jovian atmospheric dynamics: an update after galileo and...

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 68 (2005) 1935–1996 doi:10.1088/0034-4885/68/8/R06

Jovian atmospheric dynamics: an update after Galileoand Cassini

Ashwin R Vasavada1,3 and Adam P Showman2

1 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive,Pasadena, CA 91109, USA2 Department of Planetary Sciences and Lunar and Planetary Laboratory, University ofArizona, Tucson, AZ 85721, USA

E-mail: [email protected]

Received 15 May 2005Published 11 July 2005Online at stacks.iop.org/RoPP/68/1935

Abstract

The Galileo and Cassini spacecrafts have greatly enhanced the observational record of Jupiter’stropospheric dynamics, particularly through returning high spatial resolution, multi-spectraland global imaging data with episodic coverage over periods of months to years. These data,along with those from Earth-based telescopes, have revealed the stability of Jupiter’s zonaljets, captured the evolution of vortices and equatorial waves, and mapped the distributions oflightning and moist convection. Because no observations of Jupiter’s interior exist, a forwardmodelling approach has been used to relate observations at cloud level to models of shallow ordeep jet structure, shallow or deep jet forcing and energy transfer between turbulence, vorticesand jets. A range of observed phenomena can be reproduced in shallow models, though theGalileo probe winds and jet stability arguments hint at the presence of deep jets. Many deepmodels, however, fail to reproduce Jupiter-like non-zonal features (e.g. vortices). Jupiter’sdynamics likely include both deep and shallow processes, requiring an integrated approach tofuture modelling—an important goal for the post-Galileo and Cassini era.

(Some figures in this article are in colour only in the electronic version)

3 Author to whom any correspondence should be addressed.

0034-4885/05/081935+62$90.00 © 2005 IOP Publishing Ltd Printed in the UK 1935

http://dx.doi.org/10.1088/0034-4885/68/8/R06

mailto: [email protected]

http://stacks.iop.org/rp/68/1935

1936 A R Vasavada and A P Showman

Contents

Page1. Introduction and overview 19372. Galileo and Cassini at Jupiter 19403. Zonal jets 1942

3.1. Recent observations 19423.2. Vertical structure 19423.3. Stability of the flow: implications for vertical structure 19453.4. Mechanisms for pumping the jets 1947

3.4.1. Two-dimensional non-divergent models 19473.4.2. Shallow-water models 19583.4.3. Multi-layer models 19623.4.4. Deep models 1966

3.5. Observational constraints on modes of energy transfer 19724. Vortices 1974

4.1. Dynamics, stability, behaviour and structure 19754.2. Cyclones and anticyclones 19764.3. The Great Red Spot 19784.4. Mergers of the White Ovals 1980

5. Discrete storms 19825.1. Lightning statistics 19835.2. Lightning and moist convection 19855.3. Moist convection as a probe of Jupiter’s vertical structure 1986

6. Equatorial features 19877. Discussion 1989

Acknowledgments 1990References 1990

Jovian atmospheric dynamics 1937

1. Introduction and overview

The planet Jupiter reigns as the largest and most dynamic world in our solar system. Alongwith Saturn, Uranus and Neptune, Jupiter is one of the gas giants, those planets characterizedmore by their massive fluid envelopes than their relatively small solid cores. They stand in starkcontrast to the small, rocky inner planets that formed from the meagre amount of refractorymaterial available near the young Sun. At the distances of the gas giants, the temperatureof the proto-planetary disk dropped below the freezing points of water and other volatiles,providing additional solid material for accretion. There, planetary embryos grew massiveenough to capture the nebular gas itself. Jupiter and Saturn, and to a lesser extent Uranus andNeptune, completed much of their formation before the early Sun dissipated the nebular gas.As a result, Jupiter is an immense world, composed mostly of fluid hydrogen and helium withsome volatiles and refractory materials, largely mirroring the bulk composition of the solarnebula. Its interior may contain a rocky or icy core, though it is probably less than 1/10th ofits ∼140 000 km diameter (i.e. roughly the size of Earth) or is absent altogether.

Within the outer skin of Jupiter’s fluid envelope, what appears as the planet’s ‘surface’on images, pressure and temperature fall within ranges similar to those found in Earth’stroposphere and stratosphere. Temperature decreases with altitude in the convectivetroposphere, resulting in aerosol formation when gaseous chemical species reach saturationand condense. Most of the observed features are thought to be formations of ammonia orammonium hydrosulfide (NH4SH) clouds located near the 1 bar pressure level in the uppertroposphere. The contrast and sharpness of these features on images is reduced slightly bymoderate-opacity hazes in the upper troposphere and throughout the stratosphere, thoughthese hazes lack structure at small scales. A water cloud layer near the 5 bar level is predictedfrom equilibrium condensation models, but its depth below the main cloud deck makes itsdetection difficult. The changing appearance of the atmosphere results from the formation andevolution of these clouds and hazes (along with trace chemicals that colour them), and theirinteraction with the ambient winds.

For centuries humans have marvelled at the varying shapes and colours of Jupiter’s cloudysurface. The most salient features of Jupiter’s appearance are zonal (east–west) bands, ovalssuch as the Great Red Spot, and small, active storms (figures 1 and 2); many of these featurescan be monitored through modest telescopes. The visual banding is roughly correlated with∼30 zonal jets (figure 3), akin to jet streams on Earth though more confined in latitude. Regionswhere the latitudinal shear in the zonal winds is cyclonic (rotating with the planet, i.e. counter-clockwise in the northern hemisphere) are called belts, while anticyclonic regions are calledzones. Within and between these bands are hundreds of vortices of varying sizes and colours,both cyclonic and anticyclonic in each hemisphere. The Great Red Spot is the largest and mostlong-lived vortex. The significance of the third class of features, that is, storms marking thesites of intense but localized convection, has only recently been appreciated.

Though Jupiter’s atmosphere has been monitored telescopically for centuries, our detailedknowledge comes from images and other data returned by spacecraft. Pioneer 10 and 11, in1973 and 1974, were the first to encounter Jupiter. Jupiter also was a stop on the GrandTours of the twin Voyager spacecraft in March and July of 1979. The Galileo spacecraftspent nearly eight years in orbit around Jupiter from 1995 to 2003. Finally, the Cassinispacecraft flew by Jupiter in 2000, a prelude to its present mission at Saturn. The data fromthese spacecraft, along with those from the Hubble Space Telescope (HST) and ground-basedtelescopes, have provided ample motivation for theoreticians, experimentalists and numericalmodellers. Because of the fundamental physical processes involved, the study of Jupiter’satmospheric dynamics attracts workers from many disciplines, including physics, meteorology


Figure 1. True-colour mosaic of Jupiter acquired by Cassini. The portrait shows Jupiter’s zonalbands, the Great Red Spot and other vortices, several discrete storms within the orange band northof the equator and also west of the Great Red Spot, and two equatorial hot spots at the northernedge of the equatorial band near the planetary limb. NASA image PIA04866 (PIA images can beaccessed at NASA’s Planetary Photojournal, http://photojournal.jpl.nasa.gov).

and engineering. Jupiter’s atmosphere constitutes an immense fluid dynamics experiment of ascale that could never be achieved in the laboratory, and one that continues to challenge state-of-the-art computers. In many cases, the analogue for observed dynamical phenomena is notthe Earth’s atmosphere, but the Earth’s oceans or the outer layers of the Sun. Another majorchallenge to our scientific progress is the fact that we observe just a few hundred kilometresinto a planet with an equatorial radius of 71 492 km (to the 1 bar level). In fact, the interior isnot completely hidden, since the properties and motions at much deeper levels may influencewhat happens in the outer layers.

Jupiter’s jets, vortices and storms are studied with the goal of understanding their nature,behaviour, and roles in the planetary-scale circulation and energetics. What physics organizesthe fluid into jets and vortices, and what is the resulting three-dimensional structure? Whatprocesses power the storms and maintain the jets and vortices against dissipation? On Earth,the large-scale circulations are driven primarily by the equator-to-pole gradient in sunlightabsorbed by the surface and atmosphere. On Jupiter, the energetics are less straightforward;the equator-to-pole temperature contrast is nearly zero, and the fluxes of internal (primordial)energy and sunlight are comparable (though the total available energy is ∼20 times less than thatfor the Earth). At a more detailed level, what mechanisms, such as convection or turbulence,

http://photojournal.jpl.nasa.gov


Figure 2. Comparison of Jupiter’s appearance between 1979 and 2000. The top and bottomhalves contain colour mosaics acquired by Voyager 1 and Cassini, respectively. Both mosaicsare cylindrical maps showing all longitudes and ±60˚ latitude. Many real changes are apparent,though the overall shift in colour and contrast is attributable to the different imaging systems andprocessing methods used. Equatorial plumes and dark vortices in the band north of the equator aremore prominent in the Voyager map. The Cassini map has a brighter equatorial band and revealssmall storms within the band north of the equator and west of the Great Red Spot. NASA imagesPIA00011 and PIA02864.

are important in transferring energy vertically, between different length scales, or across linesof latitude? Are jets, vortices and storms observed at cloud level at all related to the dynamicsand energetics of Jupiter’s deep atmosphere?

Our understanding continues to progress with the growing record and quality ofobservations, and with the growing ability of theoretical and numerical studies and laboratoryexperiments to explain the observations using ever-more realistic physics and conditions.Because zonal jets and vortices can occur in a variety of natural and numerical regimes, second-order details from the observational record become the measures of success (i.e. the proof ofunique results). For example, a successful model of the zonal jets will capture the number of jetspresent in each hemisphere, their relative widths and strengths, their latitudinal confinement(lack of meandering), and the presence of a wide, prograde equatorial jet. Benchmarks forstudies of vortices include their size and frequency distribution with latitude, lifetimes, modesof appearance, disappearance and interaction, and any correlations between vorticity and size orfrequency. Other observations, such as the stability of the zonal jets, the presence of convectivestorms, and the statistical trends of turbulence, help test hypothesis about the energetics of thecirculation. Even today, fundamental aspects of Jovian dynamics lack basic observationalconstraints; for example, the motions within the deep interior are entirely unknown.

Our primary aim in this paper is to review our understanding of Jupiter’s troposphericdynamics after the Galileo and Cassini missions. Advances have occurred on four fronts:observation, theory, laboratory experiment and numerical simulation. We highlight spacecraftand telescopic discoveries of the last decade, focusing on imaging investigations since theyhave proved to be the foundry for dynamical discoveries (though there have been importantresults recently from ultraviolet, near-infrared and thermal spectrometers, as well as other typesof experiments). By limiting our scope to recent imaging observations and to atmosphericdynamics, we provide more details than possible in other excellent historical reviews (Peek1958, Rogers 1995) or those that cover the breadth of Jovian atmospheric science (e.g. Gierasch


Figure 3. Zonal winds in 1979 and 2000, convective storms, and lightning. The thick line showsthe wind speeds measured by Porco et al (2003) on Cassini images. The thin line shows windsmeasured by Limaye (1986) on Voyager 2 images. The profiles are remarkably similar, thoughsome changes in jet speed (e.g. at 24˚N) are apparent. Shaded (clear) bands mark areas of cyclonic(anticyclonic) shear in the zonal winds. Short horizontal lines at the left margin of the figure notethe latitudes of features interpreted to be convective storms in Cassini images (Porco et al 2003).Similar marks at the right margin note the latitudes of lightning strikes in Galileo and Cassiniimages (Little et al 1999, Dyudina et al 2004).

and Conrath (1993), Ingersoll (1998b), Irwin (2003), Bagenal et al (2004)). Our discussionof theory and simulation reaches back further in time and fills an important gap by focusingparticular attention on the dynamical mechanisms that produce Jupiter’s jet streams, whichhave not been extensively reviewed in the literature. As such, we seek to complement thereviews of Ingersoll (1990), Marcus (1993), Dowling (1995a) and Ingersoll et al (2004).We begin with a brief overview of the Galileo and Cassini imaging investigations at Jupiter.We then present four sections discussing the advances in observation, theory, experiment andsimulation relevant to Jupiter’s zonal jets, vortices, storms and equatorial features. Throughoutthe paper we use System III west longitudes and planetographic latitudes, calculated assumingequatorial and polar radii of 71 492 km and 66 854 km, respectively, at the 1 bar pressure level.

2. Galileo and Cassini at Jupiter

After visits from both Voyager spacecraft in 1979, only Earth-based telescopes viewed Jupiteruntil the Galileo spacecraft arrived in 1995. A multi-billion dollar mission, Galileo wasdesigned to spend years circling the planet and flying by its moons while returning data froman array of onboard experiments and a probe released into Jupiter’s atmosphere. Technical


issues including the 1986 Challenger accident delayed Galileo’s launch from 1982 until 1989and forced it to follow an extended trajectory aided by multiple gravity assists from Venusand Earth. The mission was dealt another major setback when its high-gain antenna failed todeploy after the spacecraft left the inner solar system, possibly as a result of the launch delays.Fortunately, engineers at NASA’s Jet Propulsion Laboratory devised a scheme to transmitdata through Galileo’s low-gain antenna to an upgraded set of antennae on Earth up to a rateof 160 bits s−1 (one hundred thousand times slower than a broadband internet connection).Although handicapped, the mission was salvaged and enjoyed spectacular success at Jupiterfrom 1995 until it was intentionally scuttled into Jupiter in late 2003. The Galileo spacecraftwas the first to enter into orbit around Jupiter rather than fly by it like the Pioneers, Voyagersand Cassini. Orbiting provides the capability to observe the Jovian system at close range notonly for several years, but also over a large range of geometries with respect to the Sun. Usingthis capability, Galileo was able to characterize Jupiter’s auroral emission on the night sideof the planet for the first time (Ingersoll et al 1998a, Vasavada et al 1999) and to completea global search for lightning (Little et al 1999). The Galileo mission also delivered the firstprobe into Jupiter’s (outer) atmosphere, the results of which are reviewed by Young (2003).

Image-based analyses of atmospheric dynamics attempt to maximize spatial resolution,temporal resolution and temporal coverage, all of which require the transmission of a largevolume of data. After the Galileo antenna failure, the imaging-science team developed anew strategy that addressed their major scientific objectives with a drastically reduced datavolume. Each time Galileo encountered Jupiter or a targeted moon on its eccentric orbital path,the spacecraft’s tape recorder was filled over several days. The spacecraft then transmittedthese data to Earth over the next several months, when the spacecraft was far from Jupiter.Atmospheric observations consisted of ‘feature tracking’; at each orbit one particular featureor region was imaged a few times (at 1 and 10 h intervals), at multiple wavelengths (typicallyfour), and at high spatial resolution (typically ∼30 km/pixel). Over the course of the mission,over a dozen regions were observed (Vasavada 2002).

The Cassini spacecraft, also a multi-billion dollar mission, was launched in 1997 andflew by Jupiter in late 2000 on its way to Saturn. The spacecraft never approached Jupiteras closely as the Voyagers or Galileo, but the spatial resolution of the camera (58 km/pixel atclosest approach) exceeded that achievable from HST for several months. The Cassini ImagingScience Subsystem acquired ‘movies’ of Jupiter over a six-month period as it approached andreceded from the planet. Multi-spectral, global mosaics were taken continually at 10 or 20 hintervals, except near the closest approach when a flurry of imaging sequences captured theplanet and its moons at close range (Porco et al 2003).

The characteristics of the Galileo and Cassini imaging experiments that represent thelargest advances over earlier missions are the improved spatial resolution, sensitivity andgeometric control of digital detectors and those detectors’ sensitivity to ultraviolet throughnear-infrared wavelengths. Imaging through spectrally narrow near-infrared filters permitscrude but quantitative (and high horizontal-resolution) sounding of the atmosphere by revealingthe amount of absorptive gaseous methane above different features (Banfield et al 1998,Simon-Miller et al 2001). In effect, Galileo and Cassini have produced four-dimensionalrecords of Jupiter’s dynamic atmosphere.

Finally, the continuous record of observations from ground-based telescopes and HSThave proved extremely useful for documenting decadal-scale trends in the dynamics of theatmosphere and the state of its major features in the periods between spacecraft encounters.In addition, they have captured major events that occurred when spacecraft were unable toobserve at close range, such as the impact of comet Shoemaker-Levy 9 in 1994 and the mergerof Jupiter’s White Ovals in 1998 and 2000.


3. Zonal jets

3.1. Recent observations

Implicit in the study of Jovian winds from imagery is the idea that clouds are passive tracersof the ambient winds. Cloud motions tracked between images, either manually or usingsophisticated digital algorithms, are converted to wind speeds. The longevity of Jupiter’scloud formations (relative to its ∼10 h rotation period) and the lack of topography (to whichsome clouds are tied on Earth) are factors that favour the success of this technique. However,cloud-tracked winds are subject to the effects of image resolution, uncertainties in camerapointing and rapid changes in cloud morphology. The motions of larger features may reflectan average of a shear zone in which the clouds are embedded (e.g. Beebe et al (1996)) or may beindependent of ambient winds. Finally, if cloud brightness or contrast is correlated with someaspect of the dynamics (e.g. certain wind speeds or directions), cloud tracking may give biasedresults (Limaye et al 1982, Sromovsky et al 1982). In spite of the potential for uncertaintiesin deriving winds from cloud motions, image-based measurements have produced a multi-decadal record of Jupiter’s zonal wind speeds at cloud level (Limaye 1986, Garcıa-Melendoand Sanchez-Lavega 2001, Porco et al 2003). Attempts to measure winds at altitudes otherthan cloud level have suffered from the lack of reliable features.

Remarkably constant by terrestrial standards, Jupiter’s zonal wind profile has evolved onlyslightly over time (figure 3). A decrease of 40–50 m s−1 in the speed of the eastward jet near24˚N has been confirmed by several analyses (Simon 1999, Garcıa-Melendo et al 2000, Porcoet al 2003), in addition to smaller changes in jet shape and/or magnitude near the equatorialregion and around 50˚N (Vasavada et al 1998, Garcıa-Melendo and Sanchez-Lavega 2001,Porco et al 2003). Cassini and HST revealed that zonal jets extend to ∼80˚ latitude in eachhemisphere, in spite of the lack of a banded appearance at high latitudes.

It should be noted that Jupiter’s appearance, including the brightness and latitudinal extentof its bands and the presence or absence of discrete features, has varied significantly over time(e.g. figure 2). Many studies have described these changes, some of which are quasi-periodicover multiyear time scales. The Cassini mosaic in figure 2 reveals the absence of large, brownvortices and discrete storms associated with the equatorial plumes relative to the Voyager era.Storms in the band just north of the equatorial zone are more prominent during the Cassiniera. Other aspects, such as the wake region west of the Great Red Spot, the presence of severalmid-latitude ovals, and the general appearance of the high latitudes, are quite similar betweenthe two encounters.

3.2. Vertical structure

Indirect (infrared) measurements suggest that Jupiter’s jets decay with height above cloudlevel. The inference derives from observations of air temperature at pressure from ∼1 bar toless than 0.1 bar (Conrath et al 1981, Pirraglia et al 1981). Regions of cyclonic shear in thelatitudinal jet profile are warm above the clouds relative to their surroundings (and anticyclonicregions are cold above the clouds). Rapidly rotating planets such as Jupiter are expected toobey the well-known thermal-wind equation (Holton 1992, p 73), which describes the verticalstructure of a fluid in geostrophic balance

∂u

∂ log p= R

f

∂T

∂y, (1)

where u is eastward velocity, p is pressure (here a proxy for height), R = 3600 J kg−1 K−1 isthe universal gas constant divided by molecular mass, f ≡ 2� sin θ is the Coriolis parameter,


Figure 4. Jupiter’s zonal winds at 7.4˚N latitude obtained by Doppler tracking of the Galileo Probesignal (Atkinson et al 1997). The thick curve is the nominal wind profile and the thin curves boundthe uncertainty envelope.

� = 1.74 × 10−4 s is Jupiter’s angular rotation rate, T is temperature, θ is latitude and y isnorthward distance. The equation relates latitudinal changes in temperature to vertical changesin velocity.

The vertical structure of the jets below the clouds remains a major unknown. The GalileoProbe provided our deepest sounding to date. The probe entered at a latitude of 7.4˚N, nearthe northern edge of the equatorial jet shown in figure 3, and measured winds from 0.4 to22 bars, corresponding to an altitude span of 150 km. At the 0.4 bar level, the probe measureda velocity of 90 m s−1 (Atkinson et al 1997, 1998), similar to the cloud-top winds that hadbeen inferred from Voyager cloud tracking (Limaye 1986). The winds increased with depth to180 m s−1 at 5 bars and remained nearly constant thereafter (figure 4). Though they indicatethat winds increase below the cloud level, these measurements are not deep enough to revealthe full vertical structure of Jupiter’s jets.

Two endpoint hypotheses have evolved regarding the vertical structure of the jets. The‘weather layer’ hypothesis has the winds confined to a thin region near cloud level (also theregion where sunlight is absorbed). Latitudinal thermal contrasts within a few scale heightsbelow the clouds, if any, would cause vertical wind shear via the thermal-wind equation(Ingersoll and Cuzzi 1969, Ingersoll et al 1984). A likely mechanism for producing thesethermal contrasts is latent heat release from condensation of water, which occurs at altitudesabove the 5–10 bar pressure level, depending on the water abundance. Thermal contrasts of5–10 K are possible from this mechanism (Gierasch 1976, Gierasch et al 1986). The thermalwinds arising from such temperature contrasts can explain the approximate magnitudes ofJupiter’s jets even if the interior deeper than ∼10 bars has no winds (Ingersoll and Cuzzi1969). This model requires cyclonic regions (CR) to be cold and anticyclonic regions to bewarm below the clouds, which is the opposite of the observed temperature structure above theclouds (no contradiction is implied).


In the ‘deep winds’ hypothesis, the observed jets extend throughout the ∼104 km-thickmolecular region of Jupiter’s interior. This hypothesis follows naturally from the fact thatthe interior is probably largely convective and hence, adiabatic. On this basis, Busse (1976)suggested that the Taylor–Proudman theorem (Taylor 1923) holds throughout the interior. Ina geostrophically balanced fluid where the friction force is negligible, the equation governingthe three-dimensional fluid vorticity can be written (Pedlosky 1987, p 43) as

2(� · ∇)v − 2�∇ · v = −∇ρ × ∇p

ρ2, (2)

where� is the planetary rotation vector, ρ is density, p is pressure andv is the three-dimensionalvelocity vector in the rotating reference frame of the planet. This equation can be viewed asa three-dimensional cousin of equation (1); writing the equation in component form allowsrecovery of the thermal-wind equation (Pedlosky 1987, pp 42–54). The term on the rightside (called the baroclinic term) is nonzero when density varies on constant-pressure surfaces.In Jupiter’s deep interior, however, convective mixing is expected to lead to nearly constantentropy, in which case density does not vary along isobars and ∇ρ × ∇p equals zero (i.e. thefluid is barotropic). This leads to the compressible-fluid generalization of the Taylor–Proudmantheorem (Pedlosky 1987, pp 43–44)

(� · ∇)v − �∇ · v = 0. (3)

In component form, using a Cartesian coordinate system (x∗, y∗, z∗ axes) with z∗ aligned alongthe rotation axis and the x∗ and y∗ axes lying in the equatorial plane, the equations become

∂u∗∂z∗

= ∂v∗∂z∗

= 0, (4)

∂u∗∂x∗

+∂v∗∂y∗

= 0, (5)

where u∗ and v∗ are the velocity components along the x∗ and y∗ axes, respectively. Theequations state that the wind components in planes perpendicular to the rotation axis areindependent of the coordinate parallel to it, and furthermore that the divergence of these windsin the plane perpendicular to the rotation axis is zero. The fluid then moves in columns, calledTaylor columns, whose axes are aligned with the rotation axis. Within a spherical planet,Taylor columns cannot move towards or away from the rotation axis, because doing so wouldforce them to stretch or contract (such motions imply a convergence or divergence in thex∗–y∗ plane that are disallowed by the theorem). Instead, any motions and the dynamicalpressure gradients that accompany them must extend throughout the planet along cylinderscentred on, and parallel to, the rotation axis. Busse (1976) presented this hypothesis as apossible explanation for why Jupiter and Saturn have zonal jet profiles that are approximatelysymmetric across their equators. The jets at corresponding latitudes in the northern andsouthern hemispheres would simply be the regions where a given cylinder rotating at a givenspeed intersects the surface. The existence of multiple, nested cylinders, each rotating atdifferent speeds, would produce multiple jets varying with latitude. Although the scenario doesnot hold in the metallic region that exists at pressures exceeding ∼1–3 Mbar (correspondingto about 0.7–0.9 of Jupiter’s radius), deep flow along cylinders remains a viable possibility inthe outer molecular region.

Jupiter’s actual jet structure probably lies between these endpoint scenarios. The argumentthat Jupiter’s interior is convective and hence barotropic is strong (Stevenson and Salpeter 1976,Guillot et al 2004), so it is hard to escape the conclusion that the Taylor–Proudman theoremholds throughout the bulk of the molecular envelope. However, Voyager, Galileo and Cassini


infrared observations prove that the atmosphere is not barotropic near cloud level, and GalileoProbe observations suggest small, but finite, static stability even as deep as 20 bars (Magalhaeset al 2002, Young 2003). This suggests that the outer boundary of the Taylor–Proudmanregion (i.e. the tops of the Taylor columns) lies below the clouds by a scale height or more.Furthermore, observed atmospheric waves require static stability to propagate (Flasar andGierasch 1986, Allison 1990, Ortiz et al 1998, Showman and Dowling 2000), again consistentwith the existence of a stably stratified baroclinic region near cloud level (e.g. Allison (2000)).Although alternative theories are not excluded by our current level of understanding, Jupiter’sjet structure probably consists of deeply seated, concentric cylinders that obey the Taylor–Proudman theorem in the molecular interior, with a baroclinic thermal-wind region superposedin the outermost layers. This hybrid structure helps explain the near (but imperfect) symmetryacross the equator in the wind profile (figure 3). The pressure level at which the transitionbetween the barotropic and baroclinic regions occurs remains a major unknown.

These endpoint scenarios for the jet structure result purely from the known theoretical linkbetween winds and temperatures, given plausible assumptions about the deep temperature–pressure structure. As applications of geostrophy, the Taylor–Proudman theorem and thermal-wind equation simply relate one aspect of spatial structure (the temperature field) to another(the wind field) at a given time. These equations say nothing about the mechanisms thatdrive the jets or whether a hypothetical wind profile is dynamically stable. In particular,equations (1)–(5) say nothing about the widths or strengths of the jets.

3.3. Stability of the flow: implications for vertical structure

An important inference regarding Jupiter’s winds comes from measuring the curvature of thezonal velocity profile versus latitude and comparing it with β, the gradient of planetary vorticity.On a spherical planet, β = 2� cos θ/a, where a is the planetary radius. Observations showthat the absolute vorticity gradient changes sign at the latitudes of westward jets and that thecurvature exceeds β by a factor of 2.5–3 at these jets (Ingersoll et al 1981, Limaye 1986). Thisis puzzling, because such fluid motions violate the barotropic stability criterion, which statesthat a two-dimensional fluid is stable provided that

β − uyy > 0, (6)

where the second term on the left is the second derivative of the zonally averaged zonal velocity(i.e. the curvature of the zonal velocity profile) with respect to northward distance, y.

Although violation of this criterion does not guarantee that the fluid is unstable (‘violation’simply implies that the criterion provides no information), numerical simulations in shallowor two-dimensional systems always produce jets with curvatures smaller than β (e.g. Williams(1978), Vallis and Maltrud (1993), Cho and Polvani (1996a), Nozawa and Yoden (1997a),Huang and Robinson (1998), Danilov and Gryanik (2004)). If an appropriate flow in a deeplayer underlying the observed cloud-level winds is invoked, however, the cloud-level windsbecome stable (Ingersoll and Cuong 1981, Dowling 1995a, 1995b). This provides another hintthat Jupiter’s winds may be deep.

Ingersoll and Pollard (1982) and Ingersoll and Miller (1986) considered barotropic shearinstabilities for a deep, adiabatic atmosphere in approximate geostrophic balance, where thesteady-state flow consists of differentially rotating concentric cylinders obeying the Taylor–Proudman theorem. They found that the column stretching that must accompany meridionalmotions of Taylor columns in spherical geometry alters the instabilities relative to the two-dimensional case, leading to a modified shear-instability criterion where flows are stable if thecurvature is less than ∼3β. Most of Jupiter’s jets that violate the barotropic stability criterion


of equation (6) do not violate this altered criterion. However, the fact that most of Jupiter’sTaylor columns would intersect the metallic hydrogen region was not considered, and theproper instability criterion in this more realistic case remains unclear.

Little work has been done to determine whether fully three-dimensional baroclinic flows(e.g. shallow flows that die off in the interior) can be stable while violating equation (6) atthe cloud tops. Williams (2002) and Read et al (2004) describe three-dimensional shallowflows that appear to violate the barotropic stability criterion. Gierasch (2004) performed astability analysis on a model where horizontal temperature contrasts allow the geostrophicallybalanced winds to decrease with depth following equation (1), reaching zero at a relativelyshallow pressure of 102–103 bars. The novel aspect of the model was the assumption thatvertical stratification is small, so that constant-entropy surfaces are nearly vertical. Gieraschfound that this geometry discourages instabilities and allows jets with curvatures up to ∼3β

to be stable, consistent with Jupiter’s jets. The unstable modes studied by Gierasch, while notbarotropic, had substantial amplitude throughout the vertical extent of the domain. It appearsthat the vorticity ‘felt’ by these modes corresponds to a vertically averaged vorticity (in fact,the system is governed by the vertically averaged vorticity equation). Because the winds,and hence vorticity, increase with height, the system can therefore remain stable—sensitiveonly to the modest vertically averaged vorticity—while experiencing large vorticity (and largezonal-wind curvature) at the top. One might wonder whether unstable modes focused nearthe top of the domain would be sensitive to the large vorticity there, and hence, capable ofdestabilizing jets with large zonal-wind curvature at the top, but apparently the system does notexhibit such shallow modes (Gierasch 2004). This trait probably results from the geostrophicsuppression of vertical motion in the model system, since strongly baroclinic modes might beexpected to induce substantial vertical motion in a system with highly tilted entropy surfaces.The scenario of Gierasch (2004) provides a promising alternative to Ingersoll and Pollard(1982) and Ingersoll and Miller (1986). However, it remains unclear how the scenario wouldbe established. Furthermore, Gierasch made several assumptions that could affect the results(e.g. geostrophic flow, rigid upper and lower boundaries). It would be worth seeing whethera more general model that relaxes some of these assumptions also allows stable flows thatviolate equation (6) at the top.

Some caveats exist regarding the application of stability theorems to real-world flows. Inparticular, analytic stability theorems such as equation (6) are typically based on flows withno longitudinal structure. However, eddies (e.g. vortices or waves) might potentially helpto stabilize a flow that would otherwise be unstable. Furthermore, a zonal average across astreet of vortices can produce u(y) profiles that violate equation (6) even though the vortices areperfectly stable; equation (6) is irrelevant to this situation. Discernment is needed to determinewhether the zonal average of a given flow represents zonal jets, to which equation (6) applies,or vortices, to which it does not. Strongly forced flows can potentially also produce jets thatviolate the relevant stability criterion, although this presumably requires the characteristictimescale for pumping the jets to be less than the timescale for the barotropic instability thatwould rob energy from the jets (a condition that Jupiter probably does not satisfy). It has alsobeen suggested that friction may alter the stability criterion (Galperin et al 2001), and somedynamical processes such as breaking Rossby waves can cause transient violations of it (e.g.Juckes and McIntyre (1987)). Nevertheless, the fact that numerous numerical simulationsof two-dimensional turbulence all satisfy equation (6) in the presence of eddies, forcing andfriction suggest that these caveats are not serious problems. Jupiter’s violation of equation (6)is therefore probably a real phenomenon that needs explaining.

In summary, the observed curvatures of the zonal wind at cloud level, uyy , provide apowerful clue that the cloud-level winds are stabilized by nonzero winds below the clouds


(Dowling 1995a, 1995b). However, the vertical structure of those winds implied by theobserved cloud-level uyy (in particular, whether the winds are shallow or deep) remainsobscure. Further theoretical and numerical work on the dynamical stability of various three-dimensional flows should help rule out some configurations, thereby better constraining theactual jet structure.

3.4. Mechanisms for pumping the jets

Two classes of models have been invoked to explain the formation of the jets observed atJupiter’s cloud level. At the outset, we emphasize that one must distinguish between deepversus shallow models for the structure of the jets from deep versus shallow models for theforcing that leads to the formation and/or maintenance of the jets. In the ‘shallow-forcing’scenario, the jets are pumped at their outer margin by turbulence injected at cloud level by moistconvection, horizontal contrasts in solar heating (either equator-to-pole or band-to-band), orother weather-layer processes. Although the forcing is shallow, the jets may or may not bedeep. In the ‘deep-forcing’ scenario, convection cells that extend throughout the molecularhydrogen region (a depth of perhaps ∼104 km) drive differential rotation in the interior thatmanifests as jets at the cloud level. Because the convection extends through a deep region, themomentum forcing that drives the jets occurs throughout the fluid depth.

The assumption is often made that shallow forcing implies shallow jets (confined to within∼100 km of the cloud deck), while deep jets can arise only from deep forcing. On this basis, thefast winds measured by the Galileo probe down to 22 bars have sometimes been interpreted asfavouring deep-convective forcing for the jets (e.g. Atkinson et al (1996), Folkner et al (1997),Seiff et al (1997, 1998), Christensen (2001), Young (2003)). However, it is worth emphasizingthat a barotropic, geostrophic fluid that obeys the Taylor–Proudman theorem will have windsthat penetrate deeply on cylinders parallel to the rotation axis regardless of whether the forcingis shallow or deep (e.g. Atkinson et al (1997)). In fact, the classic laboratory experiment usedto demonstrate the Taylor–Proudman theorem (Taylor 1923) illustrates this fact. A rotatingtank of liquid water containing an obstacle at the bottom of the tank develops two-dimensionalflow according to the Taylor–Proudman theorem. The fluid column overlying the obstacleremains fixed over the obstacle, and other fluid columns avoid the obstacle throughout theirdepth despite the fact that the forcing (the obstacle) extends throughout only a fraction of thefluid depth. Similarly, wind stress or stirring at the top of the tank would produce movingTaylor columns (i.e. jets) that penetrate throughout the fluid. On Jupiter, then, it is possiblethat deep jets obeying the Taylor–Proudman theorem could form even if the momentum forcingthat drives the jets occurs only near the top.

Here we describe the hierarchy of models for the generation of Jupiter’s jets. We beginwith the simplest, purely two-dimensional models with no divergence, then progress to two-dimensional models that allow column stretching (with its associated horizontal divergence),and next discuss full multi-layer models of shallow-atmosphere turbulence. These shallowapproaches are followed by a description of models that investigate deep forcing of the jets.

3.4.1. Two-dimensional non-divergent models. Although localized regions of ascent andhorizontal divergence (e.g. thunderstorms) have been observed in Jupiter’s cloud layer, Jupiter’scloud-top flows appear to be predominantly two-dimensional, confined to the horizontal plane(Gierasch 1976). Such horizontal confinement may result from two basic mechanisms. First,several lines of evidence suggest the existence of static stability within the cloud region (Flasarand Gierasch 1986, Allison 1990, Ingersoll and Kanamori 1995, Showman and Dowling 2000).Such static stability would naturally arise from latent heating associated with condensation of


water from ∼1–10 bars (Achterberg and Ingersoll 1989, Nakajima et al 2000), and wouldinhibit vertical motions. Second, rapid rotation reduces vertical velocities by a factor of orderRo relative to that suggested by geometrical considerations alone (i.e. the vertical velocityis of order UHRo/D rather than UH/D, where U is the characteristic horizontal wind speed,Ro = U/�D is the Rossby number and H and D are the characteristic vertical thickness andhorizontal width, respectively). For Jupiter’s large-scale circulations, Ro ≈ 0.02, and the twoconstraints together suggest that average horizontal speeds exceed vertical speeds in the cloudlayer by factors of 103 or more. Combined with the early successes in explaining Jupiter’sjets using shallow-layer structure models (Ingersoll and Cuzzi 1969), these considerationshave led to a long history of two-dimensional and quasi-two-dimensional models for Jupiter’scloud-layer dynamics, many of which are surprisingly effective in explaining properties ofJupiter’s jets and vortices. These studies include those motivated both directly by Jupiter andbasic questions of two-dimensional turbulence in planar or spherical geometry. The simplestof such models consists of pure two-dimensional motion (representing the horizontal flow atJupiter’s cloud level) with no horizontal divergence. The basis is the conservation of potentialvorticity q of fluid parcels, which for a shallow fluid layer of thickness h is defined as (Pedlosky1987, p 64)

q = ζ + f

h, (7)

where ζ ≡ k · ∇ × v is the vertical component of relative vorticity and f is the Coriolisparameter. For a purely two-dimensional fluid, h can be considered constant, so this equationreduces to conservation of the absolute vorticity, ζ + f , following the flow. Vorticity sourcesand sinks (i.e. forcing and dissipation) are typically added, leading to the non-divergent, two-dimensional vorticity equation

d(ζ + f )

dt= F − D, (8)

where F is forcing, D is dissipation and d/dt = ∂/∂t + v · ∇, where v and ∇ are the two-dimensional fluid velocity and gradient operator, respectively. If we express the velocity usinga stream function, ψ , such that u = −∂ψ/dy and v = ∂ψ/dx, then equation (8) can bere-expressed in Cartesian geometry as

∂ζ

∂t+ J (ψ, ζ ) + vβ = F − D, (9)

where u and v are the eastward and northward speeds, x and y are the eastward and northwardcoordinates, and J is the Jacobian operator, defined as

J (A, B) ≡ ∂A

∂x

∂B

∂y− ∂A

∂y

∂B

∂x. (10)

Given this framework, how might the flow evolve? The types of flow structuresgenerated in a turbulent fluid depend on which of the many possible nonlinear vortexinteractions predominate. In a three-dimensional fluid, vortex stretching, which producessmaller structures, tends to dominate over processes such as vortex merger that produce largerstructures. The result is the well-known cascade of kinetic energy from large to small scales(e.g. Frisch (1995)). Column stretching cannot occur in a two-dimensional fluid, however,and vortex merger therefore assumes a prominent role, forcing energy to undergo an inversecascade from small to large scales (e.g. Pedlosky (1987), Read (2001)). Numerous numericaland laboratory studies of two-dimensional turbulence have demonstrated the reality of theinverse cascade, and under the simplest possible conditions (constant Coriolis parameter f ),the turbulence tends to generate large, isolated vortices separated by relatively quiescent regions


(e.g. Basdevant et al (1981), McWilliams (1984)). Although the dynamics of these simulatedvortices share similarities with Jovian vortices, these simulations lack jets and fail to generatea Jupiter-like general circulation.

A major breakthrough in the study of jets came when Rhines (1975) realized that thevariation in f with latitude, characterized by the parameter β, introduces anisotropy, causingelongation of structures in the east–west direction relative to the north–south direction. Rhines(1975) showed that under certain circumstances, the energy reorganizes into jets with acharacteristic wavenumber

kβ ≈(

β

U

)1/2

. (11)

The corresponding length scale, Lβ ≈ π/kβ , is now called the Rhines scale. Williams (1978)first demonstrated the relevance of β and the inverse cascade for producing jets on Jupiter, andmany subsequent studies of two-dimensional turbulence affected by β have been published inthe fluid-dynamics literature.

The mechanism by which the Rhines scale can interact with the inverse cascade to producejets is as follows. The Rhines scale can be interpreted as a transition scale between the regimeswhere turbulence and Rossby-wave activity dominate. For length scales smaller than theRhines scale, the second term on the left side of equation (9) dominates, corresponding tononlinear vorticity advection (perhaps the defining property of turbulence). For length scaleslarger than the Rhines scale, the linear term, vβ, dominates over the nonlinear Jacobian term,and Rossby waves are the primary solutions to the equation. Therefore, energy at the smallestscales exists dominantly as turbulence and is driven by the inverse cascade to larger and largerlength scales. As the wavenumber containing the bulk of the energy reaches kβ , however, thedynamics shifts from one dominated by turbulence to one dominated by Rossby waves. Atthe Rhines wavenumber, the characteristic Rossby-wave frequency ω ≈ −β/k for a typicallyoriented wave with wavenumber k matches the characteristic turbulent advection frequencyUk, but larger-scale Rossby waves at k < kβ have faster frequencies than the turbulence, sothese waves are inefficiently forced by the turbulence (the Rossby-wave frequency results fromthe dispersion relation, ignoring the distinction between zonal and meridional wavenumbers;the advection frequency is simply one over the advection timescale L/U ). As a result, energycannot easily cascade to length scales larger than the Rhines scale, so energy piles up at kβ

and the inverse cascade slows down. Weak wave-turbulence interactions will eventually drivethe energy to wavenumbers smaller than kβ (i.e. into the regime dominated by Rossby waves),but before this happens there is often an extended phase when kβ contains more energy thanany other wavenumber (implying a predominance of structures with size Lβ).

By itself, this argument shows how the inverse cascade can produce structures at a scalenear Lβ , but it does not explain the formation of zonally elongated structures. Rhines (1975,1994), Williams (1978), Vallis and Maltrud (1993), and others have pointed out that anisotropyin the Rossby-wave dispersion relation can cause anisotropy in the inverse cascade at scalesnear Lβ . The Rossby-wave dispersion relation can be written (Vallis and Maltrud 1993, Rhines1994) as

ω = − βkx

k2x + k2

y

= −β cos φ

|k| , (12)

where |k| =√

k2x + k2

y is the total-wavenumber magnitude and φ gives the angle between

the Rossby-wave propagation direction and east. Equating this frequency with the turbulentfrequency U |k| leads to an anisotropic Rhines scale

k2β = β

U|cos φ|. (13)


(a) (b)

Figure 5. Temporal evolution of the two-dimensional energy spectrum in freely evolving, two-dimensional turbulent simulations from Vallis and Maltrud (1993). The simulations solve equation(8) starting with a turbulent initial condition. They contain no forcing or large-scale dissipation.Abscissa and ordinate correspond to the wavenumbers kx and ky , respectively, and the grid coversa range of −1.8 to 1.8 kβ along each axis. The contours represent the amplitude of kinetic energy.The initially isotropic spectrum of small-scale turbulence in (a) evolves inward towards the Rhinesbarrier in (b).

This relationship traces out a dumbbell pattern in the kx–ky wavenumber space. Outsidethe dumbbell, turbulence dominates, whereas inside the dumbbell Rossby waves dominate.Figure 5 shows a freely evolving calculation, from Vallis and Maltrud (1993), illustrating howthe flow responds to this anisotropic barrier. The initial condition contains high-wavenumberisotropic turbulence, which appears as a ring in wavenumber space (the initial turbulence hasthe same wavenumber magnitude |k| regardless of the wavevector orientation φ, figure 5(a)).The inverse cascade drives energy towards smaller wavenumbers, i.e. closer to the origin inthe kx–ky plane. The energy cannot penetrate the dumbbell, however, because the Rossby-wavefrequency there exceeds the turbulence frequency U |k|. Because the flow seeks to minimize itstotal wavenumber (i.e. distance from the origin), the energy concentrates along the y-axis. Thecorresponding flows have nearly zero kx and small, yet finite ky (figure 5(b)). These correspondto zonally elongated structures, which in many cases form jets. Vallis and Maltrud (1993) andothers have shown that, when jets form, the number of jets correlates with the Rhines scale.Nevertheless, details of the model formulation, forcing and dissipation can strongly influencethe extent to which robust jets form.

Several investigations have now shown conclusively that Jupiter-like jets do not result fromfreely evolving (i.e. unforced) two-dimensional, non-divergent turbulence on a sphere. Thesestudies use a space-filling turbulent initial condition with energy confined to small scales,which then undergoes an inverse cascade to larger structures as the integrations proceed.In addition to zero forcing, these studies usually include no large-scale dissipation. Thedissipation consists solely of a numerical hyperviscosity ν∇2pζ , where p = 1, 2, 3, 4 or 8,that smoothes grid-scale structures and helps maintain numerical stability. Williams (1978)solved equation (8) on a portion of a sphere with parameter values expected for Jupiter. Inintegrations lasting ∼115 Earth days, he found that, although the inverse cascade producessubstantial anisotropy, no bands or regular jets formed (figure 6). Yoden and Yamada (1993)and Yoden et al (1999) revisited the dynamics of freely evolving (unforced) turbulence startingfrom an initial condition containing small-scale energy, using full spherical geometry ratherthan a sector as used by Williams (1978). Their simulations became dominated by strongcircumpolar vortices containing homogenized absolute vorticity; although zonal anisotropydeveloped at middle and low latitudes, containing some jet-like structures, the robust, highly


(a) (b)

Figure 6. Panel (a) shows the stream-function at 115 Earth days in a freely evolving (unforced)solution to equation (8), from Williams (1978). The trace in (b) shows the zonally averagedzonal wind versus latitude and demonstrates that although the simulation produces some zonallyelongated structures, no jets form.

zonal jets characteristic of Jupiter did not occur. Cho and Polvani (1996a) furthermore showedthat Yoden and Yamada (1993), Yoden et al (1999) and Williams (1978) did not integrate longenough to reach a final equilibrium state. In long-time integrations, Cho and Polvani (1996a)found that the mid- and low-latitude jet-like structures eventually disappear, leading to a finalconfiguration consisting solely of two large, circumpolar vortices (figure 7). This configurationlooks even less like Jupiter than the intermediate, non-equilibrium states achieved by Yodenand Yamada (1993), Yoden et al (1999) and in Williams’ (1978) freely evolving simulations.The bottom line is that freely evolving, non-divergent, two-dimensional turbulence has notbeen able to explain Jupiter’s jets.

Interestingly, however, many investigations have demonstrated that robust zonal jets canoccur in two-dimensional, non-divergent forced turbulence on a sphere or β-plane (i.e. aCartesian geometry where β is assumed constant with latitude). The forcing generally consistsof continual, random (e.g. Markovian) stirring of the vorticity simultaneously over all spatialregions and over a narrow range of wavenumbers. Because the energy introduced by theforcing cascades upscale where it is relatively unaffected by the hyperviscosity, the meanflow would continuously accelerate if hyperviscosity were the only source of dissipation.Therefore, these investigations generally also include a large-scale drag, −ζ/τ , that allows theflows to equilibrate to a turbulent, fluctuating state characterized by an approximately steadymean vorticity (or wind speed). Here τ is the e-folding timescale for the drag to damp thevorticity towards zero; on Jupiter this might result from drag associated with three-dimensionalprocesses such as turbulent mixing or wave breaking. Williams (1978) pioneered this line ofstudy with simulations performed on a portion of a sphere. When integrated for several hundreddays using a forcing scale comparable to Jupiter’s known jet width, some of the simulationsdeveloped remarkably Jupiter-like jets, such as shown in figure 8. Importantly, Williams’(1978) forcing in these cases was not spatially isotropic, which may have had a major influencein allowing such robust jets. Williams (1978) furthermore found that simulations generatedsuccessful jets only when the forcing scale approximately matched the jet scale; small-scaleforcing did not lead to Jupiter-like jets. The implication is that, although the inverse cascade


Figure 7. Absolute vorticity, ζ +f , for turbulent, freely evolving simulations that solve equation (8)on a sphere from Cho and Polvani (1996a). The initially turbulent field (upper left) evolves tocontain two circumpolar vortices with homogenized absolute vorticity (lower right; note the lackof contours inside the polar vortices). The contours equatorward of the polar vortex are dominatedby the latitudinal variation of f . The simulation produces no mid-latitude jets. Reprinted withpermission from Cho and Polvani (1996a). Copyright 1996, American Institute of Physics.

played a role in creating anisotropy, his Jupiter-like simulations lacked any extent of mergingvortices at scales much smaller than the jets (i.e. there was no ‘inertial range’ in his upscalecascade).

Several additional investigations have confirmed Williams’ (1978) result that zonal jetscan occur in forced two-dimensional turbulence and shed light on the jet-pumping mechanisms.Nozawa and Yoden (1997a, 1997b) and Huang and Robinson (1998) solved equation (8) on asphere including small-scale, random Markovian vorticity stirring with parameters similar tothose in Williams (1978). Unlike Williams (1978), however, these groups used forcing that wasspatially isotropic. Nozawa and Yoden found two dynamical regimes at nonzero rotation rates.In the first, at intermediate rotation rates, strong circumpolar vortices coexisted with weaker,but still persistent, low- and mid-latitude east–west jets (figure 9). In the second regime, atthe highest rotation rates, the polar vortices dominated and the low- to mid-latitude jets wereextremely weak. At the beginning of the simulations where jets form, eddy energy at largewavenumbers cascades to eddy energy at small wavenumbers, arresting at kβ ; the eddy energyat kβ is subsequently converted into zonal energy (jets). Once jets are developed, however,


(a) (b)

(c) (d)

(e) (f)

Figure 8. Stream-function for a forced solution to equation (8), from Williams (1978), showing thedevelopment of zonal jets under Jovian conditions. The trace to the right of each panel shows thezonally averaged zonal wind versus latitude. The simulation is performed over only a longitudinalsector and then repeated for visual display.

eddy energy at the forcing scale is directly converted to zonal energy. In physical space, thesmall vortices get elongated and tilted into the jet shear, indicating an eddy flux of momentumu′v′ into the jets (see section 3.5). Here u′ and v′ are the eddy component of the eastward andnorthward winds, respectively, and the overbar indicates a zonal average (e.g. Ingersoll et al(1981)).

Huang and Robinson (1998) also found that persistent zonal jets form and remain relativelyfixed in latitude (figure 10). The number of jets scales approximately with the Rhines scale;weaker flows have more jets (figure 11). In their simulations, the jet pumping occurred directlyby momentum transfer from small-scale eddies produced by the forcing. In contrast, the largevortices (comparable in size to the jet widths) played little role in forcing the jets, althoughinterestingly these large eddies contained roughly half the total energy in the flow. Theseresults imply that the jet production is inherently non-local in wavenumber space; energy jumpsdiscontinuously from the (small) forcing scale to the (large) jet scale. This picture contrastswith the simple view that jets result from a continuous cascade of energy, via vortex mergers,into incrementally larger and larger vortices and finally into the jets themselves. Consistent withNozawa and Yoden (1997a), the spectrally non-local momentum transfer from small eddiesinto the jets resulted from straining of the small eddies by the large-scale shear associated withthe jets, which produces a u′v′ momentum-flux convergence in the jets. This behaviour implies


Figure 9. Stream-function at 420 Earth days for forced solutions to equation (8), from Nozawa andYoden (1997a). Each panel shows the results of a different simulation; the ratio of the actual rotationrate to Jupiter’s rotation rate is shown in the upper right of each panel. At Jupiter-like rotation rates,polar vortices and weak mid-latitude, jet-like structures form (d), but the mid-latitude structuresalmost totally disappear at higher rotation rates ( f ). Reprinted with permission from Nozawa andYoden (1997a). Copyright 1997, American Institute of Physics.

Figure 10. Zonally averaged zonal wind plotted versus latitude and time for forced solutions toequation (8), from Huang and Robinson (1998). Contour interval is 1 m s−1. Westward speeds areshaded. Numerous zonal jets form. Note that several jets merge before an apparent equilibrium isreached.

that the predominant background shear ‘felt’ by the small eddies must be dominated by the jetsthemselves rather than intermediate-scale eddies (otherwise, the straining of the small-scaleeddies would not correlate with the spatial locations of the jets, as needed for jet pumping tooccur). For this to occur, the dynamics must deviate from that predicted by simple scalingtheories. The standard view of two-dimensional turbulence holds that at scales between theforcing scale and the jet scale, the energy E(k) ∝ k−5/3. Enstrophy, which is the spatially


La

titu

de

Figure 11. Zonally averaged zonal winds versus latitude for a series of simulations from Huang andRobinson (1998). Each simulation uses different forcing and friction parameters and equilibrates toa different mean wind speed, ranging from fast on the left to slow on the right. Note the correlationbetween jet speed and jet width; the jets approximately scale with the Rhines width.

averaged mean-square vorticity of the flow, can be expressed in wavenumber space as k2E(k),implying that the enstrophy k2E(k) ∝ k1/3. The rate of straining is simply the mean vorticityand is given by the square root of enstrophy. This scaling implies that the largest wavenumbers(smallest scales) should dominate the straining; an eddy of wavenumber k will then be primarilystrained by eddies of only incrementally smaller wavenumber k − k rather than much largerstructures having much smaller wavenumber. In contrast to the standard scaling theories,however, Huang and Robinson’s (1998) simulations exhibit a spike in enstrophy at the Rhinesscale, indicating the dominance of the jets in the background rate of straining.

In all the simulations from Huang and Robinson (1998) and Nozawa and Yoden (1997a,1997b), the flow is much less zonal than that in Williams (1978). Williams’ (1978) stronglyzonal flows therefore presumably result from his particular (non-isotropic) forcing (a resultalready foreshadowed by the lack of jets in Williams’ freely evolving simulation; figure 6).Unlike Jupiter, all the simulations of Yoden and Yamada (1993), Vallis and Maltrud (1993),Nozawa and Yoden (1997a, 1997b), Huang and Robinson (1998), Williams (1978), Marcus et al(2000), Danilov and Gryanik (2004), and others produce only jets that satisfy the barotropic-stability criterion in equation (6). The eastward jets in these models are often sharper than thewestward jets, in agreement with Jupiter. This asymmetry results naturally from the tendencyof the fluid to form bands of homogenized potential vorticity separated by regions of sharppotential vorticity gradients (see Marcus and Lee (1998) for discussion).

Recent studies demonstrate that friction plays a key role in setting the jet scale andcontrolling whether jets form at all. In forced simulations without large-scale friction, narrowjets initially form but widen over time via mergers with nearby jets until the surviving jetsreach the largest possible scale, such as the domain width or planetary radius (Chekhlov et al1996, Manfroi and Young 1999, Huang et al 2001). Crudely speaking, this widening can beunderstood as a one-dimensional inverse cascade; the forcing causes the mean wind speedto increase continually over time, which through equation (11) implies an increase in thejet width over time. But it takes time for energy to cascade upscale, and when friction ispresent, the friction can remove energy before it reaches the largest scales. This can lead to anequilibrated flow with coherent structures smaller than the domain size. Consistent with thisidea, two-dimensional turbulence simulations without β show that, when large-scale linearfriction coexists with small-scale forcing, the peak energy of the flow resides at a wavenumber


(Marcus et al 2000, Danilov and Gurarie 2001)

kfr ∼ 50

(1

τ 3ε

)1/2

, (14)

where τ is the frictional-drag timescale and ε is the energy injection rate (m2 s−3). When β isnonzero, the behaviour depends on whether this ‘frictional’ wavenumber is smaller or greaterthan the Rhines wavenumber kβ . When kfr > kβ , friction removes energy before it can reachthe anisotropic Rossby-wave regime, so no jets form and isotropic turbulence results instead.When kfr < kβ , however, energy reaches the Rhines scale before being damped by friction,and the flow reorganizes into jets at the Rhines scale (Marcus et al 2000, Danilov and Gurarie2002). In an equilibrated flow, the mean kinetic energy per mass is τε, so in this context theRhines wavenumber can be expressed as kβ ≈ β1/2/(τε)1/4. Therefore, one might expect jetsto form if

τ >30

β2/5ε1/5. (15)

In other words, given a specified energy-injection rate, jets will form only if the drag isweaker than a specified value. For Jupiter, using ε = 10−5 m2 s−3 corresponding to a kinetic-energy generation rate of 1 W m−2 distributed over a layer 20 bars thick, this condition requiresτ > 107 s for jets to form (consistent with the actual radiative time constant of decades or longerfor this layer). The balance between friction and forcing leads to a relatively constant windspeed, so the resulting jets equilibrate at fixed widths rather than continually widening as inthe frictionless case. The key point is that, in a forced system, jets need friction to equilibrateat a finite width. The friction damps the large-scale energy as it slowly leaks past the Rhinesscale. The Rhines scale can therefore be viewed as a ‘halting’ scale only when friction is alsopresent (Danilov and Gurarie 2002).

The above arguments may help to explain why forced two-dimensional turbulence canproduce jets, while unforced turbulence cannot. The transition from turbulence to Rossbywaves causes a pileup of energy at the Rhines scale, but weak nonlinear interactions betweenRossby waves and turbulence still allow energy to gradually leak to wavenumbers smallerthan kβ . Given sufficient time, this leakage would produce a jet-free flow with an isotropicspectrum if forcing and large-scale friction are absent (Vallis and Maltrud 1993). These ideassuggest that, in long time integrations of unforced turbulence, any intermediate-stage, jet-likestructures might eventually disappear, consistent with the results of Cho and Polvani (1996a).In forced-dissipative simulations, in contrast, the friction removes energy before it can cascadeto the largest scales, helping to maintain coherent jets of finite width (e.g. Huang and Robinson(1998)).

At wavenumbers between the forcing scale and the scale where β-induced anisotropysets in, two-dimensional turbulence tends to produce an energy spectrum following the classicKolmogorov scaling E(k) ∝ k−5/3 (e.g. Maltrud and Vallis 1991, Read 2001). However,several recent studies have presented numerical evidence that at wavenumbers near the Rhinesscale, the spectrum instead follows E(k) ∝ k−5 (Chekhlov et al 1996, Smith and Waleffe1999, Marcus et al 2000, Huang et al 2001, Read et al 2004). This steep spectrum appearsmost readily when large-scale friction is weak or absent. Two-dimensional spectra show thatthe region along the ky axis (corresponding to zonally elongated modes with kx ≈ 0) followsk−5 and the remainder of the wavenumber domain (at wavenumbers exceeding the Rhineswavenumber) follows k−5/3. At small wavenumbers, the energy is almost totally dominatedby the component along the ky axis, which corresponds to the dominance of zonal jets inphysical space. Huang et al (2001) argue that this concentration of energy along the zonalaxis results from stabilization of the jets by the β effect. Qualitatively, it is easy to understand


how this effect can produce a steep spectrum. For a jet profile corresponding to a sinusoid ofwavenumber k, equation (11) suggests average wind speed u ∝ k−2 and hence total energyper mass is proportional to k−4. Nevertheless, the precise mechanisms that lead to the ‘−5’exponent remain unclear, and several papers have even challenged whether any power-lawscaling relationship is valid (Danilov and Gurarie 2004). As emphasized by Danilov andGryanik (2004), jet formation creates band of homogenized potential vorticity separated bynarrow regions of high potential vorticity gradient (Rhines and Young 1982, Marcus andLee 1998), which produces an energy spectrum with numerous peaks and valleys (i.e. not apower law).

This debate is directly relevant to Jupiter and Saturn. Galperin et al (2001) and Sukorianskyet al (2002) fit power laws to energy spectra derived from Voyager wind observations and foundthat exponents of −5±1 provided the best fit to the spectra. However, the peak-to-peak scatteraround the fit was three orders of magnitude, and Danilov and Gryanik (2004) argued that thisscatter consists of the peak-and-valley structure they might expect from a jet-dominated flow.Conceivably, long-time averaging might smooth over the peak-and-valley structure and yield abetter power-law fit (Galperin et al 2001, Sukoriansky et al 2002), but the long-term coherenceof the jet pattern (which produces the scatter) makes this unclear (Danilov and Gryanik 2004).

Jupiter and especially Saturn have equatorial jets that are wider and substantially faster (byfactors of 1.5–3) than most of their mid-latitude jets (figure 3). However, the two-dimensional,non-divergent studies described here fail to capture this important feature. The equatorial jetsproduced in these models tend to have widths and speeds comparable to those of the mid-latitude jets. Furthermore, Jupiter and Saturn’s equatorial jets are eastward, whereas most ofthe two-dimensional simulations produce westward equatorial flow. This failure of the two-dimensional models suggests that the equatorial jets on Jupiter and Saturn probably result fromother (e.g. three-dimensional) processes. However, it is worth noting that a small fraction oftwo-dimensional simulations produce eastward equatorial jets (e.g. figure 11).

A possible challenge to the application to Jupiter of two-dimensional models governed bya vorticity equation comes from the recent realization that potential vorticity is not conservedin Jupiter’s upper troposphere—even in the frictionless and adiabatic limit (Gierasch et al2004). This difficulty comes from the presence of hydrogen ortho/para variations and watervapour, which imply that entropy (potential temperature) is no longer a function of pressureand density alone, as needed for conservation of Ertel’s potential vorticity. It is probable thata modified vorticity equation still holds (Gierasch et al 2004), so that the two-dimensional,jet-pumping studies described here have general relevance. However, detailed comparisonsbetween simulations and observations may require the use of more advanced models that donot hinge solely on the conservation of potential vorticity.

In summary, forced two-dimensional, non-divergent models have provided an importantframework to understand the formation of jets in fluids that have approximately two-dimensional flow. In these models, the inverse cascade, the β effect, and their interactionto produce jets manifest in their simplest possible form. As successes, these models show thatjets can form over a wide range of parameter values for forcing and friction, and these jetshave approximately the correct width given Jupiter’s known wind speeds (that is, these modelsproduce jets with widths ∼π(U/β)1/2, which approximately equals the width of Jupiter’sjets for a characteristic Jupiter-like wind-speed of U ≈ 40 m s−1). As failures, the modelsoften develop polar vortices that are lacking on Jupiter, their jets always satisfy the barotropicstability criterion, and their equatorial jets are too weak and usually flow in the wrong direction.The details of the jets (e.g. the extent to which they meander in longitude) depend rathersensitively on the model parameters and formulation. Of course, Jupiter’s flow is not purelytwo-dimensional, and these investigations are best viewed as process models providing a


theoretical understanding of jet production in a simple environment rather than as realisticgeneral circulation models of Jupiter.

3.4.2. Shallow-water models. The non-divergent, two-dimensional vorticity equation,while informative, lacks key aspects of dynamics that are potentially crucial for atmospheres.In particular, column stretching cannot occur, which excludes gravity (buoyancy) waves and ahost of other phenomena. The simplest physical system that includes full column stretching isthe shallow-water model, which governs the evolution of a single layer of variable-thickness,constant-density fluid. The equations are derived under the assumption that (i) the horizontalscales of interest greatly exceed the layer thickness and (ii) the dynamics does not dependcritically on details of the vertical structure (see Pedlosky (1987), chapter 3, for a derivation).The equations can be written as

dv

dt= −g∇h − f k × v, (16)

dh

dt= −h∇ · v, (17)

where h and v are the layer thickness and horizontal velocity, respectively, and g is gravitationalacceleration. As before, d/dt = ∂/∂t + v · ∇. The dependent variables are horizontal position(e.g. latitude and longitude) and time, so the shallow-water equations still constitute a two-dimensional model. However, in contrast to equations (8) or (9), the horizontal divergence isnonzero and can cause changes in the fluid thickness through equation (17). These thicknesschanges are the shallow-water model’s form of vertical motion.

The relevance of the shallow-water model to giant planet atmospheres stems fromconsidering a two-layer model, where the upper layer represents a low-density weather layerand the lower layer represents the denser deep interior. In the limit where the lower layerbecomes infinitely deep (representing an adiabatic, convectively adjusted giant-planet interior)and lower-layer winds and pressure gradients remain steady in time (which requires theupper layer to be isostatically balanced), the two-layer system collapses to the shallow-waterequations for the upper layer, with one modification: the gravity in equation (16) is multipliedby the fractional density difference between the layers (see Gill (1982), Dowling and Ingersoll(1988,1989) for discussion). For Jupiter, the buoyancy of the weather layer relative to the deepinterior may result from the 5–10 K latent-heat release associated with condensation of water,in which case the ‘reduced’ gravity would be a few per cent of the actual gravity.

In addition to the Rhines length (which exists when f varies spatially), the shallow-waterequations contain another intrinsic length scale, the deformation radius Ld = √

gh/f thatresults from the interaction of planetary rotation with the gravitational relaxation of thicknessvariations. Roughly speaking, gravitational relaxation tends to produce structures with awidth near the deformation radius. The two-dimensional equations (8) and (9) have an infinitedeformation radius.

Several shallow-water investigations have been published that focus on the interactionbetween slow-moving vortex structures and higher-frequency gravity waves and the extent towhich these interactions affect coherent-structure formation and the inverse cascade (Farge andSadourny 1989, Spall and McWilliams 1992, Polvani et al 1994, Yuan and Hamilton 1994).However, only a few turbulent shallow-water simulations have been published that address theformation of jets.

Cho and Polvani (1996a, 1996b) performed freely evolving shallow-water simulations ona sphere starting with a random, balanced distribution of small-scale turbulence. When theRossby number was small, they found that, unlike the case of freely evolving two-dimensional,


Figure 12. Freely evolving shallow-water simulations on a sphere from Cho and Polvani (1996a).The simulation uses Jupiter parameters and a deformation radius of 2000 km. Potential vorticityis plotted; positive values are solid and negative values are dotted. In the final state, note thebunching of potential vorticity contours, separated by bands of nearly homogenized potentialvorticity, indicating the formation of jets. Numerous vortices also form, in qualitative agreementwith Jupiter. The number of jets formed also approximately scales with the Rhines width. However,the equatorial jet is westward rather than eastward as observed. Numbers on plots give times inJovian rotation periods. Reprinted with permission from Cho and Polvani (1996a). Copyright1996, American Institute of Physics.

non-divergent turbulence, the turbulence could reorganize into multiple jets (figure 12). Thisappears to be the first unforced study that produced multiple bands and jets from isotropicturbulence. Cho and Polvani found that the jets form only when the ratio of deformation radiusto planetary radius is less than one-third (consistent with the lack of jets in freely evolvingtwo-dimensional turbulence, where the deformation radius is infinite). When jets formed,their number scaled with the Rhines length, and for Jovian parameters yielded ∼10–15 jets, ascompared with the ∼30 jets exhibited by Jupiter. Cho and Polvani’s turbulent initial conditionscontained equal numbers of cyclones and anticyclones (implying an initial vorticity ‘skewness’of zero), but as the flow evolved, anticyclones grew in preference to cyclones, leading to anegative skewness. This result is consistent with planar turbulence simulations in absence ofthe β effect (Arai and Yamagata 1994, Polvani et al 1994).

Interestingly, published studies of shallow-water turbulence on a sphere all produce astrong westward jet at the equator (Cho and Polvani 1996a, 1996b, Iacono et al 1999a, 1999b,Peltier and Stuhne 2002). The predominance of the equatorial jet in these simulations differs


from the two-dimensional, non-divergent case, where the equatorial jet width and speed tendsto match those of the mid-latitude jets. The generation of strong equatorial flow in shallow-water models might be viewed as a step forward from the two-dimensional, non-divergentcase, since Jupiter and Saturn also have equatorial jets much stronger than most of their mid-latitude jets. However, the glaring discrepancy is that the equatorial jets on Jupiter and Saturnare eastward, not westward (although it is worth pointing out that Uranus and Neptune havewestward equatorial flows). Iacono et al (1999b) suggested that the development of a westwardequatorial jet in shallow-water simulations results directly from the prevalence of anticyclonesin shallow water. Stokes’ theorem applied to a hemisphere implies that∫

hemisphereζ dA =

∫equator

u dl, (18)

where the area integral is performed over the hemisphere, the line integral is performedaround the equator, A is area, and u is the east–west wind speed. The equation states that,if the hemisphere exhibits net anticyclonic vorticity, then this must manifest as westwardflow at the equator. Because the shallow-water system evolves to favour anticyclones (seesection 4.2), westward equatorial flow naturally results (in this context, it is interesting thatnon-divergent, two-dimensional simulations, which lack a cyclone–anticyclone asymmetry,sometimes produce eastward jets; see figure 11). These results seem to imply that the shallow-water model lacks sufficient physics to capture the equatorial jets on Jupiter and Saturn, andhence these equatorial jets results from three-dimensional processes (e.g. vertical momentumtransport by waves or convection) that are not included in the shallow-water model.

Several studies have shown that a finite deformation radius can inhibit the formation ofjets. This phenomenon seems counter-intuitive, because the deformation radius provides anadditional length scale, besides the Rhines scale, that under some conditions can slow down orhalt an inverse cascade, and some studies have even suggested that the deformation radius helpsto set the jet width on Jupiter (Ingersoll et al 2004). The effect is easiest to illustrate usinga one-layer quasi-geostrophic model, which retains a finite deformation radius but (unlikethe shallow-water model) formally constrains the dynamics to follow geostrophic balance(Pedlosky 1987, chapter 3). Okuno and Masuda (2003) and Smith (2004) pointed out that,under the influence of finite deformation radius, the Rossby-wave dispersion relation becomes

ω = − βkx

k2x + k2

y + k2d

, (19)

where kd is the wavenumber associated with the deformation radius. The Rhines scale isobtained by equating this frequency with the turbulence frequency U |k|, which yields amodified Rhines scale

k2β = β

U|cos φ| − k2

d, (20)

where φ is the angle between the Rossby-wave propagation direction and east (compare withequations (12) and (13)). The deformation radius slows down the Rossby-wave oscillation,which moves the Rhines scale closer to the origin on the kx–ky plane. This effect impliesthat flows with finite deformation radius have wider jets than two-dimensional, nondivergentflows for the same values of U and β. Clearly, if kd is large enough, the ‘dumbbell’ in kx–ky

wavenumber space shrinks to zero size, which implies that all wavenumbers are turbulencedominated. In this case, the Rhines scale does not exist and the flow is fully governed byisotropic turbulence in the presence of finite deformation radius. Consistent with this argument,Okuno and Masuda (2003) show that the deformation radius reduces the strength of the linearβ term relative to the nonlinear Jacobian term in the governing equation, and in the limit of


Figure 13. Forced shallow-water simulations as described in Showman (2004). Greyscale showslayer thickness, h, and arrows show wind speeds over a domain extending 120˚ in longitude and70˚ in latitude. Simulation results are shown at ∼4200 Earth days (more than 10 000 Jupiterrotations). Initially, the simulation contained no winds, but localized thunderstorms with diametersof ∼1700 km (or 1.5˚) repeatedly occur that generate multiple anticyclones which undergo aninverse cascade. Interaction with the β effect leads to a robust westward jet at the equator andan eastward jet at 30˚ latitude. At higher latitudes, however, the flow remains dominated by asingle large anticyclone that continually sweeps up the smaller anticyclones that form. No jetsform poleward of 30˚ latitude. The peak wind speed is 96 m s−1, similar to that of Jupiter. Thedeformation radius is about 1500 km in mid-latitudes.

zero deformation radius (kd → ∞), the β term formally drops out of the equation. Quasi-geostrophic numerical simulations confirm that, for small deformation radii, no jets form andisotropic turbulence results instead—even when the β effect is strong (Okuno and Masuda2003, Smith 2004).

Showman (2004) presented the first forced shallow-water simulations aimed atdetermining whether jets can result from a realistic representation of Jovian turbulence.Consistent with the suggestion of Ingersoll and Cuong (1981), Ingersoll et al (2000) andothers, he assumed that the cloud-layer turbulence results from the episodic thunderstormsobserved by Voyager, Galileo and Cassini (see section 5). Based on the idea that stormstransport mass from the deep atmosphere into the cloud layer, the storms were represented usinglocalized, episodic mass pulses added to the shallow-water layer. These mass pulses underwentgeostrophic adjustment that produced a population of vortices (primarily anticyclones), whichthen underwent an inverse cascade. Jets formed near the equator, but the mid-latitudes tendedto be dominated by large anticyclones rather than jets (figure 13). This result is broadlyconsistent with the idea that the deformation radius inhibits the β effect and the formation ofjets. Nevertheless, the existence of winds in the deep interior, which can be represented byadding a term to equation (16), altered the dynamics and sometimes allowed jets to form inthe weather layer.

The jet-suppression phenomenon described here raises issues about whether shallowturbulence can produce Jupiter-like jets under realistic conditions, particularly since thedeformation radius on Jupiter is thought to be only ∼2000 km (i.e. small enough to havean important effect). Nevertheless, three-dimensional shallow flow could behave differentlyfrom either purely two-dimensional or one-layer, shallow-water turbulence, and morethorough investigation of three-dimensional shallow models is needed for a full assessment.Furthermore, it remains unclear how the freely evolving simulations of Cho and Polvani (1996a,1996b) were able to form jets despite the jet-suppression mechanism that occurs at small


deformation radius. Interestingly, shallow-water turbulence (Cho and Polvani 1996a, 1996b,Showman 2004) seems able to produce isolated, long-lived, Jupiter-like vortices more easilythan two-dimensional turbulence (e.g. Marcus et al (2000)), which perhaps results from thefinite Ld in shallow water.

3.4.3. Multi-layer models. Multi-layer studies of shallow jet forcing have primarilyfocused on the hypothesis that jets result from baroclinic instabilities triggered by horizontaltemperature contrasts (e.g. induced by sunlight or moist convection) at cloud level. Baroclinicinstabilities are a dominant process in Earth’s troposphere because the equator is >20 K warmerthan the poles, and these instabilities help to drive Earth’s upper-tropospheric, mid-latitude jetstreams. But on Jupiter, the poles and equator are nearly equal in temperature, and the primaryupper tropospheric temperature contrasts (which are typically ∼5–10 K above the clouds) occurbetween belts and zones rather than pole and equator. These facts argue against jet pumpingcaused solely by an equator-to-pole temperature contrast. Nevertheless, differences in solarabsorption or latent heating could cause temperature contrasts between belts and zones belowthe clouds (e.g. Ingersoll and Cuzzi (1969)), and it is possible that baroclinic instabilities tapthese temperature contrasts to pump the jets. No models have yet attacked this problem, inpart because it requires full coupling between radiation, dynamics, cloud formation and moistconvection. Instead, most studies to date have used simpler (e.g. smoothly varying equator-to-pole) heating contrasts. Despite their simplicity, these studies have provided considerableinsight into how baroclinic instabilities can drive jets.

Williams (1979) performed a pioneering two-layer, quasi-geostrophic study on a β-plane.The lower layer was assumed to be denser than the upper layer, with vertical static stabilitiesvaried over a range appropriate for Jupiter’s atmosphere. The forcing consisted of heating thatvaried with latitude, which produced a latitudinal temperature contrast between the north andsouth boundaries ranging from 10 K to more than 60 K, depending on the simulation. Thestatic stabilities and layer thicknesses were small enough that the expected vertical wind shearresulting from the thermal-wind equation (1) is <1 m s−1 for the simulations with the smallestequator-to-pole temperature contrast. Consistent with this expectation, Williams’ simulationsproduced wind-speed differences between the layers of less than a few metres per second.However, the simulations developed a strong barotropic component of the flow that organizedinto multiple zonal jets with wind speeds of 50–80 m s−1. These jets apparently resulted froma two-dimensional inverse cascade, energized by the baroclinic instabilities, that arrested at theRhines scale, in analogy with many of the one-layer models discussed earlier. It is importantto emphasize that, unlike the one-layer investigations discussed earlier, no external sources ofturbulence (e.g. momentum or vorticity forcing) were imposed here; the turbulence resultednaturally from the baroclinic instabilities triggered by the imposed heating contrast. Panetta(1993) performed a similar two-layer, quasi-geostrophic β-plane study, focusing on analysinglong-time behaviour and the dynamics that maintained the equilibrium state. Robust jets wereobtained, although these meandered somewhat in latitude.

The strong jets obtained by Williams (1979) and Panetta (1993) imply the existence ofstrong horizontal pressure contrasts, which in these models are supported by the rigid upperand lower boundaries. On a giant planet, however, there is no solid surface to support suchpressure variations. It may be that a deep layer of static stability, if any, could help support suchvariations, or that the tendency of the flow to become barotropic would drive Taylor columnsthat would extend throughout the planetary interior, but these ideas have yet to be demonstrated.The relevance of these two-layer investigations to Jupiter’s jets therefore remains unclear.

Williams (2003a) revisited the issue of whether Jupiter-like jets can result from latitudinalheating contrasts using a 20-layer, primitive-equation model, which is much more sophisticated


Figure 14. Latitude–altitude cross-sections through three-dimensional simulations from Williams(2003a). Shown are contours of zonally averaged zonal wind, with eastward in solid curvesand westward in dashed. Each panel shows the results of a different simulation with differentassumptions about the radiative heating profile. In each case, ∼6–8 zonal jets form.

than the one- and two-layer models discussed previously. The primitive equations formthe basis for most Earth–atmosphere, general-circulation models and include the full rangeof gravity-wave and geostrophically balanced motions. Convection cannot occur, however,because these models assume that individual fluid columns are hydrostatically balanced, whicheliminates the vertical accelerations necessary for convective overturn. This dynamical regimeis appropriate for stably stratified fluid systems with horizontal dimensions that greatly exceedvertical dimensions (see Holton (1992)). Williams adopted a domain 15 000 km deep andintegrated the equations on a sector of a sphere using the Jovian radius and rotation rate.The forcing consisted of a Newtonian-radiation scheme that relaxed the temperature towardsan assumed radiative-equilibrium temperature profile, which varied smoothly from hot at theequator to cold near the poles. This heating/cooling was confined to the top few hundredkilometres of the domain; the underlying layers formed a neutrally stratified region intendedto represent the Jovian interior. As in the two-layer studies, the temperature contrasts drovebaroclinic instabilities that pumped momentum into numerous zonal jets of approximately thecorrect width (figure 14). The speeds at the top of the model reached ∼80 m s−1, comparable toJovian wind speeds. However, no barotropic flow existed; the winds at the top resulted purelyfrom thermal-wind shear associated with horizontal temperature contrasts. In the simulations


shown in figure 14, where the equator-to-pole temperature difference is ∼6 K, the wind speedsat the top only reach Jovian values because the temperature contrasts extend over hundredsof kilometres depth (see equation (1)). On Jupiter, sunlight absorption occurs primarily atpressures less than ∼3 bars, however. If the thermal winds were confined to this layer, theflows would be much weaker (∼m s−1). Therefore, this baroclinic instability model can explainJupiter’s jets only if the dynamics can somehow force the thermal winds to depths far belowthe layer where heating contrasts are confined. However, it has yet to be demonstrated whetherthis is possible.

Williams (2003a, 2003b, 2003c) spent considerable effort determining whether three-dimensional shallow flows driven by baroclinic instabilities can produce an eastward equatorialjet, as observed on Jupiter and Saturn. Heating profiles that generated only weak latitudinaltemperature gradients near the equator (as expected for solar heating with uniform albedo)produced westward equatorial flow similar to most of the one-layer models discussed earlier.However, when a heating function was chosen that induced a large latitudinal temperaturegradient near the equator, an eastward equatorial jet formed instead. Apparently, in thesecases, the inevitable baroclinically driven mid-latitude eastward jets could form closer to theequator than in cases with relatively constant equatorial temperatures. When the jets closestto the equator formed at a latitude less than a critical value, they experienced a barotropicinstability that drove eddy energy and momentum to the equator, producing an eastward flowright on the equator. This process bears some similarities to a mechanism suggested by DelGenio et al (1993) and Del Genio and Zhou (1996) for the super-rotation on Venus and Titan (inthose cases, however, the off-equatorial jets that become barotropically unstable are producedby the near-global Hadley circulation rather than baroclinic instabilities; this difference resultsfrom the much slower rotation rates of Venus and Titan as compared with that of Jupiter). ForJupiter, latent heating in the equatorial zone, if any, might be able to produce the latitudinaltemperature gradients required to drive the eastward equatorial jet. It remains to be seenwhether Jupiter’s equatorial zone actually behaves this way.

Numerical and laboratory experiments of baroclinic instability show that, when the fluid’stemperature profile varies monotonically with latitude (e.g. from equator to pole), large vorticessuch as the Great Red Spot and White Ovals do not form between the jets (Williams 1979,2003a, Read 1986). In these experiments, substantial longitudinal turbulent structure occurs,but this structure tends to fluctuate in time and does not exhibit the long-lived, fluid-entrappingbehaviour typical of stable vortices (the large ‘gyre’ in Williams (1979) appears to be a wavelikestructure because it does not correspond to a local maximum or minimum of potential vorticity).In contrast, laboratory experiments and numerical simulations that generate nonmonotonictemperature profiles versus latitude (i.e. alternating hot and cold latitudinal bands) do producelarge, fluid-entrapping vortices analogous to the Great Red Spot and White Ovals (Readand Hide 1983, 1984, Read 1986, Hide et al 1994, Williams 2002). This difference invortex-generating behaviour contrasts with the fact that either type of profile (monotonic ornonmonotonic temperature versus latitude) can lead to jets. These results support the idea that,if baroclinic instabilities play a role in jet formation, the instabilities may result not simply froman equator-to-pole variation in solar flux (since this always leads to monotonic temperatureprofiles versus latitude) but from local temperature contrasts between belts and zones. Suchcontrasts might result from belt-zone albedo or latent-heating differences. In this context, itis worth remembering that the observed temperatures above the clouds are not monotonic inlatitude.

An alternative to the baroclinic/barotropic-instability model for Jupiter’s superrotatingequatorial jet was proposed by Yamazaki et al (2005), who suggested that absorption ofupwardly propagating Kelvin waves could lead to eastward equatorial flow in the upper


troposphere. The source region of the Kelvin waves (underneath the eastward jet) wouldbe accelerated westwards, so this model predicts that the eastward jet is shallow and possiblyunderlain by westward equatorial flow. Given the Galileo probe observation of fast eastwardwinds down to 20 bars, this source region must lie deeper than 20 bars. Kelvin waves requirestatic stability to propagate, so the model therefore requires a statically stable layer extendingto at least 20 bars pressure. The mechanisms that would force the Kelvin wave remain to beidentified, although the interaction of moist convection with the large-scale circulation is onepossibility. A superimposed Hadley cell can provide the anticyclonic ‘cusp’ at the equator(Yamazaki et al 2005).

Few three-dimensional studies yet exist that investigate whether jets can result fromsmall-scale turbulence (as opposed to baroclinic instability) in Jupiter’s shallow weather layer.Kitamura and Matsuda (2004) presented two-layer, primitive-equation simulations of freelyevolving turbulence in a stably stratified fluid on a sphere. The initial condition contained small-scale, space-filling turbulence; damping consisted solely of a hyperviscosity that maintainednumerical stability but had minimal effect on the flow’s energy. As in the shallow-watersimulations of Cho and Polvani (1996a), they found that multiple jets emerged at the Rhinesscale when the Rossby number was much less than one and deformation radius was smallcompared with a planetary radius. As in analogous two-dimensional investigations (Yodenand Yamada 1993, Cho and Polvani 1996b, Yoden et al 1999), a strong circumpolar vortexdeveloped at each pole. The jets were largely barotropic except at the equator, where the toplayer developed an eastward jet and the bottom layer developed a westward jet. This resultmay have relevance in understanding equatorial superrotation on Jupiter and Saturn.

Laboratory experiments by Read et al (2004) and Aubert et al (2002) show that whensmall-scale turbulence is induced in a shallow rotating tank of water with a sloping bottom(representing the β effect), multiple zonal jets can result. The resulting jet widths scale with theRhines length π(U/β)1/2. However, the jets in both experiments had greater longitudinal andtime variability than Jupiter’s jets. Interestingly, the jets in the Read et al (2004) experimentviolated the standard (two-dimensional) barotropic stability criterion. It is possible that thestrong turbulent forcing and eddy activity help to maintain such a violation, although it mayalso simply reflect the fact that three-dimensional fluids are subject to different stability criteriathan two-dimensional fluids (e.g. Dowling (1995a)). In any case, the experiments demonstratethe reality of jet production from an inverse cascade in a three-dimensional, real-life setting.

On balance, the shallow-forcing models discussed here and in the two precedingsubsections show promise in explaining Jupiter’s mid- and high-latitude jets. These modelstend to produce zonal jets of approximately the correct width over a wide range of modelconditions, indicating that jet formation by the interaction of the inverse cascade with theβ effect is a robust process, independent of model details. However, the specifics, suchas whether jets migrate in latitude and time, the sharpness of the jets, and the extent towhich energy partitions between vortices and jets depend on the model. The one-layer (two-dimensional and shallow-water) studies fail to explain Jupiter and Saturn’s eastward equatorialjets. Multilayer-shallow dynamics can overcome this failure for some parameter regimes, butit remains to be seen whether these are the actual parameter regimes occupied by Jupiterand Saturn. Injection of turbulence into the cloud layer by convection (e.g. thunderstorms)currently provides the most promising energy mechanism for driving an inverse cascade nearcloud-level. Baroclinic instabilities triggered by belt-zone temperature contrasts are anotherpromising possibility. In contrast, baroclinic instabilities triggered by an equator-to-poletemperature contrast are probably inhibited by the similar temperatures between the polesand equator (and if such instabilities occur, they may not inject much energy into the flow).The extent to which realistic thunderstorm turbulence can generate Jupiter-like jets has never


been investigated with a multi-layer model, and this is an important avenue for future research.Another possible shallow-energy source that could help drive the jets is band-scale overturningcirculations (e.g. Hadley cells) driven by variations in latent heating or sunlight absorptionbetween adjacent bands (Ingersoll and Cuzzi 1969, Barcilon and Gierasch 1970, Gierasch1973, 1976, Gierasch et al 1973). Under this hypothesis, energy is emplaced into the cloudlayer directly at a scale comparable to the sizes of the largest jets and vortices. By itself, thisband-scale forcing cannot easily generate zonally elongated structures, but in concert with theβ effect and an inverse cascade driven by turbulence, such band-scale forcing could play amajor role in shaping the cloud-level jet structure. The next generation of Jupiter circulationmodels will start to include these effects.

3.4.4. Deep models. The alternate paradigm for formation of Jupiter’s jets hypothesizes thatthe jets result from convection throughout Jupiter’s interior. This hypothesis originated withBusse (1976, 1970), who argued from linear analysis and laboratory experiments that thermalconvection in a spherical shell, such as Jupiter’s molecular envelope, will drive nested Taylor–Proudman cylinders that manifest as jets at the cloud layer. For many years, progress in thisarea involved linear or weakly nonlinear analyses of convection and waves in spherical shells,and debate has long existed over the applicability of these results to the strongly nonlinearregime of Jupiter’s interior (e.g. Yano (1998)). We refer the reader to Yano (1994, 1998)and Busse (1994, 2002) for reviews of this early literature. The past few years have seengreat advances in testing the deep-convection hypothesis, as computers are now fast enough tointegrate the three-dimensional fluid equations at parameter regimes and spatial resolutions ofgreater relevance for giant planets (though the simulated parameter regimes are still far fromthe Jovian regime). Here we focus on qualitative aspects and the recent literature.

Consider a convecting spherical shell that is heated from the inside and cooled from theoutside. When viscous effects are small and rotation dominates the dynamics, the Taylor–Proudman theorem still approximately holds despite the convective motions. The primaryviolation of the Taylor–Proudman theorem will occur at the solid boundaries, where thin,frictionally dominated Ekman layers form. Nevertheless, a secondary circulation developsdue to the buoyant convective forces, which causes exchange of fluid between the nestedcylinders, as depicted schematically in figure 15. When the Taylor–Proudman theorem holds,this convective overturning (figure 15) must be weak (much weaker than Jupiter’s observed jetspeeds); otherwise strong divergence or convergence will occur in the plane perpendicular tothe rotation axis, in violation of the Taylor–Proudman constraint.

The basic mechanism by which convective rolls can cause differential rotation is theReynolds stresses that result from tilting of the convection cells. When convection cells arenot tilted, the net vertical momentum transport averages to zero, so no zonal flow develops.In contrast, imagine a columnar convection cell whose convective motions are parallel tothe equatorial plane (figure 15). Let u′ be the east–west convective velocity and w′ be theconvective velocity towards or away from the planetary rotation axis. When the convectioncells become tilted to the east (figure 15(a)), the outward branches of the convective motions(which transport fluid away from the rotation axis, w′ > 0) have net eastward momentum(u′ > 0), while the inward branches (which transport fluid toward the rotation axis, w′ < 0)have net westward momentum (u′ < 0). In both cases, u′w′ > 0, implying a net outwardtransport of eastward momentum (inward transport of westward momentum) by the circulation.This transport induces a mean flow, eastward at the outer regions and westward at the innerregions, which further tilts the convection cells. A positive feedback ensues; infinitesimaltilting induces differential zonal flow that strengthens the tilting, which further strengthensthe zonal flow. Internal viscous stresses (which are stronger for stronger differential rotation)


(a) (b)

Figure 15. Schematic diagram of Taylor columns and differential rotation in a cylindrical annulusfrom Busse (2002). In panel (a), the concave-inward curvature of the surface forces the convectioncells to tilt east and out, which drives eastward flow at the outer edge and westward flow at theinner edge. In panel (b), the convex-inward curvature causes convection cells to tilt west and out,which leads to westward flow at the outer edge and eastward flow at the inner edge. Reprinted withpermission from Busse (2002). Copyright 2002, American Institute of Physics.

counteract the process, and equilibration occurs when these viscous stresses balance the u′w′

Reynolds stresses. Analogously, convection cells that tilt to the west would drive westwardflow at the outer regions and eastward flow at the inner regions (figure 15(b)).

Theory and experiments to date suggest that Rossby waves provide a key mechanism thatbreaks the degeneracy shown in figure 15 and determines the direction of differential rotation(Busse 2002). In a thin atmosphere, Rossby waves propagate to the west (see equation (12)),which results from the fact that poleward and equatorward motion induce anticyclonic and cy-clonic relative vorticity, respectively (see Holton (1992), p 216, for a discussion of this point). Inthe deep-cylinder context, however, motion of a Taylor column toward the rotation axis inducescyclonic vorticity while motion away from the rotation axis induces anticyclonic vorticity—the opposite of the thin-atmosphere case (because of the spherical geometry, inward-movingTaylor columns get stretched, which through conservation of potential vorticity spins them up;outward-moving Taylor columns get compacted, which spins them down). This sign reversalmeans that Rossby waves propagate eastward in a deep-spherical geometry. In a sphere, agiven deflection of a vortex column towards or away from the rotation axis induces only minorcolumn stretching when the column is close to the rotation axis, but great column stretchingwhen the column is far from the rotation axis. This implies that Rossby waves propagateeastward faster for columns further from the rotation axis. Because Rossby-wave dynamicstends to induce propagation of vortices (westward in the thin-atmosphere context; eastward inthe deep-spherical context), outward-moving branches of convection cells, which are basicallyTaylor vortex columns moving away from the rotation axis, get deflected to the east. Eastwardzonal flow at the top, and westward flow at the bottom, results. This process is illustrated fora rotating cylindrical annulus in figure 15(a). In contrast, if Taylor columns terminate againstboundaries that are concave rather than convex (figure 15(b)), Rossby waves propagate east-ward faster for fluid columns closer to the rotation axis. Outward-moving vortex columns thenget deflected west while inward-moving vortex columns get deflected east. The result is west-ward flow at the top and eastward flow at the bottom. Using a two-dimensional model throughJupiter’s equatorial plane, Yano et al (2003) confirm that, as expected, the turbulent motion ofTaylor columns inside a sphere readily allows the formation of eastward flow at the equator.

Recent three-dimensional numerical simulations of the fluid equations demonstrate thatzonal flows indeed result from convection in spherical shells, and they show that eastward


flow inevitably occurs at the equator, consistent with the qualitative considerations above.These investigations solve the Navier–Stokes momentum equations subject to the Boussinesqapproximation, which allows density variations (associated with temperature variations) toinfluence the momentum equation through a buoyancy term, but assumes that these densityfluctuations are small enough for the incompressible continuity equation to apply. Theseequations are

E

(dv

dt− ∇2v

)+ 2z × v = −∇p +

(RaE

Pr

)g

g0T r, (21)

dT

dt= 1

Pr∇2T , (22)

∇ · v = 0, (23)

where E = ν/�D2 is the Ekman number (i.e. the ratio of frictional to Coriolis forces),Pr = ν/κ is the Prandtl number (ratio of kinematic viscosity to thermal diffusivity), andRa = αg0 T D3/κν is the Rayleigh number (a measure of convective vigor). Here v, T

and p are three-dimensional velocity, temperature and pressure, z is the unit vector parallel tothe rotation axis, r is the radial unit vector, ν is kinematic viscosity, κ is thermal diffusivity, T is the temperature drop between the inner and outer boundaries, � is the rotation angularfrequency, α is the thermal expansion coefficient, g is gravity (which typically increases linearlywith radius in these studies), g0 is the reference gravity (e.g. at the outer boundary) and D

is the thickness of the spherical shell (i.e. the difference between the outer and inner radii).The equations have been made non-dimensional using D for length, D2/ν for time, ν/D

for velocity, ρν� for pressure and T for temperature. This practice is standard, althoughvariations are also in use (e.g. Christensen (2002)).

Solution of these equations at Ekman numbers of 10−3–10−5 show that several dynamicalregimes exist, depending on the Rayleigh number. At weakly supercritical Rayleigh numbers,the flow forms broad columnar convection cells (with axes parallel to the rotation axis)outside the tangent cylinder that are steady in time except for a constant longitudinal drift ofthe convection patterns. In this regime, the zonal flow is weaker than the convective flow andthe total heat flux only slightly exceeds the conductive flux. Increases in Rayleigh number up totwo times the critical Rayleigh number cause these convection cells to become time-dependent,first periodic and then chaotic in time (Sun et al 1993). Because the critical Rayleigh numberinside the tangent cylinder exceeds that outside, these weakly nonlinear regimes generallyexhibit no convection inside the tangent cylinder.

At Rayleigh numbers that are a few times critical, the flow outside the tangent cylinderbecomes fully turbulent, with numerous, narrow columnar convective plumes that ascend anddescend from the inner and outer boundaries, respectively. When no-slip boundary conditionsare adopted (i.e. the tangential component of the velocity vector is zero at the boundaries), littlezonal flow forms. When free-slip boundary conditions are used, however, a strong zonal flowdevelops that can dominate over the non-zonal flow by a factor of ten or more, depending on E

(Grote et al 2000, Aurnou and Olson 2001, Christensen 2001). In this regime, the convectionand zonal flow tend to be strongly episodic in time (Grote and Busse 2001). Weak zonalflow permits domain-crossing convective plumes that produce large Reynolds stresses andacceleration of the flow. Once these Reynolds stresses lead to strong zonal flow, however, theflow shears the convective plumes apart before they can traverse the layer and so the Reynoldsstresses (and flow acceleration) weaken. Viscous stresses then cause a deceleration of theflow. Once the zonal flow decreases substantially, the convective plumes can again traversethe layer, causing the cycle to repeat (a similar oscillatory behaviour also occurs in the simpler


(a) (b) (c)

Figure 16. Results of a three-dimensional Boussinesq convection simulation from Christensen(2001). Simulation used E = 3×10−5, Ra = 108 and inner-to-outer radius ratio of 0.6. (a) Radialvelocity midway through the spherical shell. (b) Temperature in the equatorial plane. Red is hotand blue is cold. (c) Zonally averaged zonal wind in the shell, in a plane parallel to the rotation axis.Red is eastward and blue is westward. (Christensen 2001). Copyright 2001 American GeophysicalUnion. Reproduced by permission of American Geophysical Union.

periodic solutions at lower Rayleigh number; see Sun et al (1993)). Simulations by Aurnouand Olson (2001) at an Ekman number of 3 × 10−4, Rayleigh number of six times critical,and an inner-to-outer radius ratio of 0.75 show that the flow pattern in this regime consists ofthree broad jets with speeds comparable with Jupiter’s wind speeds—an eastward equatorialjet and broad westward jets at high northern and southern latitudes. Similar simulations withan inner-to-outer radius ratio of 0.35 show that substantial jets do not form, however.

At Rayleigh numbers greater than 10–20 times critical, convection fills the entire shell,including the regions inside the tangent cylinder (figure 16). At constant Ekman number,further increases in the Rayleigh number cause a decrease in the ratio of zonal to non-zonal flow(Christensen 2001). The zonal-flow speed matches that on Jupiter for Rayleigh numbers of ∼10to ∼50 times critical. Simulations by Christensen (2001) show that at these higher Rayleighnumbers (>10 critical Ra), jets seem able to form over a wider range of shell thicknessesthan they can at lower Rayleigh numbers (<10 critical Ra). Ekman numbers of 3 × 10−4

lead to three jets (westward high-latitude jets in each hemisphere and an eastward jet at theequator) for inner-to-outer radius ratios of either 0.35 or 0.6. The flow also contains threejets for E = 3 × 10−5 and an inner-to-outer radius ratio of 0.35. For E = 3 × 10−5 and aninner-to-outer radius ratio of 0.6, however, seven jets form, a strong equatorial jet and threehigh-latitude jets in each hemisphere (figure 17).

A major defect of these simulations, from the perspective of explaining Jupiter, is that theyall have jets that are too few and too broad. In these simulations, the flow outside the tangentcylinder tends to contain the eastward equatorial jet and half of the adjacent higher-latitudewestward jets (so that the tangent cylinder intersects the surface at the peaks of the westwardjets closest to the equator). Figure 17 illustrates this phenomenon. When Busse (1976) firstproposed that Jupiter’s jets result from deep convection, he envisioned that the fluid outsidethe tangent cylinder would develop multiple nested differentially rotating cylinders (i.e. thatmany jets would form outside the tangent cylinder). However, no numerical simulation hasever produced such a phenomenon (Christensen 2002). Most of the published simulationsuse inner-to-outer radius ratios of 0.3–0.75, which forces the tangent cylinder to intersectthe surface at latitudes from 74˚ to 43˚ and leads to an equatorial jet that is much too wide.


Figure 17. Zonally averaged zonal wind (thick solid line) at the surface from the same simulationshown in figure 16. Also shown is the root-mean-square velocity of the non-zonal winds (thickdashed line). The horizontal thin dashed lines show the latitudes of the tangent cylinder.

Motivated by the apparent control of the shell thickness on the jet width, Aurnou and Heimpel(2004b) and Heimpel and Aurnou (2004) performed numerical simulations in a thin shell, withan inner-to-outer radius ratio of ∼0.9, which implies that the tangent cylinder intersects thesurface at 27˚ latitude. Their simulations developed a strong, Jupiter-like equatorial jet withthe correct width and numerous mid-latitude jets that resemble Jupiter more closely than anyearlier simulation.

For Jupiter, the motivation for considering convection in a spherical shell rather than afull sphere results from the proposal that the transition region between molecular and metallichydrogen acts as a barrier that inhibits convection across this interface (Stevenson and Salpeter1977). The depth of this transition remains poorly known but probably lies between 0.7 and 0.9Jupiter radii, at pressures of ∼1–3 Mbar (Guillot et al 2004). Convection would occur in bothregions, as is necessary to transport the observed heat flux, but the strong zonal flows mightonly be expected in the molecular region, because the Lorentz force in the metallic regionacts to brake any zonal flows there (Kirk and Stevenson 1987, Grote et al 2000, Busse 2002).Therefore, the thin shells adopted by Aurnou and Heimpel (2004b) and Heimpel and Aurnou(2004) may indeed be appropriate for Jupiter if the molecular–metallic transition occurs at ∼0.9Jupiter radii. Recent laboratory work indicates that the transition from molecular to metallichydrogen may not be sharp in density or conductivity (Weir et al 1996). This possibility mayhave dynamical implications and should be considered in future models of the deep atmosphere.

Standing back from the details, do these studies support the idea that Jupiter’s jets resultfrom deep convection? Although cutting-edge simulations have achieved jets with speedssimilar to Jupiter’s, most published convection simulations produce jets that are too few andtoo broad. Only convection in an extremely thin shell (inner-to-outer radius ratio >0.9) appearscapable of producing Jupiter-like jet patterns with ∼20 jets. Despite their mixed success inreproducing the multiple, mid-latitude jets, however, most deep-convection numerical studiesrobustly produce eastward flow at the equator, as required to explain Jupiter. The easewith which these models produce eastward equatorial flow over a wide range of Rayleighnumbers, Ekman numbers, and inner-to-outer radius ratios is an attractive feature of thesemodels, and lends credence to the possibility that Jupiter and Saturn’s equatorial jets actuallyresult from such deep convection. This major success starkly contrasts with the failure of


one-layer, shallow-water models to produce eastward equatorial flow (nevertheless, multi-layer investigations of turbulence in the stably stratified atmosphere can produce eastwardequatorial flow under some circumstances; more work is needed to determine the relevance ofthese circumstances to Jupiter). In contrast to the deep-convection models, however, the forcedshallow-atmosphere models are remarkably successful in producing multiple, mid-latitude jetsregardless of the exact parameter settings and formulation of the models.

One possible synthesis that might explain Jupiter is a hybrid scenario where the eastwardequatorial jet results from deep-convective Reynolds stresses (as described in this section) andthe numerous mid-latitude jets result from an inverse cascade of turbulence injected into thecloud layer (as described in sections 3.4.1–3.4.3). This scenario combines the most robust(least model-dependent) aspects of each model: deep convection’s ability to produce eastwardequatorial flow and shallow-atmosphere turbulence’s ability to produce multiple, mid-latitudejets regardless of model details. An advantage of this hybrid scenario is that the convectionregion need not be thin. In this scenario, it is unclear whether the mid-latitude jets wouldpenetrate through the molecular region. Although the shallow-atmosphere jet forcing couldpotentially drive deep mid-latitude jets that extend as Taylor columns throughout the entiremolecular envelope, it is also possible that the deep-convective turbulence would disrupt anysuch jets in the interior. If so, the mid-latitude jets might penetrate only partway through themolecular envelope.

Alternatively, perhaps the molecular region and its convection are indeed confined to athin shell, as proposed by Aurnou and Heimpel (2004b) and Heimpel and Aurnou (2004),and the jets predominantly result from convection in this layer. It is worth pointing out,however, that the distinction between the deep-convection and shallow-turbulence hypothesesblurs for such thin shells; although Reynolds stresses acting on convecting fluid columnsdrive the eastward equatorial jet, the higher-latitude jets in these simulations may result froman inverse cascade modified by the β effect, exactly as proposed by the shallow-atmosphereturbulence investigators (Williams 1978), with the distinction that the turbulence that drivesthe inverse cascade results directly from the dry convection throughout the depth of the thinshell rather than moist convection, baroclinic instabilities, or other processes confined to thecloud layer.

Nevertheless, the deep-convection simulations published to date are not yet in a Jovianregime, so their resulting jet profiles must be viewed as extremely tentative. First, the Ekmannumber of Jupiter’s molecular envelope is ∼10−15 (Guillot et al 2004), whereas the simulationsto date all use E > 10−5 (this is a computational limitation, because small Ekman numberslead to fine-scale turbulent structure that requires high numerical resolution to resolve). It is anopen issue whether the convective behaviour, in particular the jet widths and speeds, remainssimilar over this ten-order-of-magnitude range of Ekman number. In fact, linear theory (e.g.Roberts (1968), Busse (1970)) predicts the widths of convective columns in rotating convectionscales as E1/3, which argues that the Taylor columns become infinitely narrow as E tends tozero. This scaling predicts that the Taylor columns when E ≈ 10−15 would be several ordersof magnitude narrower than Jupiter’s actual jets. If so, deep convection might be unable toproduce Jupiter-like jets. However, it remains unclear whether the predicted scaling remainsrelevant in the fully turbulent, nonlinear convection regime above ∼10 times critical Ra. Thesimulations of Aurnou and Olson (2001), Christensen (2001, 2002) and Aurnou and Heimpel(2004a) show a tendency for finer convective structure inside the tangent cylinder as the Ekmannumber is decreased, in qualitative agreement with the linear scaling. On the other hand, noneof these simulations produce numerous, nested alternating jets outside the tangent cylinder(Christensen 2002) as the Ekman number is reduced from 10−3 to 10−5. The range of Ekmannumber accessible to current numerical simulations is insufficient to know with confidence


how the convection behaves when E approaches 10−15. Unfortunately, reaching substantiallylower Ekman number must await major advances in computational methods and hardware.

Second, all the investigations described here adopted the Boussinesq approximation, whichassumes an incompressible continuity equation (23). Ascending or descending plumes thattraverse the layer retain essentially constant density. However, on Jupiter the density increasesby a factor of ∼1000 between the cloud layer and the bottom of the molecular envelope dueto the compressibility of fluid hydrogen. This density variation implies that ascending ordescending fluid parcels expand or contract, respectively, by a factor of ∼1000 as they traversethe molecular envelope. Such density variations tend to discourage coherent plumes fromtraversing the entire layer, which would drastically alter the convective planform and the natureof any Reynolds stresses that exist (at moderate Rayleigh number, the Boussinesq simulationssuggest that the strongest Reynolds stresses—hence acceleration of the jets—occur whenplumes retain coherency throughout the convecting layer). No rotating three-dimensional,spherical-shell calculations have yet been published that include compressibility, and it remainsunclear whether the jet-pumping dynamics elucidated in the Boussinesq models are relevantfor the compressible system. Two-dimensional simulations by Evonuk and Glatzmaier (2004)including compressibility have already confirmed, however, that convection indeed behavesdifferently in the Boussinesq case compared with compressible cases with top-to-bottomdensity contrasts of ∼100-fold. They solved the anelastic equations, which are equivalentto equations (21)–(23) except that the continuity equation (23) is replaced with the alternatecontinuity equation

∇ · (ρ0v) = 0, (24)

where ρ0(r) is the background density profile, which is a specified function of the radialcoordinate, r . Full three-dimensional anelastic simulations of rotating convection in sphericalshells are now computationally possible, and current progress suggests that such anelasticinvestigations will be carried out within the next few years.

A believable model for convection in Jupiter’s interior must eventually adopt morerealistic boundary conditions than the impermeable inner and outer spherical walls usedso far. Columnar jets that develop in the simulations by Christensen (2001), Aurnou andOlson (2001) and others produce geostrophic pressure variations that are supported by theimpermeable boundaries, but such boundaries do not exist on Jupiter. A long-term goal isto study the entire interior, both molecular and metallic layers, simultaneously. Finally, it isworth mentioning that none of the deep-convection simulations have produced the multitudeof compact vortices that exist at Jupiter’s cloud layer. Shallow models, however, can easilyreproduce such vortices. This suggests that, even if the jet-forcing is deep, shallow models arestill needed to understand the dynamics of these vortices (section 4).

3.5. Observational constraints on modes of energy transfer

The ubiquitous occurrences of vortex mergers (Mac Low and Ingersoll 1986, Li et al 2004),as well as the approximate correspondence of the latitudinal jet width to the Rhines scale(Cho and Polvani 1996a), provide tentative observational evidence that an inverse cascade isactually occurring in Jupiter’s cloud layer. Vortices with diameters smaller than ∼2000 kmtend to be circular, while larger vortices are elongated in the east–west direction (Mac Low andIngersoll 1986), which confirms the influence of β at scales exceeding ∼2000 km and supportsthe inference that β can modify the inverse cascade to produce jets. Nevertheless, it is notobvious that the inverse cascade (mergers) occurring among the individual vortices plays a rolein pumping the jets. Classical two-dimensional turbulence theory predicts that vortex mergers


produce incrementally larger vortices that eventually reach the jet scale. However, Jupiter’slargest vortices, such as the Great Red Spot and White Ovals, have never been observed toelongate into jet-like structures or merge with existing jets. In fact, the observations followthe opposite trend; through recent decades, the White Ovals, the CR between the White Ovals,and the Great Red Spot have been shrinking with time in the east–west direction (Rogers 1995,Simon-Miller et al 2002), directly contrary to a simple manifestation of the inverse cascade.Indeed, these large vortices may result from the jets via barotropic or baroclinic instabilities,hence depleting the jets of energy rather than pumping them up (Williams and Wilson 1988,Dowling and Ingersoll 1989, Williams 1996).

By tracking small-scale clouds in Voyager and Cassini images, several authors haveattempted to determine observationally whether turbulence at cloud level pumps the zonal jets(Beebe et al 1980, Ingersoll et al 1981, Salyk et al 2004). The method involves calculation ofthe correlation between small-scale fluctuations in eastward and northward velocities, u′v′,associated with individual cloud motions (here, u′ and v′ are the deviations from zonalaverage of the zonal and meridional winds). If the product of their zonal average, u′v′, ispositive (negative), then northward (southward) transport of eastward momentum occurs. Ifthe latitudinal variations in momentum transport and zonal jet velocity are correlated, it maybe taken as evidence that turbulent eddies are forcing the zonal flow. Limaye et al (1982) andSromovsky et al (1982) criticized the approach of Beebe et al (1980) and Ingersoll et al (1981)on the basis that the clouds were tracked by hand and may have been a statistically biasedsample. The Cassini results (Salyk et al 2004) benefit from automated cloud tracking on auniform grid and higher-quality images. While the latter contain some evidence for momentumtransfer, the authors remain concerned that certain cloud morphologies may cause spuriousmeasurements (Colette Salyk, personal communication).

These observationally inferred momentum transfers have been widely cited as implyingan acceleration of the jets by turbulence at cloud level (Beebe et al 1980, Ingersoll et al 1981,1990, 2004, Salyk et al 2004). However, Read (1986) suggests that caution is warranted.By expanding the dynamical variables into zonal-mean and deviation (eddy) components,A = A + A′, and zonally averaging the equations of motion, we obtain the Eulerian-meanequation for the evolution of the zonal-mean flow, u, over time in Cartesian geometry usingpressure as a vertical coordinate (Holton 1992, pp 313–17, Andrews et al 1987, p 124)

∂u

∂t= −v

∂u

∂y− ω

∂u

∂p+ f v + X − ∂(u′v′)

∂y− ∂(u′ω′)

∂p, (25)

where y, p and t are northward distance, pressure and time, v and ω are the northwardand vertical velocities, f is the Coriolis parameter and the overbar denotes a zonal average.On the right side, the first and second terms represent changes in the zonal-mean flow byadvection of momentum in latitude and height, respectively. The third term represents Coriolisaccelerations by the mean meridional flow and the fourth term represents accelerations byfriction. The final two terms represent mean-flow acceleration associated with the horizontaland vertical eddy motions. Clearly, one must know all the terms in the equation, not just thevalue of −∂(u′v′)/∂y, to fully evaluate the kinematics of jet pumping. Although the valuesof −∂(u′v′)/∂y inferred by Beebe et al (1980), Ingersoll et al (1981) and Salyk et al (2004)are suggestive, they do not automatically imply that jet acceleration occurs at cloud level.In particular, Charney and Drazin (1961), Andrews and McIntyre (1976, 1978) and othershave shown that frictionless, adiabatic steady waves/eddies do not accelerate the zonal-meanflow (a result called the ‘non-acceleration theorem’). However, these same waves may inducenonzero values of −∂(u′v′)/∂y. The reason for the non-acceleration is that such waves alsoinduce a mean-meridional flow, v, that exactly cancels the value of −∂(u′v′)/∂y (see Plumb


(1983), White (1986) and Holton (1992) pp 316–22 for analytical examples and discussion ofthis phenomenon). Strong nonlinearities (e.g. wave breaking or absorption), radiative effectsand friction violate the non-acceleration theorem and allow the waves to affect the zonal-mean flow, but even in this case the waves induce a mean-meridional wind whose Coriolisacceleration largely cancels the value of −∂(u′v′)/∂y. The key point is that the effect of theeddies resides not only in the u′v′ and u′ω′ terms but also in the f v term; one must know allthese terms for a full assessment of the eddies on the mean flow. Diagnostic analysis of Earth’sgeneral circulation shows that the magnitude and spatial pattern of the actual eddy-induced,zonal-mean-flow acceleration differs substantially from the magnitude and spatial pattern of−∂(u′v′)/∂y. The near-constancy of Jupiter’s jets, the long radiative time constants, and thepresumably weak friction all suggest that Jupiter could be close to non-acceleration conditions,in which case the eddy-induced, zonal-mean-flow acceleration might differ substantially from−∂(u′v′)/∂y on Jupiter, too (Read 1986).

In summary, then, tentative evidence supports the idea that an inverse cascade amongvortices exists on Jupiter. Small-scale turbulence may also drive the jets via a shear-straining mechanism at cloud level, but the current measurements are insufficient for a fullassessment of this hypothesis. If it turns out that −∂(u′v′)/∂y provides an approximately validrepresentation of the eddy-induced jet pumping, then the Voyager and Cassini observationswould constitute extraordinary evidence for an inverse cascade that drives the jets at cloudlevel. The measurements would further suggest that jet pumping is inherently non-local inwavenumber space: turbulence at the smallest scales transfers energy directly into the jetsvia a shear-straining mechanism. Observations and models suggest that the eddy-induced jetpumping (if any) and the inverse cascade among the vortices could be distinct processes. Theobserved history of the White Ovals, as well as a variety of numerical investigations, favourthe formation of Jupiter’s largest vortices from a jet instability that robs energy from the jets(see sections 4.2–4.4). However, we cannot rule out the possibility that the Great Red Spotor other large vortices instead resulted directly from an inverse cascade (i.e. gradual growthby merger of small vortices). This broad picture coincides with the numerical simulations ofHuang and Robinson (1998), where the jets were accelerated directly by small-scale vorticesthat tilted into the shear, without any intermediate set of vortex mergers. In their simulations,an inverse cascade also produced large vortices that interacted with the jets on short timescalesbut played minimal role in pumping the jets.

4. Vortices

In addition to the zonal jets, Jupiter’s atmosphere contains a wealth of circulating featuresranging from compact vortices to filamentary structures that extend many tens of degrees inlongitude (figures 1 and 2). Up to several hundred vortices are present among Jupiter’s zonalwinds at any instant in time. Cyclones rotate in the same sense as the planet (i.e. counter-clockwise in the northern hemisphere, clockwise in the southern hemisphere) and anticyclonesrotate the other way. Anticyclonic spots often appear bright and oval-shaped, although somesmall anticyclones are darker than their surroundings. Small cyclones appear as dark, irregularovals. Larger CR can extend many tens of degrees in longitude but remain confined in latitudeby Jupiter’s zonal jet structure. Observers have described CRs as chaotic, filamentary, folded,multi-lobed or ribbon-like, but they display an overall cyclonic sense of shear. Figure 18 showsthree anticyclonic ovals, an irregularly shaped cyclonic vortex and a small, multi-lobed CR.How all these features appear on images is determined by a complex interplay between theirhorizontal circulation, vertical motions and the concomitant advection, creation and destructionof cloud material. Small vertical motions strongly influence aerosol opacity, but the dynamics


Figure 18. Galileo false-colour mosaic of a region containing White Ovals DE (left) and BC(right), an intervening teardrop-shaped cyclonic vortex, and a smaller anticyclonic oval below BC.The structure in the lower left is a multi-lobed cyclonic region. In this representation, reddish cloudsare deep, greenish clouds are within Jupiter’s main cloud deck, and white clouds are high and thick.

are probably overwhelmingly horizontal (typical diameters of 103–104 km can be comparedwith the pressure scale height of ∼20 km).

4.1. Dynamics, stability, behaviour and structure

It is widely agreed that Jupiter’s vortices are free vortices, with closed streamlines of flowaround their centres and little exchange of fluid with the zonal winds. Voyager imagingrevealed the tendency for vortices of the same sign and present within the same shear zone tomerge when they approach one another. Merging is the expected behaviour of free vortices(called vortices hereafter), as opposed to nonlinear wave structures (solitons) that are morelikely to pass through one another (though with some interaction). Vortices within theoreticalmodels, numerical simulations, and laboratory experiments exhibit a wealth of behaviours.An enormous fluid-dynamics literature exists on this subject; readers are referred to Flierl(1987), McWilliams (1991), Marcus (1993), Hopfinger and van Heijst (1993) and Carton(2001) for entries into the literature. Even in simplified one-layer models (e.g. the purely two-dimensional, quasi-geostrophic, or shallow-water systems), vortices can merge, fragment, ejectfilaments, eject small vortices, orbit each other, oscillate in shape or location, radiate wavesand interact with jets in a variety of ways. Vortices in three-dimensional models exhibit allthe two-dimensional behaviours as well as modes of fragmentation, oscillation, merging, waveinteractions and instabilities (e.g. baroclinic instabilities) that are inherently three-dimensional.

In planetary atmospheres, where β is nonzero, isolated vortices tend to lose energy byradiating Rossby waves. Because Rossby waves are dispersive (equation (12)), a weak vortexcan be fully disrupted by dispersion. For typical Jovian parameters (β ≈ 4×10−12 m−1 s−1 andvortices a few thousand kilometres across), the time scale to completely disperse a vortex is afew days. Nonlinear advection can resist the dispersion, allowing vortices to remain stable overlonger periods, although isolated vortices will still slowly decay over time. Vortices can exhibitany of several possible nonlinear balances that help them survive against dispersion, dependingon the vortex size and other parameters (e.g. Williams and Yamagata (1984), Williams (1985)).Interestingly, numerical simulations show that the existence of zonal jets can slow or halt the


Rossby-wave-induced gradual vortex decay (Achterberg and Ingersoll 1994). This result helpsexplain the longevity of the many Jovian vortices that survive for years or longer. Nevertheless,the longest-lived vortices may require an energy source to help maintain them against dispersionand friction; possibilities include latent-heat release, sloping convection, and interactions withother vortices. Near the equator, where β is greatest, the dispersion is so strong that vorticesgenerally cannot survive. Numerical simulations by Williams and Wilson (1988), Showmanand Dowling (2000) and others show that, under Jovian conditions, coherent structures at theequator tend to take the form of nonlinear waves (e.g. solitons) rather than vortices. Consistentwith these ideas, observations show that no vortices exist within ∼10˚ latitude of Jupiter’sequator.

In theoretical studies, isolated vortices tend to migrate westward at a substantial fractionof the Rossby-wave phase speed, and in the absence of strong background winds they alsodrift equatorward (anticyclones) or poleward (cyclones) because of secondary circulations inthe near-vortex environment that are induced by the vortex itself (e.g. Achterberg and Ingersoll(1994), LeBeau and Dowling (1998), Yamazaki et al (2004)). Equatorward migration of largeanticyclones has been observed on Neptune (Smith et al 1989), but north–south migrationdoes not seem to occur for any vortices on Jupiter. Numerical studies demonstrate that strong,Jupiter-like zonal jets can prevent the equatorward migration by modifying the vortex-inducedenvironmental circulation that would cause migration in the absence of the jets (Achterbergand Ingersoll 1994, Williams 1996). Furthermore, three-dimensional simulations by Williams(1996) show that the migration rate and long-term stability of shallow vortices overlying aquiescent deep interior depends strongly on the vortex thickness. Thinner vortices migrate inlatitude more slowly, and are halted more easily by jets, than thick vortices.

The thickness of Jupiter’s vortices is unknown, but dynamical arguments suggest that theyare thin compared with a planetary radius. The rapid evolution and short dynamical lifetimesof most Jovian vortices argue against them being Taylor columns that penetrate through themolecular envelope. Furthermore, the non-zonal airflow that occurs in these vortices wouldviolate the Taylor–Proudman theorem, especially for the largest vortices such as the Great RedSpot, if they were Taylor columns. Upper tropospheric and lower stratospheric temperaturemeasurements imply that most large vortices extend only 2–4 scale heights (40–80 km) abovethe clouds (Conrath et al 1981). A recent series of multi-layer, quasi-geostrophic numericalsimulations by Dritschel and colleagues (Dritschel and de la Torre Juarez 1996, Dritschel et al1999, Reinaud et al 2003) show that three-dimensional, geostrophically balanced vorticestend to be baroclinically unstable if their thickness exceeds their width by a factor greaterthan ∼f/N , where f is the Coriolis parameter and N is the Brunt–Vaisala frequency. Inthe sub-cloud troposphere of Jupiter, the latent heating associated with condensation of watermight produce a characteristic Brunt–Vaisala frequency ∼0.002 s−1. If so, vortices such asthe Great Red Spot and White Ovals probably extend less than ∼500 km below the clouds.

4.2. Cyclones and anticyclones

Li et al (2004) recorded the evolution, on Cassini’s 70-day approach movie, of over 500spots (excluding CR) with diameters exceeding 700 km. Of these, 306 were bright and 211were dark (note that the former includes vortices and 31 convective storms, which we discussseparately in section 6). They were able to determine the vorticity of the largest 100 spotsand found that all were anticyclonic. Over 70 days, they observed 393 appearances and 363disappearances. These numbers are statistically similar, suggesting that the number of spotsremains roughly constant with time. Some vortices were detected as they developed contrastwithin an otherwise featureless region (221), while others emerged from filamentary, turbulent


regions (141), usually a CR. The latter clearly represent real vortex-formation events, butthe former might represent brightening of pre-existing vortices via formation of an overlyingcloud layer rather than vortex formation. Of 126 vortex interactions observed by Li et al , 119resulted in mergers while 7 resulted in spots passing around each other. The remaining vorticesobserved to disappear were destroyed when they encountered a CR (90) or simply faded awaywithout any apparent interaction (147). The latter may not represent the destruction of vortices,but simply changes in their cloud properties. All the spots that survived throughout the 70days were large (diameter > 2000 km), although many large spots also disappeared. Thestatistical lifetime of a spot was 16.8 days (calculated without including convective storms).Using HST images acquired over a six-year period, Morales-Juberıas (2002a) found that largeanticyclones (north–south extents of 1000–6000 km) had lifetimes of 1–3 years.

Numerical turbulence studies show that, among other processes, vortices can form bybarotropic or baroclinic instability of unstable large-scale structures (e.g. larger vortices), roll-up of pre-existing filaments into coherent spots, and assembly from small-scale turbulence.These mechanisms are broadly consistent with the second observed mode of vortex appearancedescribed above. Numerical simulations that represent thunderstorms as localized sourcesof mass injected into the cloud layer show that vortices, primarily anticyclones, naturallyform during the geostrophic-adjustment process that follows the convection events (Showman2004). This mode of vortex formation has been speculated (e.g. Ingersoll et al (2000)) butnot clearly observed on Jupiter. However, Cassini may have witnessed this phenomenon onSaturn (Porco et al 2005).

Voyager, HST, Galileo and Cassini observations show that all long-lived Jovian vorticeshave relative vorticity with the same sign as that of the jets in which they are embedded.Consistent with this observation, numerical simulations show that vortices can exist stably injets when this condition is satisfied. On the other hand, vortices embedded in jets with theopposite sense of vorticity are quickly sheared apart unless their relative vorticity has a magni-tude at least ∼6 times greater than the jet (Moore and Saffman 1971, Kida 1981, Marcus 1990,Meacham et al 1990, Achterberg and Ingersoll 1994, Marcus et al 2000). Most Jovian vorticeshave relative vorticities only a few times greater than that of the jets in which they reside, whichis consistent with the observed lack of stable cyclones (anticyclones) in anticyclonic (cyclonic)shear zones on Jupiter. Recently, Morales-Juberıas et al (2002a) found that most vortices driftwestward relative to the ambient winds, consistent with numerical simulations by Achterbergand Ingersoll (1994), Williams and Wilson (1988) and others. Interestingly, most large vorticesreside at latitudes where the zonal velocity is near zero; the significance of this result remainsunclear. Vortex size is unrelated to either drift velocity or the ambient wind velocity.

Several laboratory and numerical investigations are consistent with the idea that Jupiter’slargest vortices, the Great Red Spot and White Ovals, formed by barotropic or baroclinicinstability of the zonal jets (Read and Hide 1983, 1984, Read 1986, Sommeria et al 1988,1989, Williams and Wilson 1988, Dowling and Ingersoll 1989, Williams 1996, 2002). In thenumerical simulations, jet instability produces numerous (∼10) mid-sized vortices at a singlelatitude, which undergo mergers that lead to fewer, larger vortices. The formation of the WhiteOvals in 1938–40, which occurred by pinching of a zonal jet and the subsequent creation ofseveral zonally elongated vortices that evolved into the White Ovals (Rogers 1995), is broadlyconsistent with this scenario (Marcus 2004). Li et al (2004) tested the hypothesis that spots aregenerated where the zonal winds are unstable with respect to the barotropic stability criterion.Two-dimensional, barotropic flow is unstable where the curvature of the wind profile withlatitude exceeds the local value of the planetary vorticity gradient, β. Since β is positiveat all latitudes, only westward jets are susceptible to violating this criterion. They find thatbetween the Voyager, HST and Cassini eras, some westward jets developed more curvature


than β, while some decreased in curvature below β. In the northern hemisphere, spots areconcentrated at the latitudes of the westward jets, giving some support to the hypothesis. Datafrom the southern hemisphere are less conclusive.

Voyager and Cassini images show that approximately 90% of Jupiter’s compact vorticesare anticyclonic (Mac Low and Ingersoll 1986, Li et al 2004), and that nearly all anticyclonicstructures are circular or oval in shape. Although a few compact, relatively coherent cyclonicfeatures such as the ‘brown barges’ have been observed (Hatzes et al 1981, Morales-Juberıaset al 2002b), many Jovian cyclones are zonally elongated, filamentary structures, some of whichhave a multi-lobed morphology. This cyclone–anticyclone asymmetry cannot be capturedusing the two-dimensional non-divergent or quasi-geostrophic equations. Remarkably,however, a predominance of anticyclones relative to cyclones occurs naturally in the shallow-water equations, which allow the full column stretching that is precluded in the two-dimensionaland quasi-geostrophic systems. Shallow-water turbulence simulations that are initializedwith equal numbers of cyclones and anticyclones show that, as the turbulence evolves, theanticyclones become more compact and strong, while the cyclones remain disorganized (Araiand Yamagata 1994, Polvani et al 1994, Cho and Polvani 1996b). Polvani et al (1994)speculate that this asymmetry occurs because, in a geostrophically balanced fluid, anticyclonescorrespond to thick regions while cyclones correspond to thin regions. As a result, thedeformation radius (see section 3.4.2) near anticyclones exceeds that near cyclones. Thedeformation radius acts as a length scale over which vortices interact. According to thisidea, the small deformation radius in cyclones inhibits the vortices from ‘feeling’ the distantparts of themselves, leading to multiform or disorganized flow patterns. In anticyclones,the larger deformation radius increases the interaction range, promoting a more coherent,organized morphology. Nevertheless, little work has been performed to test this idea, andit remains unclear whether the effect has sufficient magnitude to account for the strength ofthe cyclone–anticyclone asymmetry observed in shallow-water simulations. Marcus (2004)further cautions that the occurrence of chaotic-appearing clouds in cyclones need not implythat the cyclones themselves are chaotic, and he presents simulations showing how compact,oval-shaped cyclones can generate filamentary, disorganized cloud patterns (while equal-shaped anticyclones tend to form more compact, organized cloud patterns). Nevertheless,the overwhelming dominance of anticyclones relative to cyclones observed on Jupiter arguesthat the cyclone–anticyclone asymmetry on Jupiter is real. However, it is worth rememberingthat the extent of the asymmetry could depend on altitude. Laboratory experiments by Read(1986), for example, produce flows that have strong anticyclones and weak cyclones at the top,but at deeper levels, the cyclones dominate over the anticyclones. The height dependence ofthe asymmetry on Jupiter is unknown.

4.3. The Great Red Spot

The Great Red Spot (GRS) is Jupiter’s most prominent anticyclone due to its large size, distinctcolour and longevity (figures 1 and 2). Astronomers have noted the presence of a giant vortexin Jupiter’s southern hemisphere for hundreds of years, and the present GRS can be tracedback at least 100 years (Rogers 1995). Its shape is roughly elliptical, with dimensions of22 000 × 11 000 km in 1996. Zonal jets to the north (and to a lesser extent, the south) aredeflected by the GRS, resulting in a broad, turbulent wake region to the west. Unidentifiedtrace chemicals, dredged from deeper depths and/or dynamically confined by the vortex, arethought to cause the red colouration. Interestingly, it is the only anticyclonic vortex to displaya quiescent centre. The central region of the GRS, to about half its radial dimension, ischaracterized by low-velocity (5–10 m s−1) turbulent eddy motions at various length scales.


Spatially averaged, the vorticity there either approaches zero or is slightly cyclonic. The bulkof the GRS velocity and vorticity is concentrated in a collar surrounding the central region(Sada et al 1996, Vasavada et al 1998). In other large Jovian anticyclones, tangential velocityincreases more linearly with radial distance (vorticity remains roughly constant).

To the casual observer, the GRS seems unchanged over the two decades it has been closelymonitored by spacecraft and HST (e.g. figure 2). However, these data have revealed significantchanges in its dynamics. The GRS is shrinking in the longitudinal direction and the collar windspeeds are increasing; the change is obvious when recent spacecraft images are compared withcentury-old telescopic images. Simon-Miller et al (2002) demonstrated that the longitudinalextent of the GRS has decreased from ∼35˚ in the late 1800s to ∼21˚ during the Voyager era,to ∼18˚ during the Galileo era. They also found that the maximum tangential velocity (alongthe GRS meridian) increased from ∼120–140 m s−1 during the Voyager era, to ∼150 m s−1

measured on Galileo images from 1996 and to 190 m s−1 measured on Galileo images from2000. The shrinkage of the GRS over time is an intriguing puzzle. It could result from anincrease in the vortex nonlinearity over time, which would then more strongly balance Rossby-wave dispersion and allow compaction of the GRS. Alternatively, it could result from erosionof the vortex’s outer layers by external strains caused by the zonal jets, the cyclonic turbulentwake to the west of the GRS, or close encounters between the GRS and small vortices (someof which are destroyed during such interactions) (Mariotti et al 1994).

The long-term energetics and evolution of the GRS are poorly understood. The GRSperiodically ingests smaller (∼2000 km) anticyclones encountering it from the east and shedsvorticity to the west. The details of the latter process are convoluted by the turbulent wake.But vortices that emerge at the western margin of the turbulent wake eventually drift westwardaround the planet and merge with the GRS from the east. Several authors have suggestedthat, by swallowing these small vortices, the GRS gains energy that helps maintain it againstdispersion and friction (Ingersoll 1990, Dowling 1995a, Ingersoll et al 2004). During theVoyager 2 encounter, a large cyclonic feature just southeast of the GRS deflected the incomingvortices, reducing the ingestion rate by a factor of 2.5 relative to Voyager 1. When thiscircumstance is viewed in light of the decreased velocity of the northern collar and the slightlysmaller longitudinal aspect of the GRS measured by Voyager 2, it suggests that the reducedingestion rate of anticyclones had driven the GRS to a new steady state regime (Sada et al1996). However, it is not a foregone conclusion that mergers between the GRS and smallvortices would strengthen the GRS. If the smaller vortex has a lower magnitude of potentialvorticity than the GRS, then merger would cause a weakening, not a strengthening, of theGRS (i.e. it would decrease the mean potential vorticity inside the GRS). In a similar vein,Read (1992) describes a numerical simulation where the merger of two like-signed vorticescauses entrainment into the main vortex of a filament with vorticity of the opposite sign; thenet effect is a weakening and increase in area of the larger vortex. Naively, one might expectmergers between the GRS and small vortices to increase the size of the GRS, so the GRSshrinkage then seems even more puzzling. At a minimum, the GRS shrinkage suggests that itis not in a state of long-term equilibrium, so perhaps no external energy source is required tomaintain it.

Much of the attempt to understand the GRS has taken a forward-modelling approach;laboratory experiments or numerical simulations are performed for a range of conditions,and if simulations can match the observations only for specific parameter values, then thosevalues are inferred to apply to the GRS. Laboratory experiments by Read and Hide (1983,1984) and Read (1986) produced large vortices whose vorticity was confined to a ring at theperiphery of the vortex, as is the case with the Great Red Spot; these authors showed thatthis result has implications for the deformation radius. Marcus (1988) performed one-layer,


quasi-geostrophic simulations in cylindrical geometry. His study also reproduced the quiescentinterior and active collar of the GRS, but these features could only be reproduced if theradius of deformation in the cloud layer was ∼1300 km (also see Marcus and Lee 1994).This provided one of the first estimates of this parameter with an uncertainty better than afactor of 3.

Cho et al (2001) continued this line of inquiry using the three-dimensional, quasi-geostrophic equations, which they solved using a contour-advection technique that allowsextremely high horizontal resolution. They integrated the equations near the cloud level overa domain 1.25 scale heights deep. The Brunt–Vaisala frequency ranged from zero at thebottom to a maximum value, N∗, and a corresponding deformation radius L∗

d = N∗H/f , atthe top (here H is scale height and f is Coriolis parameter). The initial condition containedthe GRS and surrounding jets, with a barotropic structure, and the subsequent evolution wascompared with observations. Cho et al found that the simulations could only match the GRS ifL∗

d ≈ 2000 km (independent of the exact vertical profile of the stratification). Values a factorof two smaller or larger produced simulated behaviour that deviated wildly from the actualbehaviour of the GRS. The successful simulations captured the confinement of the vorticityto the GRS collar, the formation of filaments and deflection of the jets outside the GRS, andthe existence of numerous small ∼1000 km wide coherent structures inside the quiescent GRScore that counter-rotate with respect to the motion in the collar (as noted on Galileo imagesby Vasavada et al (1998)). Broadly speaking, the inferences by Cho et al are consistent withthose of Marcus (1988). Marcus’ one-layer model effectively provides a vertically averagedestimate of Ld. In the simulations of Cho et al , L∗

d substantially exceeds the vertical averageof Ld throughout the domain.

Dowling and Ingersoll (1988) performed measurements of the absolute vorticity aroundstreamlines in the GRS active collar. They found that the absolute vorticity (ζ + f ) variessubstantially around streamlines. From conservation of the potential vorticity, q = (ζ +f )/h,where h is the column thickness, the measurement implies that the thickness of fluid columnsvaries by up to a factor of ∼2 as the columns circle the GRS. Dowling and Ingersoll (1988,1989) presented arguments showing that these variations result from nonzero deep zonal windunderlying the GRS. Given assumptions about the deformation radius, this deep wind can thenbe determined. Regardless of the precise deep wind profile, the inference that columns stretchand contract by a factor of two has major implications. Quasi-geostrophic theory explicitlyassumes that columns undergo only small fractional stretching. Dowling and Ingersoll’s (1988,1989) measurements and simulations imply that non-quasi-geostrophic dynamics are crucialfor explaining the detailed character of the GRS. Nevertheless, the fact that Ertel’s potentialvorticity is not conserved in Jupiter’s upper troposphere (Gierasch et al 2004) suggests cautionin the detailed use of vorticity dynamics as a tool to derive atmospheric properties fromobservations of the GRS or other features.

4.4. Mergers of the White Ovals

An unprecedented opportunity to witness the interaction of Jovian vortices occurred whentwo of Jupiter’s White Ovals (WO) merged in 1998 and the remaining two merged in 2000(figure 19, Sanchez-Lavega et al 1999, 2001, Youssef and Marcus 2003). The WO were threeanticyclonic vortices (called BC, DE and FA) at 33˚S latitude observed to form in 1939–1940.They were the largest anticyclones after the Great Red Spot, with major axes of ∼10 000 km(BC was 9800×7400 km in 1998). Voyager and Galileo imaging revealed that their tangentialvelocities reach a maximum of 120 m s−1 (Mitchell et al 1981, Simon et al 1998). Angularvelocity and vorticity are roughly constant in their interiors, consistent with their uniform


Figure 19. Four images acquired by the HSTs Wide-Field/Planetary Camera (WFPC2) showingthe evolution of Jupiter’s White Ovals over a three-year period. White Ovals DE and BC mergedto form BE. Subsequently, FA and BE merged to form BA. Feature o1 is a cyclonic vortex thatinteracted with the White Ovals during these years. NASA image PIA02823.

appearance. No change in velocity was noted between Voyager and Galileo, though WO BCbecame noticeably more round by decreasing in longitudinal extent and slightly increasing inlatitudinal extent (Simon et al 1998).

The interaction between vortices must be studied within a larger dynamical context, asexemplified by the band containing the WO and the cyclonic band to the north. Both bandscontain structures whose meridional dimensions exceed the width of the band. Cyclonicstructures (vortices and filamentary regions) intrude and deflect zonal winds southward intothe anticyclonic band. As a consequence, cyclonic features (sometimes many tens of degreesin longitude) significantly influence interactions between WO. Throughout much of their∼60-year lifetime, the WO drifted relative to one another in longitude. As two WO approachedone another, the intervening cyclonic structure decreased in size. Instead of merging, the WOstalled and reversed course, as if repelled by the close encounter. However, a new behaviourbecame evident by 1988, when the centres of DE and BC approached to within 17˚ longitudeand remained tightly spaced until their merger in 1998. This final decade was characterized bythe confinement of DE and BC, and more recently FA, within a system of up to six cyclone–anticyclone pairs, with BC at the eastern extreme (Simon et al 1998).

Although high-resolution spacecraft images of the WO were taken only before and after themergers, telescopic observers recorded the properties and dynamics of the vortices throughoutthe events. In fact, telescopic observers alerted the Galileo Imaging Team when the first mergerhad occurred. Galileo images taken just before the first merger show a cyclonic structure tightlycompressed between DE and BC (figure 18). The post-merger oval, BE, was larger in bothaxes than either DE or BC, though total area was not conserved. The detection of a cyclonicstructure after the merger, called O1, may indicate its survival through the merger, thoughSanchez-Lavega et al (1999) speculate that it may instead be a product of the merger, citingthe fact that its area and that of BE equal the sum of the areas of DE and BC. By 1999, theremaining ovals FA and BE approached each other, separated by O1. Earth-based monitoringshowed that O1 then moved southward into the anticyclonic band, possibly pushed by the GRSas it drifted by to the north. Sanchez-Lavega et al (2001) suggest that O1 was subsequentlysheared apart, apparently allowing the direct interaction of FA and BE.


Numerical models of vortex interactions (simulated as ∼2D geostrophic turbulence usingthe two-dimensional, quasi-geostrophic, or shallow-water equations) successfully reproducethe fact that like-sign Jovian vortices that approach each other within the same zonal bandtend to merge. Eventually a single, large vortex dominates the band. Throughout most oftheir history, the WO violated this empirical rule by failing to merge and drifting apart. Assuggested by the observations described above, the behaviour and stability of the WO werestrongly influenced by interactions with the cyclonic band to the north.

Youssef and Marcus (2003) explored the role of cyclone–anticyclonic interactions inpreventing mergers. Their theoretical framework considers the evolution of a system ofanticyclones and cyclones, similar to a classical Karman vortex street. Using heuristic modelsand two-dimensional numerical simulations, they conclude that in their final decade, theWO were part of a tightly packed but stable configuration of vortices confined in longitudewithin the troughs of a planetary Rossby wave. As long as neighbouring vortices differedin sign, the configuration was stable (i.e. at least partially self-correcting when perturbed).However, the destruction of a cyclone due to some external forcing, such as the passage of theGRS, permitted the direct interaction and merger of two anticyclones.

Morales-Juberıas et al (2003) used a multi-layer numerical model to understand thealtitude-dependent behaviour observed telescopically during the merger of WO BE and FA. In890 nm images sensitive to variations in an upper haze layer between 150–200 mb pressure, thevortices orbited each other with an anticyclonic sense of rotation before eventually merging.In 840 nm images sensitive to the main cloud deck in the 400–1000 mb pressure range, thevortices ‘appeared to translate into each other while exhibiting small changes to their shape’.Three-dimensional simulations such as theirs require the input of poorly constrained quantities,such as vertical profiles of temperature and winds and their variation around jets and vortices,all of which significantly affect the results. However, using available data and reasonableassumptions, they search the input parameter space for simulations that reproduce the verticaldichotomy in the merger. Their primary conclusion is that the orbiting behaviour at higheraltitude can be reproduced only if the WO winds decay with height at a rate comparable to orless than the zonal winds. Though still shallow compared with Jupiter’s radius, the vorticeswould be taller than required by the purely two-dimensional simulations that capture otheraspects of their behaviour.

5. Discrete storms

Galileo’s observations of lightning and discrete storms arguably have changed our view ofJupiter more than any other recent result. This advance partly results from the uniquecapabilities of the Galileo mission. As an orbiter, Galileo could conduct multiple surveys of theplanet’s night side at relatively high spatial resolution, resulting in many lightning detections.Furthermore, the camera’s sensitivity to near-infrared radiation provided the ability to assessvertical cloud structure at the spatial resolution of the images. Bright, expanding clouds hadbeen noted on Voyager images as candidates for moist convection. However, Galileo revealedthem to be uniquely tall structures with roots at levels where water is the most likely condensate(Banfield et al 1998, Gierasch et al 2000). Meanwhile, Galileo’s near-infrared mappingspectrometer found that relative humidity varies greatly in the lower troposphere, reachingsaturation near the location of a storm (Roos-Serote et al 2000). The combination of theiractivity (vigorous production of cloud material), vertical range from the water condensationlevel to above the 0.5 bar cloud level, association with lightning, and association with highrelative humidity make a strong case for the presence of moist convection within Jupiter’sbelts. Calculations and numerical models of Jovian convective storms suggest that they may


Figure 20. False-colour, night-side images of Jupiter’s clouds showing clusters of lightning strikes,acquired by Galileo. The images were taken 75 min apart in a broad, visible-wavelength filter. Theclouds are lit by moonlight from Jupiter’s moon, Io, and have been coloured red. The ∼1 minexposures capture many lightning strikes within each storm cluster. Latitude ranges from theequator to 50˚N.

transport the bulk of Jupiter’s internal heat flux vertically through the troposphere (Gieraschet al 2000, Hueso et al 2002).

5.1. Lightning statistics

Our understanding of Jovian lightning, though first observed on Voyager images (Smith et al1979, Cook et al 1979, Magalhaes and Borucki 1991, Borucki and Magalhaes 1992), wasgreatly advanced by Galileo and Cassini. Galileo imaged Jupiter’s night side at mediumresolution (67–134 km/pixel) from the midnight meridian on its tenth and eleventh orbitsand viewed selected regions at high spatial resolution (∼25 km/pixel) on subsequent orbits.Imaging strategies included the use of long (90–180 s) exposures to capture multiple flashes anddeliberate smearing of images in order to separate (on the image array) multiple flashes comingfrom a single location. Most Galileo lightning searches were conducted using a broadbandvisible-wavelength filter. Cassini searched Jupiter’s night side for lightning as it receded fromthe planet. Although it surveyed a much larger area than Galileo or Voyager, it did so froma farther distance and at a moderate phase angle, resulting in images with increased levelsof light scattered from the sunlit hemisphere. Cassini imaged lightning using a narrow filtercentred around the Hα line in order to mitigate the scattered light contamination. However,lightning emission at this wavelength was much weaker than expected (perhaps due to its depthas discussed below), resulting in few detections.

Most lightning strikes on Galileo and Cassini images cluster in storms that range in sizefrom ∼100 to ∼1700 km, with typical separations of ∼104 km between storms. Multiplestrikes define the shape of a storm on time-exposure images (figure 20). Galileo imagesshowed 26 unique storms during the primary mission (Little et al 1999), with an additionalthree in the extended mission (Gierasch et al 2000, Dyudina et al 2004). Cassini observedthree unique storms in nightside images (Dyudina et al 2004). The storms, many of whichwere imaged more than once, span the latitude range of 56˚S to 60˚N. When their locations are


Figure 21. False-colour, day-side view of storms northwest of Jupiter’s Great Red Spot, acquiredby Galileo, with insets showing night-side lightning detections co-located with the storm. Inthis false-colour representation, bluish clouds are deep and white clouds are high and thick. Theday-side storms are ∼1000 km across and are located near 16˚S (right) latitude.

plotted against the profile of zonal winds, a surprising pattern emerges: they fall within regionsof cyclonic shear in the zonal winds (the belts) or the locations of westward jets (figure 3).Lightning has been detected within many CR. Belts near 51˚N and 56˚S displayed a higherfrequency of lightning strikes, though the analysis for the latter latitude may not be statisticallysignificant.

Individual lightning strikes appear as bright patches spanning several pixels on high-resolution images (∼25 km/pixel), probably much larger than their actual size (figure 21). Thefavoured explanation is that the lightning occurs below optically thick cloud and haze layersand illuminates them from beneath (or within). Photons are scattered as they traverse the cloudsand hazes, making the strikes appear as resolved spots whose brightness falls off with distancefrom a central point. This property allows them to be easily distinguished from cosmic rayhits on the detector, which brighten discrete pixels or lines of pixels (though lightning strikesimaged without intervening clouds might fail to be recognized for the same reason). A lowerlimit on the energy associated with individual lightning flashes was derived from calibratedimages by assuming that the strike emitted energy isotropically from below a cloud deck andby neglecting any backward scattering within the cloud. The most energetic flash measured1.6 × 1010 J, a few times larger than the largest observed on Earth. The optical power perunit area from Galileo data is 3 × 10−7 W m−2, about the same as on Earth (4 × 10−7 W m−2)and little changed from Voyager 2 (3.2 × 10−7 W m−2). Voyager 1 and Galileo data wereextrapolated to global flash rates of 4 × 10−3 and 4.2 × 10−3 flashes km−2 yr−1, respectively,much smaller than the terrestrial rate of 6 flashes km−2 yr−1.

The depth to the lightning has been estimated by comparing images of illuminated cloudswith the results of radiative transfer calculations. Dyudina et al (2002) simulated six well-resolved bright patches imaged by Galileo, five of which were found to have a distributionof brightness that is elongated along the line of sight when corrected for the oblique viewingangle. The elongation suggests that the patches are three-dimensional, perhaps due to the


vertical aspect of the lightning bolts or the illumination of discrete, vertically extended clouds.In their Monte Carlo simulation, photons emitted isotropically from a point or vertical linesource are scattered and absorbed by clouds with optical properties representative of thosemeasured by spacecraft. The elongated brightness distribution can be reproduced only whena non-plane-parallel cloud is used, though the cloud structure may be axisymmetric aboutthe vertical axis (e.g. a dome-shaped cloud). The depths of all six strikes were found to be35–171 km below the top of the continuous cloud near ∼0.5–1 bar, which corresponds to depthswhere NH4SH or water is expected to condense. One strike appears to be at a depth consistentonly with a water cloud. The large derived depths require clouds that extend continuouslyfrom the water condensation level to the top of the troposphere, giving further evidence fortheir association with powerful convection.

5.2. Lightning and moist convection

While lightning is an interesting physical process in its own right, added significance comesfrom its association with moist convection. On Earth, atmospheric convection is intensifiedby the release of latent heat from condensing water vapour. Multiple collisions between icyparticles (both the solid and liquid phases are thought to be important) generate charge thatis separated by convective motions and differential gravitational fall rates, and is released bycloud–cloud or cloud–surface lightning strikes. By analogy, Jovian lightning strikes may marklocations of moist convection and yield insight into the spatial distribution of water and its rolein atmospheric energetics (Gibbard et al 1995, Yair et al 1995, 1998).

Voyager, Galileo and Cassini observations show that convective storms also occurprimarily within the belts, as with the lightning observations (figure 3). The most prominentstorms occur in the turbulent wake region west of the Great Red Spot (figure 21). Movies of thisregion show storms appearing, vigorously producing bright cloud material and getting shearedapart by the zonal flow over the course of several days. Other prominent outbursts occur withinthe North Equatorial Belt. Convective storms at higher latitudes are less salient, either becausethey are less powerful (and do not extend as high or create as much cloud material) or becauseof the less favourable viewing geometry. The distribution of lifetimes of 31 convective stormson Cassini images follow a decaying exponential with a mean of 3.5 days (Li et al 2004).

Both Galileo and Cassini were able to correlate night-side lightning detections withclouds on dayside images taken nearby in time. Although uncertainties in image navigationsometimes prevented exact co-registration of the images, there were often small (∼500–2000 km), unusually bright clouds near the locations of lightning strikes (Little et al 1999,Gierasch et al 2000, Dyudina et al 2004). These clouds are sufficiently rare (only several arepresent over Jupiter’s surface at any instant) so that the correlation with the lightning distributionwas often obvious. In several cases, lightning-bearing clouds were observed on the daysidethrough the NIR filters and shown to be both bright and vertically extended (figures 21 and 22,Gierasch et al (2000), Dyudina et al (2004)). Modelling of the visible and NIR spectra suggestthat these clouds are denser and contain larger particles than other Jovian clouds (Banfieldet al 1998, Gierasch et al 2000, Dyudina et al 2001, Irwin and Dyudina 2002). In a few cases,lightning was associated with deep clouds that did not exhibit a tall structure. However, tallbright clouds were present nearby (Dyudina et al 2004).

Yair et al (1995) found that the minimum water abundance required to produce lightningis 0.5 times the solar value (a mole fraction of about 7×10−5). In their model, Jovian lightningoccurs within large cumulonimbus type clouds with vertical extents up to 50 km. Gibbard et al(1995) argue that moist convection extends from 4–6 bars to as high as the ∼0.5 bar level,while lightning occurs most readily at the 3- or 4-bar level. They further note that below about


Figure 22. False-colour mosaic of a discrete storm located ∼10 000 km northwest of the GreatRed Spot acquired by Galileo. In this representation, reddish clouds are deep, while white cloudsare high and thick. The bright storm is ∼1000 km across and extends ∼75 km from base to top(Banfield et al 1998). NASA image PIA01639.

5 bars, temperatures do not favour the ice phase and charge separation would be inefficient.Additional work is needed to understand the inconsistency of the latter with the >5 bar depthsinferred from lightning emissions (Dyudina et al 2004).

5.3. Moist convection as a probe of Jupiter’s vertical structure

The occurrence of strong thunderstorms on Jupiter implies the existence of a large reservoirof convective available potential energy, or CAPE (Emanuel 1994, pp 169–71), that can bereleased by moist-convective motions. The requirement is that the density profile is stable todry convection but unstable to moist convection (and hence the actual density profile between∼1–6 bars lies between the moist and dry adiabats). A key component is the presence of a stablelayer that underlies the condensation region, which would inhibit dry convection below thewater-condensation level but allows violent, episodic convection when plumes occasionallyascend to the condensation level (Showman and de Pater 2005). In atmospheres, CAPE isproduced over radiative timescales (∼years on Jupiter) by absorption of sunlight and thermalcooling to space. Moist convection reduces CAPE. The stable layer below the condensationlevel prevents CAPE from being completely depleted on a convective timescale (∼hours onJupiter) and allows an equilibrium between the rates at which CAPE is produced and destroyed.

The confinement of thunderstorms to belts (and their absence in zones) is a major puzzlethat needs explaining. Showman and de Pater (2005) described two endpoint scenarios thatprovide such an explanation. Either CAPE in zones is nearly zero, or the stable layer in zones isso thick that moist plumes cannot reach the condensation level. The first scenario (zero CAPE)would occur if large-scale ascent occurred throughout the zones, resulting in temperatures thatfollow a moist adiabat. Individual moist plumes would therefore have no buoyancy (i.e. theyalso follow a moist adiabat, so the density difference between them and the surroundingswould be zero), so violent thunderstorms would be impossible. This scenario appears tobe ruled out, however, by the observation that the ammonia abundance from ∼1–5 bars isseveral-fold less than the deep-interior abundance of 3–4 times solar measured by the Galileoprobe. Widespread environmental ascent should transport the deep abundance (minus thesmall amount lost to formation of NH4SH at 2 bars) to the 0.6 bar ammonia condensation


Figure 23. Galileo mosaic showing the region north of Jupiter’s equator (about 1–21˚N) in a near-infrared continuum filter. The dark feature near the centre of the mosaic is an equatorial hot spot,characterized by a clearing in the clouds. The upper half of the mosaic shows a small anticyclonewith two discrete storms to its south. The leftmost storm (just northeast of the hot spot) was shownto be associated with very high relative humidity, consistent with a moist convective origin. NASAimage PIA00604.

level. Therefore, the sharp decrease in ammonia abundance between ∼6 and ∼1 bar suggeststhat minimal ascent or mixing occurs from 1–6 bars, at least over widespread areas.

However, the second scenario for inhibiting zone thunderstorms, that zones havethick stable layers extending below the condensation level, matches available observations(Showman and de Pater 2005). In particular, such a thick stable layer extending below thecondensation altitude would naturally inhibit vertical mixing, helping to maintain a verticalgradient in ammonia abundance below the cloud-level (condensation and rainout of ammoniaat the 0.6 bar ammonia-condensation level provide the source of ammonia-free air at thetop). Furthermore, an attractive feature of this scenario is that thick stable layers naturallyresult from large-scale dynamics in anticyclonic regions (Dowling 1990, Ingersoll 1997). Ona rapidly rotating planet where winds are geostrophically balanced, anticyclones are high-pressure centres. If these anticyclonic motions are confined to a shallow weather layer thatcan be described by the shallow-water equations or its multilayer equivalent, then anticyclonesare thick regions where the shallow-water thickness, h, is large (see section 3.4.2). On a giantplanet, the weather layer may therefore penetrate deeper into the interior in anticyclones thanin cyclones, discouraging moist air from the deep interior from reaching the condensation leveland thereby preventing thunderstorms in anticyclones.

6. Equatorial features

The northern margin of Jupiter’s bright equatorial band (7˚–10˚N) is interrupted by a seriesof dark patches, intermittent in longitude (figures 2 and 23). These patches have been called‘5 µm hot spots’ because their relatively low cloud opacity allows 5 µm thermal emissionfrom deeper atmospheric levels to escape (the 5 µm wavelength band is unusual because itis a spectral window with extremely low gaseous absorption; radiation at most other infraredwavelengths escapes to space from pressures <0.5 bars regardless of the cloud opacity). Inspite of their prominent appearance, hot spots are poorly understood features. For example,


does the opacity vary as a result of vortices, mesoscale weather systems, or planetary-scalewaves? Hot spots defy easy classification. They do not produce bright cloud material orlightning, and their relationship to the ambient winds is not clear. Most vortices have arounded appearance and roll like ball bearings within a meridional wind shear, deflecting fluidaround their perimeter. Hot spots, however, are quasi-rectangular and translate without anyvortex-like rotation (Vasavada et al 1998). Cloud tracers reveal that winds that encounter thehot spot from certain directions are deflected, while winds from other directions appear totranslate through the hot spot. Hot spots occur at quasi-regular longitude intervals, suggestinga wave analogy. Other planetary-scale waves have been observed in Jupiter’s stratosphere. Butaerosol perturbations associated with hot spots are confined to the lower troposphere (Banfieldet al 1998, Orton et al 1998).

Much of our understanding of the temporal evolution of hot spots comes from Earth-basedstudies conducted to provide context for the Galileo and Cassini missions (Ortiz et al 1998,Orton et al 1998). Individual hot spots vary in shape, brightness at 4.78 µm and reflectivity atred/near-IR wavelengths (which is anti-correlated with the 4.78 µm brightness) over several-month intervals. A significant result of these studies is that, over a period of months to years, hotspots translate as a group with a fairly constant eastward velocity and longitudinal separation.Between 1993 and 1997, this velocity varied by ∼4%. When the velocity increased (from 99.6to 103.5 m s−1), so did the total number of hot spots (from 8 to 10). The translation rate of anindividual hot spot sometimes strays from the average, resulting in the merging or splitting ofsome features.

The study of hot spots took on increased significance after the Galileo probe descended intothe southern edge of one (Orton et al 1998). The probe measured temperature, composition,and opacity for about 150 km down to a level of about 22 bars. In addition, tracking of theProbe by the Galileo orbiter and Earth-based antennae gave a vertical profile of the horizontalwind speed (figure 4). Analyses of these data are complicated by their collection within a hotspot. For example, do measurements represent global values or only the conditions within hotspots? The probe found NH3 and H2S to be depleted to depths of 8–15 bars, well below theircondensation levels (Young 2003). The abundance of water increased towards the end of theProbe’s lifetime, but never reached its expected value. Surprisingly, probe tracking found thateastward winds increased dramatically from 90 m s−1 near 0.5 bars to 180 m s−1 near 5 barsand subsequently remained constant (Atkinson et al 1998). The former is close to the cloud-tracked velocity of ∼100 m s−1 at the 0.7 bar cloud level at the probe-entry latitude. It is notknown whether the increase in wind speed with depth is a characteristic of the equatorial jetor is locally driven by hot spot dynamics.

By analogy with Earth’s atmosphere, initial explanations invoked local convergence withinhot spots near the 1 bar level and subsequent down-welling to maintain volatile-poor regions(Baker and Schubert 1998, Showman and Ingersoll 1998, Atreya et al 1999). This ‘drydowndraft’ hypothesis was bolstered by Galileo imaging data showing advection of cloudmaterial towards hot spots (Vasavada et al 1998). However, a massive downdraft from 1 to20 bars would have to overcome the convectively stable temperature profile inferred from probedata as well as the large vertical shear in the horizontal winds.

Other studies have considered whether hot spots might be the manifestation of a planetaryscale wave trapped at equatorial latitudes (Ortiz et al 1998, Friedson and Orton 1999, Showmanand Dowling 2000). In this view, individual hot spots are the troughs of the wave structure.Ortiz et al (1998) find some evidence for dispersion in their power spectra of the hot-spotlatitude band. Showman and Dowling (2000) were able to simulate several characteristicsof hot spots using a multi-layer model that solves the fully nonlinear primitive equations.They interpret the features as being high-amplitude (and nonlinear) Rossby waves propagating


westward at 70 m s−1 relative to the eastward equatorial jet. Air parcels are deflected to greaterdepths (via column stretching) and experience up to a factor of two increases in pressure as thewave passes through them. The wave model accounts for the observed opacity changes and thelayered structure in the depleted volatiles. In addition, the simulations create an anticyclonicflow pattern (in the hot spot rest frame) extending from the centre of the hot spot to the equator.Advection of cloud material by this flow would resemble cloud motions observed by Galileo.Finally, simulations show vertical wind shear similar in magnitude to the observed values,which suggests that the change in wind measured by the Probe from 1 to 5 bars results fromlocal dynamics associated with hot spots and may not be representative of Jupiter as a whole.

7. Discussion

Observations reveal that Jupiter’s atmospheric dynamics contains a wealth of processes thatinteract over many temporal and spatial scales. Small, energetic convective storms withlifetimes measured in days are sheared apart near cloud-level, perhaps releasing their energyinto small-scale eddies that eventually drive the zonal jets. What role vortices play in theenergetics is less clear. Vortices, having lifetimes of months to years, appear to grow bymerging with other vortices, supporting the idea of an inverse cascade of energy. However,we have yet to witness a vortex expanding longitudinally and merging with a jet; the largestvortices, such as the Great Red Spot and White Ovals, appear to become more compact overtime, and some of these vortices may form by robbing energy from the jets rather than pumpingthe jets up. Whether the jets are driven within the shallow weather layer or the deep interiorremains unclear; numerical simulations show that each scenario has strengths and weaknesses.The zonal velocity profile is almost symmetric across the equator (supporting the idea of jetspenetrating the planet on deep, cross-equatorial cylinders), and almost invariant with time(consistent with a high-inertia, deep structure). Yet the data are not definitive in either case.As mentioned above, tentative observations suggest that small eddies at cloud level pumpthe jets via a shear-straining mechanism. Purely shallow models of Jupiter’s dynamics aresuccessful at explaining a wide range of observed phenomena. However, the Galileo probewinds and considerations of jet stability hint at a role for deeper structures.

In spite of the continuing uncertainties, much progress has been made. Theoretical,laboratory and numerical studies have now explored a wide range of configurations andparameters, revealing those that best match the details of the observational record. Perhapsthe goal for the post-Galileo and Cassini era is to move from piecewise studies towards anintegrated understanding of Jupiter’s dynamics. For example, modellers conduct separatestudies of deep and shallow jet structure and deep and shallow jet forcing. In reality, thedistinctions are artificial. A hybrid approach might hypothesize that Jupiter’s wide, progradeequatorial jet is the cloud-level manifestation of deep convection on cross-equatorial cylinders.But poleward of the equatorial zone, perhaps the dynamics of jets and vortices are best describedby shallow (i.e. two-dimensional) processes, which have enjoyed better success than deepmodels in reproducing both Jupiter-like, mid-latitude zonal jets and vortices.

The forward modelling approach to understanding Jupiter’s dynamics described in thisarticle has limitations; models can be compared with data only to the extent that the appropriateobservations exist and are of sufficient quality. Because of the loss of Galileo’s antenna andthe flyby nature of other missions, no single spacecraft to date has conducted the definitiveimaging experiment, acquiring multi-spectral, high spatial and temporal resolution, globaldata continuously over many years. Cassini offers this opportunity at Saturn, in some waysa more difficult planet to observe due to its more subtle cloud features and large seasonalcycle. However, in the case of Jupiter, it is likely that data of other types than imaging will


yield the greatest insights. Remote sounding experiments (e.g. infrared or microwave) coulddramatically improve our understanding of the vertical profiles of temperature, pressure andwind, and their spatial variations. Multiple entry probes could retrieve similar information insitu, along with information about composition and aerosols. Any observations of the deeperatmosphere would be revolutionary, such as that from a mission that could map Jupiter’s gravityfield to high order (Hubbard 1999). The prospect of such missions is uncertain given theirexpense and NASA’s present focus on the Moon and Mars. Interestingly, some additionalmotivation comes from the discoveries of ‘hot Jupiters’ orbiting very close around other stars(large planets at small orbital distances is a configuration favoured by current observationaltechniques). Such discoveries only add to our desire to understand the larger family of gasgiants, including the four in our solar system and those of a range of sizes and environmentalconditions around other stars, or adrift in interstellar space.

Acknowledgments

Work by ARV was carried out at the Jet Propulsion Laboratory, California Institute ofTechnology, under contract with the National Aeronautics and Space Administration. APSwas supported by NASA Planetary Atmospheres grant NAG5-13329. We thank Sarah Horstand Ulyana Dyudina for help in preparing the manuscript and an anonymous reviewer forproviding comprehensive and extremely helpful comments.

References

Achterberg R K and Ingersoll A P 1989 A normal-mode approach to Jovian atmospheric dynamics J. Atmos. Sci. 462448–62

Achterberg R K and Ingersoll A P 1994 Numerical-simulation of baroclinic Jovian vortices J. Atmos. Sci. 51 541–62Allison M 1990 Planetary waves in Jupiter’s equatorial atmosphere Icarus 83 282Allison M 2000 A similarity model for the windy Jovian thermocline Planet. Space Sci. 48 753–74Andrews D G and McIntyre M E 1976 Planetary waves in horizontal and vertical shear: the generalized Eliassen–Palm

relation and the mean zonal acceleration J. Atmos. Sci. 33 2031–48Andrews D G and McIntyre M E 1978 Generalized Eliassen–Palm and Charney–Drazin theorems for waves on

axisymmetric mean flows in compressible atmospheres J. Atmos. Sci. 35 175–85Andrews D G, Holton J R and Leovy C B 1987 Middle Atmosphere Dynamics (San Diego: Academic)Arai M and Yamagata T 1994 Asymmetric evolution of eddies in rotating shallow water Chaos 4 163–75Atkinson D H, Pollack J B and Seiff A 1996 Galileo Doppler measurements of the deep zonal winds at Jupiter Science

272 842–3Atkinson D H, Ingersoll A P and Seiff A 1997 Deep zonal winds on Jupiter: update of Doppler tracking the Galileo

probe from the orbiter Nature 388 649–50Atkinson D H, Pollack J B and Seiff A 1998 The Galileo probe Doppler wind experiment: measurement of the deep

zonal winds on Jupiter J. Geophys. Res. 103 22911–28Atreya S K, Wong M H, Owen T C, Mahaffy P R, Neimann H B, de Pater I, Drossart P and Encrenaz T 1999

A comparison of the atmosphere of Jupiter and Saturn: deep atmospheric composition, cloud structure, verticalmixing and origin Planet. Space Sci. 47 1243–62

Aubert J, Jung S and Swinney H L 2002 Observations of zonal flow created by potential vorticity in a rotating fluidGeophys. Res. Lett. 29 1876

Aurnou J M and Olson P L 2001 Strong zonal winds from thermal convection in a rotating spherical shell Geophys.Res. Lett. 28 2557–9

Aurnou J M and Heimpel M H 2004a Zonal jets in rotating convection with mixed mechanical boundary conditionsIcarus 169 492–8

Aurnou J M and Heimpel M H 2004b Generating banded zonal flows on Jupiter and Saturn American GeophysicalUnion Fall Meeting 2004, San Francisco, CA

Bagenal F, Dowling T E and McKinnon W B (ed) 2004 Jupiter: Planet, Satellites and Magnetosphere (Cambridge:Cambridge University Press)

Baker R D and Schubert G 1998 Deep convective entrainment by downdrafts in Jupiter’s atmosphere Icarus 136 340–3


Banfield D, Gierasch P J, Bell M, Ustinov E, Ingersoll A P, Vasavada A R, West R A and Belton M J S 1998 Jupiter’scloud structure from Galileo imaging data Icarus 135 230–50

Barcilon A and Gierasch P J 1970 A moist, Hadley cell model of Jupiter’s cloud bands J. Atmos. Sci. 27 550–60Basdevant C, Legras B and Sadourny R 1981 A study of barotropic model flows- intermittency, waves and predictability

J. Atmos. Sci. 38 2305–26Beebe R F, Ingersoll A P, Hunt G E, Mitchell J L and Miller J P 1980 Measurements of wind vectors, eddy momentum

transports, and energy conversions in Jupiter’s atmosphere from Voyager 1 images Geophys. Res. Lett. 7 1–4Beebe R F, Simon A A and Huber L F 1996 Comparison of Galileo probe and Earth-based translation rates of Jupiter’s

equatorial clouds Science 272 841Borucki W J and Magalhaes J A 1992 Analysis of Voyager 2 images of Jovian lightning Icarus 96 1–14Busse F H 1970 Differential rotation in stellar convection zones Astrophys. J. 159 629–40Busse F H 1976 Simple model of convection in the Jovian atmosphere Icarus 29 255–60Busse F H 1994 Convection driven zonal flows and vortices in the major planets Chaos 4 123–34Busse F H 2002 Convective flows in rapidly rotating spheres and their dynamo action Phys. Fluids 14 1301–14Carton X 2001 Hydrodynamical modeling of oceanic vortices Surveys Geophys. 22 179–263Charney J G and Drazin P G 1961 Propagation of planetary-scale disturbances from the lower into the upper atmosphere

J. Geophys. Res. 66 83–109Chekhlov A, Orszag S A, Sukoriansky S, Galperin B and Staroselsky I 1996 The effect of small-scale forcing on

large-scale structures in two-dimensional flows Physica D 98 321–34Cho J Y-K and Polvani L M 1996a The emergence of jets and vortices in freely evolving, shallow-water turbulence

on a sphere Phys. Fluids 8 1531–52Cho J Y-K and Polvani L M 1996b The morphogenesis of bands and zonal winds in the atmospheres on the giant

outer planets Science 273 335–7Cho J Y-K, de la Torre Juarez M, Ingersoll A P and Dritschel D G 2001 A high-resolution, three-dimensional model

of Jupiter’s Great Red Spot J. Geophys. Res. 106 5099–105Christensen U R 2001 Zonal flow driven by deep convection in the major planets Geophys. Res. Lett. 28 2553–6Christensen U R 2002 Zonal flow driven by strongly supercritical convection in rotating spherical shells J. Fluid Mech.

470 115–33Conrath B J, Flasar F M, Pirraglia J A, Gierasch P J and Hunt G E 1981 Thermal structure and dynamics of the Jovian

atmosphere II—Visible cloud features J. Geophys. Res. 86 8769–75Cook A F, Duxbury T C and Hunt G E 1979 First results on Jovian lightning Nature 280 794Danilov S and Gurarie D 2001 Forced two-dimensional turbulence in spectral and physical space Phys. Rev. E 63

061208-1–061208-12Danilov S and Gurarie D 2002 Rhines scale and spectra of the β-plane turbulence with bottom drag Phys. Rev. E 65

067301-1–067301-3Danilov S and Gurarie D 2004 Scaling, spectra, and zonal jets in beta-plane turbulence Phys. Fluids 16 2592–603Danilov S and Gryanik V M 2004 Barotropic beta-plane turbulence in a regime with strong zonal jets revisited J. Atmos.

Sci 61 2283–95Del Genio A D and Zhou W 1996 Simulations of superrotation of slowly rotating planets: sensitivity to rotation and

initial condition Icarus 120 332–43Del Genio A D, Zhou W and Eichler T P 1993 Equatorial superrotation in a slowly rotating GCM: implications for

Titan and Venus Icarus 101 1–17Dowling T E and Ingersoll A P 1988 Potential vorticity and layer thickness variations in the flow around Jupiter’s

Great Red Spot and White Oval BC J. Atmos. Sci. 45 1380–96Dowling T E and Ingersoll A P 1989 Jupiter’s Great Red Spot as a shallow-water system J. Atmos. Sci. 46

3256–78Dowling T E 1990 A new constraint on Jupiter’s winds Bull. Am. Astron. Soc. 22 1065Dowling T E 1995a Dynamics of Jovian atmospheres Ann. Rev. Fluid Mech. 27 293–334Dowling T E 1995b Estimate of Jupiter’s deep zonal-wind profile from Shoemaker-Levy-9 data and Arnold second

stability criterion Icarus 117 439–42Dritschel D G and de la Torre Juarez M 1996 The instability and breakdown of tall columnar vortices in a quasi-

geostrophic fluid J. Fluid Mech. 328 129–60Dritschel D G, de la Torre Juarez M and Ambaum M H P 1999 The three-dimensional vortical nature of atmospheric

and oceanic turbulent flows Phys. Fluids 11 1512–20Dyudina U A, Ingersoll A P, Danielson G E, Baines K H, Carlson R W and the Galileo NIMS and SSI Teams 2001

Interpretation of NIMS and SSI images on the Jovian cloud structure Icarus 150 219–33Dyudina U A, Ingersoll A P, Vasavada A R and Ewald S P 2002 Monte Carlo radiative transfer modeling of lightning

observed in Galileo images of Jupiter Icarus 160 336–49


Dyudina U A, Del Genio A D, Ingersoll A P, Porco C C, West R A, Vasavada A R and Barbara J M 2004 Lightningon Jupiter observed in the H-alpha line by the Cassini imaging science subsystem Icarus 172 24–36

Emanuel K 1994 Atmospheric Convection (New York: Oxford University Press)Evonuk M and Glatzmaier G A 2004 2D studies of various approximations used for modeling convection in giant

planets Geophys. Astrophys. Fluid Dyn. 98 241–55Farge M and Sadourny R 1989 Wave-vortex dynamics in rotating shallow water J. Fluid Mech. 206 433–62Flasar F M and Gierasch P J 1986 Mesoscale waves as a probe of Jupiter’s deep atmosphere J. Atmos. Sci. 43

2683–707Flierl G 1987 Isolated eddy models in geophysics Ann. Rev. Fluid Mech. 19 493–530Folkner W M, Preston R A, Border J S, Navarro J, Wilson W E and Oestreich M 1997 Earth-based radio tracking of

the Galileo probe for Jupiter wind estimation Science 275 644–6Friedson A J and Orton G S 1999 A dynamical model of Jupiter’s 5-micron hot spots Bull. Am. Astron. Soc. 31 1155Frisch U 1995 Turbulence (New York: Cambridge University Press)Galperin B, Sukoriansky S and Huang H-P 2001 Universal n−5 spectrum of zonal flows on giant planets Phys. Fluids

13 1545–8Garcıa-Melendo E and Sanchez-Lavega A 2001 A study of the stability of Jovian zonal winds from HST images:

1995–2000 Icarus 152 316–30Garcıa-Melendo E, Sanchez-Lavega A, Gomez J M, Lecacheux J, Colas F, Miyazaki I and Parker D 2000 Long-lived

vortices and profile changes in the 23.7˚N high-speed Jovian jet Icarus 146 514–24Gibbard S, Levy E H and Lunine J I 1995 Generation of lightning in Jupiter’s water cloud Nature 378 592Gierasch P J 1973 Jupiter’s cloud bands Icarus 19 482–94Gierasch P J, Ingersoll A P and Williams R T 1973 Radiative instability of a cloudy planetary atmosphere Icarus 19

473–81Gierasch P J 1976 Jovian meteorology—Large-scale moist convection Icarus 29 445–54Gierasch P J, Magalhaes J A and Conrath B J 1986 Zonal mean properties of Jupiter’s upper troposphere from Voyager

infrared observations Icarus 67 456–83Gierasch P J and Conrath B J 1993 Dynamics of the atmosphere of the outer planets—Post-Voyager measurements

objectives J. Geophys. Res. 98 5459–69Gierasch P J, Ingersoll A P, Banfield D, Ewald S P, Helfenstein P, Simon-Miller A A, Vasavada A R, Breneman H H,

Senske D A and the Galileo Imaging Team 2000 Observation of moist convection in Jupiter’s atmosphere Nature403 628–30

Gierasch P J 2004 Stability of jets on Jupiter and Saturn Icarus 167 212–19Gierasch P J, Conrath B J and Read P L 2004 Nonconservation of Ertel potential vorticity in hydrogen atmospheres

J. Atmos. Sci. 61 1953–65Gill A E 1982 Atmosphere–ocean Dynamics (San Diego: Academic)Grote E, Busse F H and Tilgner A 2000 Regular and chaotic spherical dynamos Phys. Earth Planet. Int. 117

259–72Grote E and Busse F H 2001 Dynamics of convection and dynamos in rotating spherical fluid shells Fluid Dyn. Res.

28 349–68Guillot T, Stevenson D J, Hubbard W B and Saumon D 2004 The interior of Jupiter Jupiter: The Planet, Satellites

and Magnetosphere ed F Bagenal et al (New York: Cambridge University Press) pp 35–57Hatzes A, Wenkert D D, Ingersoll A P and Danielson G E 1981 Oscillations and velocity structure of long-lived

cyclonic SPOT J. Geophys. Res. 86 8745–9Heimpel M H and Aurnou J M 2004 Modeling zonal flows on the giant planets American Geophysical Union Fall

Meeting 2004, San Francisco, CAHide R, Lewis S R and Read P L 1994 Sloping convection: a paradigm for large-scale waves and eddies in planetary

atmospheres? Chaos 4 135–62Hopfinger E J and van Heijst G J F 1993 Vortices in rotating fluids Ann. Rev. Fluid Mech. 25 241–89Holton J 1992 Introduction to Dynamic Meteorology (San Diego: Academic)Huang H-P and Robinson W A 1998 Two-dimensional turbulence and persistent zonal jets in a global barotropic

model J. Atmos. Sci. 55 611–32Huang H-P, Galperin B and Sukoriansky S 2001 Anisotropic spectra in two-dimensional turbulence on the surface of

a rotating sphere Phys. Fluids 13 225–40Hubbard W B 1999 Gravitational signature of Jupiter’s deep zonal flows Icarus 137 357–9Hueso R, Sanchez-Lavega A and Guillot T 2002 A model for large-scale convective storms in Jupiter J. Geophys.

Res. 107 5075Iacono R, Struglia M V, Ronchi C and Nicastro S 1999a High-resolution simulations of freely decaying shallow water

turbulence on a rotating sphere Nuovo Cimento 22 813–21


Iacono R, Struglia M V and Ronchi C 1999b Spontaneous formation of equatorial jets in freely decaying shallowwater turbulence Phys. Fluids 11 1272–4

Ingersoll A P and Cuzzi J N 1969 Dynamics of Jupiter’s cloud bands J. Atmos. Sci. 26 981–5Ingersoll A P and Cuong P G 1981 Numerical model of long-lived Jovian vortices J. Atmos. Sci. 38 2067–76Ingersoll A P and Pollard D 1982 Motion in the interiors and atmospheres of Jupiter and Saturn—Scale analysis,

anelastic equations, barotropic-stability criterion Icarus 52 62–80Ingersoll A P, Beebe R F, Conrath B J and Hunt G E 1984 Structure and dynamics of Saturn’s atmosphere Saturn

ed T Gehrels and M S Matthews (Tucson: University of Arizona Press) pp 195–238Ingersoll A P and Miller R L 1986 Motions in the interiors and atmospheres of Jupiter and Saturn, 2. Barotropic

instabilities and normal modes of an adiabatic planet Icarus 65 370–82Ingersoll A P 1990 Atmospheric dynamics of the outer planets Science 248 308–15Ingersoll A P and Kanamori H 1995 Waves from the collisions of comet Shoemaker-Levy-9 with Jupiter Nature 374

706–8Ingersoll A P 1997 Convective parameterization in a Jovian shallow water model Bull. Am. Astron. Soc. 29 1018Ingersoll A P, Beebe R F, Mitchell J L, Garneau G W, Yagi G M and Muller J P 1981 Interaction of eddies and

mean zonal flow on Jupiter as inferred from Voyager 1 and Voyager 2 images J. Geophys. Res.—Space Phys. 868733–43

Ingersoll A P, Vasavada A R, Little B, Anger C D, Bolton S J, Alexander C, Klaasen K P and Tobiska W K 1998aImaging Jupiter’s aurora at visible wavelengths Icarus 135 251–64

Ingersoll A P 1998b Atmospheres of the giant Planets The New Solar System ed J K Beatty et al (Cambridge: CambridgeUniversity Press) pp 201–20

Ingersoll A P, Gierasch P J, Banfield D, Vasavada A R and the Galileo Imaging Team 2000 Moist convection as anenergy source for the large-scale motions in Jupiter’s atmosphere Nature 403 630–2

Ingersoll A P, Dowling T E, Gierasch P J, Orton G S, Read P L, Sanchez-Lavega A, Showman A P, Simon-Miller A A andVasavada A R 2004 Atmospheric dynamics Jupiter: The Planet, Satellites and Magnetosphere ed F Bagenal et al(New York: Cambridge University Press) pp 105–28

Irwin P G J and Dyudina U 2002 The retrieval of cloud structure maps in the equatorial region of Jupiter using aprincipal component analysis of Galileo NIMS data Icarus 156 52–63

Irwin P G J 2003 Giant Planets of Our Solar System: Atmospheres, Composition, and Structure (New York: Springer)Juckes M N and McIntyre M E 1987 A high-resolution one-layer model of breaking planetary-waves in the stratosphere

Nature 328 590–6Kida S 1981 Motion of an elliptic vortex in a uniform shear-flow J. Phys. Soc. Japan 50 3517–20Kirk R L and Stevenson D J 1987 Hydromagnetic constraints on deep zonal flow in the giant planets Astrophys. J.

316 836–46Kitamura Y and Matsuda Y 2004 Numerical experiments of two-level decaying turbulence on a rotating sphere Fluid

Dyn. Res. 34 33–57LeBeau R P Jr and Dowling T E 1998 EPIC simulations of time-dependent, three-dimensional vortices with application

to Neptune’s Great Dark Spot Icarus 132 239–65Li L, Ingersoll A P, Vasavada A R, Porco C C, del Genio A D and Ewald S P 2004 Life cycles of spots on Jupiter from

Cassini images Icarus 172 9–23Limaye S S, Revercomb H E, Sromovsky L A, Krauss R J, Santek D A and Suomi V E 1982 Jovian winds from

Voyager 2 I—Zonal mean circulation J. Atmos. Sci. 39 1413–32Limaye S S 1986 Jupiter—New estimates of the mean zonal flow at the cloud level Icarus 65 335–52Little B, Anger C D, Ingersoll A P, Vasavada A R, Senske D A, Breneman H H, Borucki W J and the Galileo SSI

Team 1999 Galileo images of lightning on Jupiter Icarus 142 302–23Mac Low M-M and Ingersoll A P 1986 Merging of vortices in the atmosphere of Jupiter: an analysis of Voyager

images Icarus 65 353–69Magalhaes J A and Borucki W J 1991 Spatial-distribution of visible lightning on Jupiter Nature 349 311–13Magalhaes J A, Seiff A and Young R E 2002 The stratification of Jupiter’s troposphere at the Galileo probe entry site

Icarus 158 410–33Maltrud M E and Vallis G K 1991 Energy spectra and coherent structures in forced two-dimensional and beta-plane

turbulence J. Fluid Mech. 228 321–42Manfroi A J and Young W R 1999 Slow evolution of zonal jets on the beta plane J. Atmos. Sci. 56 784–800Marcus P S 1988 Numerical simulations of Jupiter’s Great Red Spot Nature 331 693–6Marcus P S 1990 Vortex dynamics in a shearing zonal flow J. Fluid Mech. 215 393–430Marcus P S 1993 Jupiter’s Great Red Spot and other vortices Ann. Rev. Astron. Astrophys. 31 523–73Marcus P S and Lee C 1994 Jupiter’s Great Red Spot and zonal winds as a self-consistent, one-layer, quasigeostrophic

flow Chaos 4 269–86


Marcus P S and Lee C 1998 A model for eastward and westward jets in laboratory experiments and planetaryatmospheres Phys. Fluids 10 1474–89

Marcus P S, Kundu T and Lee C 2000 Vortex dynamics and zonal flows Phys. Plasmas 7 1630–40Marcus P S 2004 Prediction of a global climate change on Jupiter Nature 428 828–31Mariotti A, Legras B and Dritschel D G 1994 Vortex stripping and the erosion of coherent structures in two-dimensional

flows Phys. Fluids 6 3954–62McWilliams J C 1984 The emergence of isolated coherent vortices in turbulent flow J. Fluid Mech. 146 21–43McWilliams J C 1991 Geostrophic vortices Nonlinear Topics in Ocean Physics Proceedings of the International

School of Physics ‘Enrico Fermi’ Course CIX ed A R Osborne (Amsterdam: Elsevier) pp 5–50Meacham S P, Flierl G R and Send U 1990 Vortices in shear Dyn. Atmos. Oceans 14 333–86Mitchell J L, Beebe R F, Ingersoll A P and Garneau G W 1981 Flow fields within Jupiter’s Great Red Spot and White

Oval BC J. Geophys. Res. 86 8751–7Moore D W and Saffman P G 1971 Structure of a line vortex in an imposed strain Aircraft Wake Turbulence and its

Detection ed J H Olsen et al (New York: Plenum Press) pp 339–54Morales-Juberıas R, Sanchez-Lavega A, Lecacheux J and Colas F 2002a A comparative study of Jovian anticyclone

properties from a six-year (1994–2000) survey Icarus 157 76–90Morales-Juberıas R, Sanchez-Lavega A, Lecacheux J and Colas F 2002b A comparative study of Jovian cyclonic

features from a six-year (1994–2000) survey Icarus 160 325–35Morales-Juberıas R, Sanchez-Lavega A and Dowling T E 2003 EPIC simulations of the merger of Jupiter’s White

Ovals BE and FA: altitude dependent behavior Icarus 166 63–74Nakajima K, Takehiro S-I, Ishiwatari M and Hayashi Y-Y 2000 Numerical modeling of Jupiter’s moist convection

layer Geophys. Res. Lett. 27 3129–32Nozawa T and Yoden S 1997a Formation of zonal band structure in forced two-dimensional turbulence on a rotating

sphere Phys. Fluids 9 2081–93Nozawa T and Yoden S 1997b Spectral anisotropy in forced two-dimensional turbulence on a rotating sphere Phys.

Fluids 9 3834–42Okuno A and Masuda A 2003 Effect of horizontal divergence on the geostrophic turbulence on a beta-plane: suppression

of the Rhines effect Phys. Fluids 15 56–65Ortiz J L, Orton G S, Friedson A J, Stewart S T, Fisher B M and Spencer J R 1998 Evolution and persistence of

5-micron hot spots at the Galileo probe entry latitude J. Geophys. Res. 103 23051–69Orton G S et al 1998 Characteristics of the Galileo probe entry site from Earth-based remote sensing observations

J. Geophys. Res. 103 22791–814Panetta R L 1993 Zonal jets in wide baroclinically unstable regions: persistence and scale selection J. Atmos. Sci. 50

2073–106Pedlosky J 1987 Geophysical Fluid Dynamics (New York: Springer)Peek B M 1958 The Planet Jupiter (London: Faber and Faber)Peltier W R and Stuhne G R 2002 The upscale turbulent cascade: shear layers, cyclones, and gas giant bands

Meteorology at the Millennium ed R P Pearce (New York: Academic) pp 43–61Pirraglia J A, Conrath B J, Allison M D and Gierasch P J 1981 Thermal structure and dynamics of Saturn and Jupiter

Nature 292 677–9Plumb R A 1983 A new look at the energy cycle J. Atmos. Sci. 40 1669–88Polvani L M, McWilliams J C, Spall M A and Ford R 1994 The coherent structures of shallow-water

turbulence: deformation-radius effects, cyclone/anticyclone asymmetry and gravity-wave generation Chaos 4177–86

Porco C C et al 2003 Cassini imaging of Jupiter’s atmosphere, satellites and rings Science 299 1541–7Porco C C et al 2005 Cassini imaging science: initial results on Saturn’s atmosphere Science 307 1243–7Read P L and Hide R 1983 Long-lived eddies in the laboratory and in the atmospheres of Jupiter and Saturn Nature

302 126–9Read P L and Hide R 1984 An isolated baroclinic eddy as a laboratory analogue of the Great Red Spot on Jupiter

Nature 308 45–8Read P L 1986 Stable, baroclinic eddies on Jupiter and Saturn: a laboratory analog and some observational tests Icarus

65 304–34Read P L 1992 Long-lived eddies in the atmospheres of the major planets Rotating Fluids in Geophysical and Industrial

Applications ed E J Hopfinger (New York: Springer) pp 283–99Read P L 2001 Transition to geostrophic turbulence in the laboratory, and as a paradigm in atmospheres and oceans

Surveys Geophys. 22 265–317Read P L, Yamazaki Y H, Lewis S R, Williams P D, Miki-Yamazaki K, Sommeria J, Didelle H and Fincham A 2004

Jupiter’s and Saturn’s convectively driven banded jets in the laboratory Geophys. Res. Lett. 31 L22701


Reinaud J N, Dritschel D G and Koudella C R 2003 The shape of vortices in quasi-geostrophic turbulence J. FluidMech. 474 175–92

Rhines P B 1975 Waves and turbulence on a beta-plane J. Fluid Mech. 69 417–43Rhines P B and Young W R 1982 Homogenization of potential vorticity in planetary gyres J. Fluid Mech. 122 347–67Rhines P B 1994 Jets Chaos 4 313–39Roberts P H 1968 On the thermal instability of a rotating-fluid sphere containing heat sources Phil. Trans. R. Soc.

Lond. Ser. A 263 93Rogers J H 1995 The Giant Planet Jupiter (Cambridge: Cambridge University Press)Roos-Serote M, Vasavada A R, Kamp L, Drossart P, Irwin P, Nixon C and Carlson R W 2000 Proximate humid and

dry regions in Jupiter’s atmosphere indicate complex local meteorology Nature 405 158–60Sada P V, Beebe R F and Conrath B J 1996 Comparison of the structure and dynamics of Jupiter’s Great Red Spot

between the Voyager 1 and 2 encounters Icarus 119 311–35Salyk C, Ewald S P, Ingersoll A P, Lorre J and Vasavada A R 2004 The relationship between eddies and zonal flow

on Jupiter: results from the Cassini flyby 36th DPS meeting, Louisville, USA (American Astronomical Society)paper 18.07

Salyk C 2005 Private communicationSanchez-Lavega A, Rojas J F, Hueso R, Lecacheux J, Colas F, Acarreta J R, Miyazaki I and Parker D 1999 Interaction

of Jovian White Ovals BC and DE in 1998 from Earth-based observations in the visual range Icarus 142116–24

Sanchez-Lavega A et al 2001 The merger of two giant anticyclones in the atmosphere of Jupiter Icarus 149 491–5Seiff A, Blanchard R C, Knight T C D, Schubert G, Kirk D B, Atkinson D, Mihalov J D and Young R E 1997 Wind

speeds measured in the deep jovian atmosphere by the Galileo probe accelerometers Nature 388 650–2Seiff A, Kirk D B, Knight T C D, Young R E, Mihalov J D, Young L A, Milos F S, Schubert G, Blanchard R C

and Atkinson D 1998 Thermal structure of Jupiter’s atmosphere near the edge of a 5-µm hot spot in the northequatorial belt J. Geophys. Res.—Planets 103 22857–89

Showman A P and Ingersoll A P 1998 Interpretation of Galileo probe data and implications for Jupiter’s dry downdraftsIcarus 132 205–20

Showman A P and Dowling T E 2000 Nonlinear simulations of Jupiter’s 5-micron hot spots Science 289 1737–40Showman A P 2004 Effects of moist convection on the large-scale circulation of Jupiter and Saturn Bull. Am. Astron.

Soc 36 1135Showman A P and de Pater I 2005 Dynamical implications of Jupiter’s tropospheric ammonia abundance Icarus 174

192–204Simon A A 1999 The structure and temporal stability of Jupiter’s zonal winds: a study of the North Tropical Region

Icarus 141 29–39Simon A A, Beebe R F, Gierasch P J, Vasavada A R, Belton M J S and the Galileo Imaging Team 1998 Global context

of the Galileo-E6 observations of Jupiter’s White Ovals Icarus 135 220–9Simon-Miller A A, Banfield D and Gierasch P J 2001 Color and the vertical structure in Jupiter’s belts, zones, and

weather systems Icarus 154 459–74Simon-Miller A A, Gierasch P J, Beebe R F, Conrath B, Flasar M F, Achterberg R K and the Cassini CIRS Team 2002

New observational results concerning Jupiter’s Great Red Spot Icarus 158 249–66Smith B A et al 1979 The Jupiter system through the eyes of Voyager 1 Science 204 951–72Sommeria J, Meyers S D and Swinney H L 1988 Laboratory simulation of Jupiter’s Great Red Spot Nature 331 689–93Sommeria J, Meyers S D and Swinney H L 1989 Experiments on vortices and Rossby waves in eastward and westward

jets. Nonlinear Topics in Ocean Physics: Proc. International School of Physics ‘Enrico Fermi,’ Course CIXed A R Osborne (Amsterdam: Elsevier) pp 227–69

Spall M and McWilliams J C 1992 Rotational and gravitational influences on the degree of balance in the shallowwater equations Geophys. Astrophys. Fluid Dyn. 64 1–29

Smith B A, Soderblom L A, Banfield D, Barnet C, Beebe R F, Bazilevskii A T, Bollinger K, Boyce J M, Briggs G Aand Brahic A 1989 Voyager 2 at Neptune—Imaging science results Science 246 1422–49

Smith L M and Waleffe F 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensionalturbulence Phys. Fluids 11 1608–22

Smith K S 2004 A local model for planetary atmospheres forced by small-scale convection J. Atmos. Sci. 61 1420–33Sromovsky L A, Revercomb H E, Suomi V E, Limaye S S and Kraus R 1982 Jovian winds fromVoyager 2 Part II—

Analysis of eddy transports J. Atmos. Sci. 39 1433–45Stevenson D J and Salpeter E E 1976 Interior models of Jupiter Jupiter: Studies of the Interior, Atmosphere,

Magnetosphere and Satellites ed T Gehrels (Tucson: University of Arizona Press) pp 85–112Stevenson D J and Salpeter E E 1977 The dynamics and helium distribution in hydrogen–helium planets Astrophys. J.

35 239–61


Sukoriansky S, Galperin B and Dikovskaya N 2002 Universal spectrum of two-dimensional turbulence on a rotatingsphere and some basic features of atmospheric circulation on giant planets Phys. Rev. Lett. 89 124501-1–124501-4

Sumita I and Olson P 2000 Laboratory experiments on high Rayleigh number thermal convection in a rapidly rotatinghemispherical shell Phys. Earth Planet. Int. 117 153–70

Sun Z-P, Schubert G and Glatzmaier G A 1993 Transitions to chaotic thermal convection in a rapidly rotating sphericalfluid shell Geophys. Astrophys. Fluid Dyn. 69 95–131

Taylor G I 1923 Experiments on the motion of solid bodies in rotating fluids Proc. R. Soc. A 104 213–8Vallis G K and Maltrud M E 1993 Generation of mean flows and jets on a beta-plane and over topography J. Phys.

Oceanogr. 23 1346–62Vasavada A R et al 1998 Galileo imaging of Jupiter’s atmosphere: the Great Red Spot, equatorial region, and White

Ovals Icarus 135 265–75Vasavada A R, Bouchez A H, Ingersoll A P, Little B, Anger C D and the Galileo SSI Team 1999 Jupiter’s visible

aurora and Io footprint J. Geophys. Res. 104 27133–42Vasavada A R 2002 Galileo: close-up views of Jupiter’s dynamic atmosphere High. Astron. 12 602–5Weir S T, Mitchell A C and Nellis W J 1996 Metallization of fluid molecular hydrogen at 140 GPa (1.4 Mbar) Phys.

Rev. Lett. 76 1860–3White A A 1986 Finite amplitude, steady Rossby waves and mean flows: analytical illustrations of the Charney-Drazin

non-acceleration theorem Q. J. R. Meteorol. Soc. 112 749–73Williams G P 1978 Planetary circulations 1: Barotropic representations of Jovian and terrestrial turbulence J. Atmos.

Sci. 35 1399–426Williams G P 1979 Planetary circulations 2: Jovian quasi-geostrophic regime J. Atmos. Sci. 36 932–68Williams G P and Yamagata T 1984 Geostrophic regimes, intermediate solitary vortices and Jovian eddies J. Atmos.

Sci. 41 453–78Williams G P 1985 Jovian and comparative atmospheric modeling Adv. Geophys. A 28 381–429Williams G P and Wilson R J 1988 The stability and genesis of Rossby vortices J. Atmos. Sci. 45 207–41Williams G P 1996 Jovian dynamics. Part I: Vortex stability, structure and genesis J. Atmos. Sci. 53 2685–734Williams G P 2002 Jovian dynamics. Part II: The genesis and equilibration of vortex sets J. Atmos Sci. 59 1356–70Williams G P 2003a Jovian dynamics. Part III: Multiple, migrating, and equatorial jets J. Atmos. Sci. 60 1270–96Williams G P 2003b Barotropic instability and equatorial superrotation J. Atmos. Sci 60 2136–52Williams G P 2003c Jet sets J. Meteorol. Soc. Japan 81 439–76Yair Y, Levin Z and Tzivion S 1995 Lightning generation in a Jovian thundercloud: results from an axisymmetric

numerical cloud model Icarus 115 421–34Yair Y, Levin Z and Tzivion S 1998 Model interpretation of Jovian lightning activity and the Galileo Probe results

J. Geophys. Res. Atmos. 103 14157–66Yamazaki Y H, Skeet D R and Read P L 2004 A new general circulation model of Jupiter’s atmosphere based on the

UKMO unified model: three-dimensional evolution of isolated vortices and zonal jets in mid-latitudes Planet.Space Sci. 52 423–45

Yamazaki Y H, Read P L and Skeet D R 2005 Hadley circulations and Kelvin wave-driven equatorial jets in theatmospheres of Jupiter and Saturn Planet. Space Sci. 53 508–25

Yano J-I 1994 A critical review on the dynamics of Jovian atmospheres Chaos 4 287–97Yano J-I 1998 Deep convection in the interior of major planets: a review Aust. J. Phys. 51 875–89Yano J-I, Talagrand O and Drossart P 2003 Outer planets: Origins of atmospheric zonal winds Nature 421 36Yoden S and Yamada M 1993 A numerical experiment on two-dimensional decaying turbulence on a rotating sphere

J. Atmos. Sci. 50 631–43Yoden S, Ishioka K, Hayashi Y-Y and Yamada M 1999 A further experiment on two-dimensional decaying turbulence

on a rotating sphere Nuovo Cimento 22 803–12Young R E 2003 The Galileo Probe: How it has changed our understanding of Jupiter New Astron. Rev. 47 1–51Youssef A and Marcus P S 2003 The dynamics of Jovian white ovals from formation to merger Icarus 162 74–93Yuan L and Hamilton K 1994 Equilibrium dynamics in a forced-dissipative f-plane shallow-water system J. Fluid

Mech. 280 369–94

jovian atmospheric dynamics: an update after galileo and...

Documents