.

The Earth's dipole moment is currently

decreasing faster than free decay rate,

i.e. if there were no dynamo! About

half its value in Roman times.

Growing reversed �ux patches under S.

Africa?


35

Reversals

Geomagnetic reversal record: last 5 million years

The Earth's �eld reverses randomly

about once every 0.3 million years.

There have been periods as long as

60 million years with no reversals

(superchrons). Reversals occur

relatively quickly ∼ 10, 000 years.During a reversal, the dipole axis traces

out a path at the Earth's surface: are

there preferred longitudes?


36

Excursions

Dipole component varies signi�cantly between reversals, the so-called

excursions.

Current decline may be an excursion rather than a reversal.


37

Other terrestrial planets

Three ideas for Mercury's weak �eld:

(i) No dynamo, but solar wind �eld ampli�ed.

(ii) As Mercury cooled, inner core grew. Remaining liquid outer core has

progressively higher impurity content, so thin shell of liquid iron. Thin shell

dynamos have high harmonic content, low dipole content.

(iii) Most of Mercury's core is liquid, but mostly stably strati�ed as heat

produced mostly carried by conduction. Dynamo only near ICB, only small

fraction of �eld penetrates stably strati�ed upper core.

Messenger should be able to decide between these theories.


38

Venus rotates only once every 243 days. But Rossby number U∗/`Ω withEarth-like U∗ would still be small.

Venus appears to have no plate tectonics. This could reduce F and hence

make Venus's core subadiabatic.

Venus seems to have been resurfaced 300 million years ago. Possibly there was

mantle convection then, and possibly a dynamo. High surface temperature

unfavourable for remanent magnetism.

Ganymede has a weak surface �eld, but a small core, so at CMB �eld is

∼ 2 Gauss, giving an Elsasser number of order unity. Surprising that there issu�cient heating in the core to make it convect.

Collapse of Martian dynamo possibly due to core becoming stably strati�ed.


39

Dynamos in the Giant Planets

Electrical conductivity is due to very high pressure ionising electrons (quantum

e�ect). Earth-like conductivity in the deep interior, gradually falling to low

values near the surface.

Giant planets are convecting, so thermal convection most natural energy

source.

Some secular variation occurs on Jupiter, suggesting core �ow ∼10−3metres/sec. Rm large.

Where is the dynamo most e�cient? Possibly nearer outside, where local Rm

not so big?

Why is Saturn's �eld so axisymmetric? Stably strati�ed region with strong

di�erential rotation outside the dynamo region? This could axisymmetrise the

observed �eld


40

Uranus and Neptune have very unusual �elds. Thin shell dynamos suggested.

Not much known about internal structure and heat �ux. Ionic conductivity is

low, so Rm cannot be very large.


41

Major Problems in Planetary Dynamo theory

(i) Why is Mercury's �eld so weak?

(ii) Why does Venus not have a magnetic �eld?

(iii) Why are most planets dipole dominated? Why do geomagnetic reversals

occur?

(iv) What powered the geodynamo before inner core formation?

(v) What killed o� the Martian dynamo?

(vi) How does Ganymede maintain a dynamo when its core is so small?

(vii) Why is Saturn's �eld so axisymmetric?

(viii) Why are the �elds of Uranus and Neptune non-dipolar?


42

Winds of Giant Planets

• Jupiter and Saturn have belts and zones associated with east-west zonal�ows: east-west �ows independent of longitude

• Also, long-lived storms such as the Great Red Spot on Jupiter

• What drives these winds? Why are they so di�erent from winds on Earth?

Are the winds just on the surface, or do they reach deep into the planet?

(iv) Winds on the Giant Planets 42/60

43

Jupiter from the Cassini Mission

Giant planets have banded structure. Also huge vortices such as the Great

Red Spot, and smaller white ovals.


44

How Jupiter rotates

Timelapse movie of Jupiter: start is in rest frame, half way through switches

to frame rotating with the planet.


jupiter1.aviMedia File (video/avi)

45

Zonal Flow on Jupiter

Banded structure of the giant planets is associated with zonal �ow. Eastward

and westward winds moving at over 100 metres/sec

Completely di�erent from Earth's wind pattern of midlatitude westerlies and

eastward trade winds


46

Winds in the Giant Planets

Jupiter zonal �ow Saturn zonal �ow

More variability in Saturn's winds than Jupiter's. Eastward (prograde) jets at

equatorward side of dark belts, westward (retrograde) jets at poleward side of

dark belts.


47

Galileo Probe

Probe entered 7◦ N, in eastward

equatorial jet. Found velocity

increases inward, supporting deep

convection model.


48

Winds on the Ice Giants

Note that the equatorial belt goes westward on the ice giants, eastward on

Jupiter and Saturn

Quite di�erent from the Solar di�erential rotation, which has a rapidly rotating

equator and slowly rotating poles


49

Storms on Outer Planets

As well as the persistent zonal �ows, also get giant storms on all the outer

planets.

Great Red Spot of Jupiter is only one we know has lasted hundreds of years.

Noted by Hooke end of 17th century.

Great White spots on Saturn last typically a few years. There are also vortices

at Saturn's poles.

Dark spot of Neptune lasted twenty years, but then disappeared.


50

The Great Red Spot

Giant storm, larger than

entire Earth. Anticyclonic

vortex that has lasted over

300 years

Maintained by absorbing

small vortices.

Why doesn't it break up?


51

Neptune's Dark Spot

Seen by Voyager, but

disappeared in 1994 as seen

by Hubble Space Telescope

White structure is the

`Scooter'


52

Storms on Saturn

White spot is a storm near equator.

Bizarre hexagonal storm at Saturn's North pole. Storms at South pole too.


53

Proudman-Taylor theorem

Proudman-Taylor theorem: �ow in rapidly rotating �uids is z-independent. It

moves in columns. So �ows in the outer planets may be largely columnar. If

an obstacle is moved through the �uid, the whole column of �uid above and

below moves with the obstacle.

However, Jupiter's Magnetic Field rotates steadily. Dipole inclined at about

10◦ to polar axis.

Suggest no fast zonal winds in magnetic �eld region i.e. the electrically

conducting region.

Suggests the convection columns cannot penetrate the metallic hydrogen

region.


54

Zonal �ows in Jupiter and Saturn

Jupiter: Large radius ratio, narrowly con�ned bands

Saturn: Smaller radius ratio, less con�ned bands

Are zonal �ows deep, 15,000 km,

driven by convection in molecular

H/He layer, or shallow, con�ned

to stably strati�ed surface layers?

Broader equatorial belt

on Saturn suggests that

the surface zonal �ow

is a�ected by the deep

structure


55

How is Jupiter's �ow driven?

Heat created at the birth of the solar system is still coming out of the planet.

It comes out by convection: hot �uid rises towards the surface, cool �uid sinks

back into interior.

Ω↑

→g →r

→tangent

→Flux

→

Flux

cylinder

s →

We need to study the pattern of

convection in a rapidly rotating

spherical shell

Proudman-Taylor theorem suggests

�ow will be approximately columnar


56

Linear theory or rotating convection

If the �uid is Boussinesq,

constant density except

for small variations to

allow buoyancy, convection

occurs as cartridge belt

rolls


57

Rapidly rotating convection at onset: varying E

Contours of axial vorticity.

Sequence with increasing

rotation E = 3 × 10−5,10−5,E = 3× 10−6, 10−6.Picture by Emmanuel Dormy.

Note convection onsets

�rst near tangent cylinder.

Convection columns get

thinner at smaller E.


58

Boussinesq spherical convection model

uφ: E = 3× 10−6, Pr = 0.1, η = 0.9, Ra ∼ 100RacritFrom Heimpel, Aurnou and Wicht, 2005

Colours denote azimuthal �ow velocity: red eastward, blue westward.


59

Zonal �ow from compressible convection model

ur Meridional section uφ Meridional Latitude variationsection of zonal �ow

Strongly compressible �ow with stress-free boundaries. Nρ = 5.0, E =6× 10−6, Pr = 0.1, η = 0.85, Ra = 23Racrit


60

Zonal jet formation: State of Play

(i) Eastward (prograde) jet near the equator is a robust feature of all

simulations. This jet is probably deep. Most observational evidence concerns

this jet.

(ii) High latitude jets occur with stress-free boundaries in compressible �ows

as well as incompressible �ows.

(iii) If jets inside the tangent cylinder interact with the magnetic �eld near the

transition region, they may be strongly damped.

(iv) Proudman-Taylor theorem strongly constrains the zonal �ow. The

convection rolls, though, have increasingly strong z-dependence as the density

variation increases.


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