Generation of equatorial jets by large-scale latent heating on the giant planets

Three-dimensional numerical simulations show that large-scale latent heating resulting from condensation of water vapor can produce multiple zonal jets similar to those on the gas giants (Jupiter and Saturn) and ice giants (Uranus and Neptune). For plausible water abundances (3-5 times solar on Jupiter/Saturn and 30 times solar on Uranus/Neptune), our simulations produce ∼20 zonal jets for Jupiter and Saturn and 3 zonal jets on Uranus and Neptune, similar to the number of jets observed on these planets. Moreover, these Jupiter/Saturn cases produce equatorial superrotation whereas the Uranus/Neptune cases produce equatorial subrotation, consistent with the observed equatorial-jet direction on these planets. Sensitivity tests show that water abundance, planetary rotation rate, and planetary radius are all controlling factors, with water playing the most important role; modest water abundances, large planetary radii, and fast rotation rates favor equatorial superrotation, whereas large water abundances favor equatorial subrotation regardless of the planetary radius and rotation rate. Given the larger radii, faster rotation rates, and probable lower water abundances of Jupiter and Saturn relative to Uranus and Neptune, our simulations therefore provide a possible mechanism for the existence of equatorial superrotation on Jupiter and Saturn and the lack of superrotation on Uranus and Neptune. Nevertheless, Saturn poses a possible difficulty, as our simulations were unable to explain the unusually high speed (∼) of that planet’s superrotating jet. The zonal jets in our simulations exhibit modest violations of the barotropic and Charney-Stern stability criteria. Overall, our simulations, while idealized, support the idea that latent heating plays an important role in generating the jets on the giant planets.     

Jupiter and Saturn rotation periods

Anderson and Schubert [2007. Saturn's Gravitational field, internal rotation, and interior structure. Science 317, 1384-1387 (paper I)] proposed that Saturn's rotation period can be ascertained by minimizing the dynamic heights of the 100 mbar isosurface with respect to the geoid; they derived a rotation period of 10 h 32 m 35 s. We investigate the same approach for Jupiter to see if the Jovian rotation period is predicted by minimizing the dynamical heights of its isobaric (1 bar pressure level) surface using zonal wind data. A rotation period of 9 h 54 m 29.7 s is found. Further, we investigate the minimization method by fitting Pioneer and Voyager occultation radii for both Jupiter and Saturn. Rotation periods of 9 h 55 m 30 s and 10 h 32 m 35 s are found to minimize the dynamical heights for Jupiter and Saturn, respectively. Though there is no dynamical principle requiring the minimization of the dynamical heights of an isobaric surface, the successful application of the method to Jupiter lends support to its relevance for Saturn.We derive Jupiter and Saturn rotation periods using equilibrium theory to explain the difference between equatorial and polar radii. Rotation periods of 9 h 55 m 20 s and 10 h 31 m 49 s are found for Jupiter and Saturn, respectively. We show that both Jupiter's and Saturn's shapes can be derived using solid-body rotation, suggesting that zonal winds have a minor effect on the planetary shape for both planets.The agreement in the values of Saturn's rotation period predicted by the different approaches supports the conclusion that the planet's period of rotation is about 10 h 32 m.     

Constraints on deep-seated zonal winds inside Jupiter and Saturn

The atmospheres of Jupiter and Saturn exhibit strong and stable zonal winds. How deep the winds penetrate unabated into each planet is unknown. Our investigation favors shallow winds. It consists of two parts. The first part makes use of an Ohmic constraint; Ohmic dissipation associated with the planet's magnetic field cannot exceed the planet's net luminosity. Application to Jupiter (J) and Saturn (S) shows that the observed zonal winds cannot penetrate below a depth at which the electrical conductivity is about six orders of magnitude smaller than its value at the molecular-metallic transition. Measured values of the electrical conductivity of molecular hydrogen yield radii of maximum penetration of 0.96RJ and 0.86RS, with uncertainties of a few percent of R. At these radii, the magnetic Reynolds number based on the zonal wind velocity and the scale height of the magnetic diffusivity is of order unity. These limits are insensitive to difficulties in modeling turbulent convection. They permit complete penetration along cylinders of the equatorial jets observed in the atmospheres of Jupiter and Saturn. The second part investigates how deep the observed zonal winds actually do penetrate. As it applies heuristic models of turbulent convection, its conclusions must be regarded as tentative. Truncation of the winds in the planet's convective envelope would involve breaking the Taylor-Proudman constraint on cylindrical flow. This would require a suitable nonpotential acceleration which none of the obvious candidates appears able to provide. Accelerations arising from entropy gradients, magnetic stresses, and Reynolds stresses appear to be much too weak. These considerations suggest that strong zonal winds are confined to shallow, stably stratified layers, with equatorial jets being the possible exception.     

Deep jets on gas-giant planets

Three-dimensional numerical simulations of the atmospheric flow on giant planets using the primitive equations show that shallow thermal forcing confined to pressures near the cloud tops can produce deep zonal winds from the tropopause all the way down to the bottom of the atmosphere. These deep winds can attain speeds comparable to the zonal jet speeds within the shallow, forced layer; they are pumped by Coriolis acceleration acting on a deep meridional circulation driven by the shallow-layer eddies. In the forced layer, the flow reaches an approximate steady state where east-west eddy accelerations balance Coriolis accelerations acting on the meridional flow. Under Jupiter-like conditions, our simulations produce 25 to 30 zonal jets, similar to the number of jets observed on Jupiter and Saturn. The simulated jet widths correspond to the Rhines scale; this suggests that, despite the three-dimensional nature of the dynamics, the baroclinic eddies energize a quasi-two-dimensional inverse cascade modified by the β effect (where β is the gradient of the Coriolis parameter). In agreement with Jupiter, the jets can violate the barotropic and Charney-Stern stability criteria, achieving curvatures ^∂2u/∂y² of the zonal wind u with northward distance y up to 2β. The simulations exhibit a tendency toward neutral stability with respect to Arnol'd's second stability theorem in the upper troposphere, as has been suggested for Jupiter, although deviations from neutrality exist. When the temperature varies strongly with latitude near the equator, our simulations can also reproduce the stable equatorial superrotation with wind speeds greater than . Diagnostics show that barotropic eddies at low latitudes drive the equatorial superrotation. The simulations also broadly explain the distribution of jet-pumping eddies observed on Jupiter and Saturn. While idealized, these simulations therefore capture many aspects of the cloud-level flows on Jupiter and Saturn.     

Turbulent convection in rapidly rotating spherical shells: A model for equatorial and high latitude jets on Jupiter and Saturn

The origin of zonal jets on the jovian planets has long been a topic of scientific debate. In this paper we show that deep convection in a spherical shell can generate zonal flow comparable to that observed on Jupiter and Saturn, including a broad prograde equatorial jet and multiple alternating jets at higher latitudes. We present fully turbulent, 3D spherical numerical simulations of rapidly rotating convection with different spherical shell geometries. The resulting global flow fields tend to be segregated into three regions (north, equatorial, and south), bounded by the tangent cylinder that circumscribes the inner boundary equator. In all of our simulations a strong prograde equatorial jet forms outside the tangent cylinder, whereas multiple jets form in the northern and southern hemispheres, inside the tangent cylinder. The jet scaling of our numerical models and of Jupiter and Saturn is consistent with the theory of geostrophic turbulence, which we extend to include the effect of spherical shell geometry. Zonal flow in a spherical shell is distinguished from that in a full sphere or a shallow layer by the effect of the tangent cylinder, which marks a reversal in the sign of the planetary β-parameter and a jump in the Rhines length. This jump is manifest in the numerical simulations as a sharp equatorward increase in jet widths—a transition that is also observed on Jupiter and Saturn. The location of this transition gives an estimate of the depth of zonal flow, which seems to be consistent with current models of the jovian and saturnian interiors.     

Interaction of moist convection with zonal jets on Jupiter and Saturn

Observations suggest that moist convection plays an important role in the large-scale dynamics of Jupiter's and Saturn's atmospheres. Here we use a reduced-gravity quasigeostrophic model, with a parameterization of moist convection that is based on observations, to study the interaction between moist convection and zonal jets on Jupiter and Saturn. Stable jets with approximately the same width and strength as observations are generated in the model. The observed zonal jets violate the barotropic stability criterion but the modeled jets do so only if the flow in the deep underlying layer is westward. The model results suggest that a length scale and a velocity scale associated with moist convection control the width and strength of the jets. The length scale and velocity scale offer a possible explanation of why the jets of Saturn are stronger and wider than those of Jupiter.     

Zonal jets in rotating convection with mixed mechanical boundary conditions

Large-scale zonal flows, as observed on the giant planets, can be driven by thermal convection in a rapidly rotating spherical shell. Most previous models of convectively-driven zonal flow generation have utilized stress-free mechanical boundary conditions (FBC) for both the inner and the outer surfaces of the convecting layer. Here, using 3D numerical models, we compare the FBC case to the case with a stress free outer boundary and a non-slip inner boundary, which we call the mixed case (MBC). We find significant differences in surface zonal flow profiles produced by the two cases. In low to moderate Rayleigh number FBC cases, the main equatorial jet is flanked by a strong, high-latitude retrograde jets in the northern and southern hemispheres. For the highest Rayleigh number FBC case, the equatorial jet is flanked by strong reversed jets as well as two additional large-scale alternating jets at higher latitudes. The MBC cases feature stronger equatorial jets but, much weaker, small-scale alternating zonal flows are found at higher latitudes. Our high Rayleigh number FBC results best compare with the zonal flow pattern observed on Jupiter, where the equatorial jet is flanked by strong retrograde jets as well as small-scale alternating jets at high latitude. In contrast, the MBC results compare better with the observed flow pattern on Saturn, which is characterized by a dominant prograde equatorial jet and a lack of strong high latitude retrograde flow. This may suggest that the mechanical coupling at the base of the jovian convection zone differs from that on Saturn.     

Shell thickness variations and the long-wavelength topography of Titan

The long-wavelength topography of Titan has an amplitude larger than that expected from tidal and rotational distortions at its current distance from Saturn. This topography is associated with small gravity anomalies, indicating a high degree of compensation. Both observations can be explained if Titan has a floating, isostatically-compensated ice shell with a spatially-varying thickness. The spatial variations arise because of laterally-variable tidal heating within the ice shell. Models incorporating shell thickness variations result in an improved fit to the observations and a degree-two tidal Love number h_2t consistent with expectations, without requiring Titan to have moved away from Saturn. Our preferred models have a mean shell thickness of ≈100 km in agreement with the observed gravity anomalies, and a heat flux appropriate to a chondritic Titan. Shell thickness variations are eliminated by convection; we therefore conclude that Titan’s ice shell is not convecting at the present day.     

A comparison of the interiors of Jupiter and Saturn

Interior models of Jupiter and Saturn are calculated and compared in the framework of the three-layer assumption, which rely on the perception that both planets consist of three globally homogeneous regions: a dense core, a metallic hydrogen envelope, and a molecular hydrogen envelope. Within this framework, constraints on the core mass and abundance of heavy elements (i.e. elements other than hydrogen and helium) are given by accounting for uncertainties on the measured gravitational moments, surface temperature, surface helium abundance, and on the inferred protosolar helium abundance, equations of state, temperature profile and solid/differential interior rotation. Results obtained solely from static models matching the measured gravitational fields indicate that the mass of Jupiter’s dense core is less than 14 M_⊕ (Earth masses), but that models with no core are possible given the current uncertainties on the hydrogen–helium equation of state. Similarly, Saturn’s core mass is less than 22 M_⊕ but no lower limit can be inferred. The total mass of heavy elements (including that in the core) is constrained to lie between 11 and 42 M_⊕ in Jupiter, and between 19 and 31 M_⊕ in Saturn. The enrichment in heavy elements of their molecular envelopes is 1–6.5, and 0.5–12 times the solar value, respectively. Additional constraints from evolution models accounting for the progressive differentiation of helium (Hubbard WB, Guillot T, Marley MS, Burrows A, Lunine JI, Saumon D, 1999. Comparative evolution of Jupiter and Saturn. Planet. Space Sci. 47, 1175–1182) are used to obtain tighter, albeit less robust, constraints. The resulting core masses are then expected to be in the range 0–10 M_⊕, and 6–17 M_⊕ for Jupiter and Saturn, respectively. Furthermore, it is shown that Saturn’s atmospheric helium mass mixing ratio, as derived from Voyager, Y=0.06±0.05, is probably too low. Static and evolution models favor a value of Y=0.11−0.25. Using, Y=0.16±0.05, Saturn’s molecular region is found to be enriched in heavy elements by 3.5 to 10 times the solar value, in relatively good agreement with the measured methane abundance. Finally, in all cases, the gravitational moment J₆ of models matching all the constraints are found to lie between 0.35 and 0.38×10⁻⁴ for Jupiter, and between 0.90 and 0.98×10⁻⁴ for Saturn, assuming solid rotation. For comparison, the uncertainties on the measured J₆ are about 10 times larger. More accurate measurements of J₆ (as expected from the Cassini orbiter for Saturn) will therefore permit to test the validity of interior models calculations and the magnitude of differential rotation in the planetary interior.     

Electrostatic waves at frequencies close to the harmonics of electron gyrofrequency

An analytical solution of the dispersion equation for electrostatic waves propagating in a plasma consisting of cold electrons and the electrons with the loss cone distribution function is obtained for wave frequencies close to the harmonics of electron gyrofrequency. Electrostatic emissions at these frequencies were observed in the magnetospheres of the Earth, Jupiter, and Saturn.     

Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2009

Every three years the IAU Working Group on Cartographic Coordinates and Rotational Elements revises tables giving the directions of the poles of rotation and the prime meridians of the planets, satellites, minor planets, and comets. This report takes into account the IAU Working Group for Planetary System Nomenclature (WGPSN) and the IAU Committee on Small Body Nomenclature (CSBN) definition of dwarf planets, introduces improved values for the pole and rotation rate of Mercury, returns the rotation rate of Jupiter to a previous value, introduces improved values for the rotation of five satellites of Saturn, and adds the equatorial radius of the Sun for comparison. It also adds or updates size and shape information for the Earth, Mars?? satellites Deimos and Phobos, the four Galilean satellites of Jupiter, and 22 satellites of Saturn. Pole, rotation, and size information has been added for the asteroids (21) Lutetia, (511) Davida, and (2867) ?teins. Pole and rotation information has been added for (2) Pallas and (21) Lutetia. Pole and rotation and mean radius information has been added for (1) Ceres. Pole information has been updated for (4) Vesta. The high precision realization for the pole and rotation rate of the Moon is updated. Alternative orientation models for Mars, Jupiter, and Saturn are noted. The Working Group also reaffirms that once an observable feature at a defined longitude is chosen, a longitude definition origin should not change except under unusual circumstances. It is also noted that alternative coordinate systems may exist for various (e.g. dynamical) purposes, but specific cartographic coordinate system information continues to be recommended for each body. The Working Group elaborates on its purpose, and also announces its plans to occasionally provide limited updates to its recommendations via its website, in order to address community needs for some updates more often than every 3 years. Brief recommendations are also made to the general planetary community regarding the need for controlled products, and improved or consensus rotation models for Mars, Jupiter, and Saturn.     

Comparative evolution of Jupiter and Saturn

We present evolutionary sequences for Jupiter and Saturn, based on new non-gray model atmospheres, which take into account the evolution of the solar luminosity and partitioning of dense components to deeper layers. The results are used to set limits on the extent to which possible interior phase separation of hydrogen and helium may have progressed in the two planets. When combined with static models constrained by the gravity field, our evolutionary calculations constrain the helium mass fraction in Jupiter to be between 0.20 and 0.27, relative to total hydrogen and helium. This is consistent with the Galileo determination. The helium mass fraction in Saturn’s atmosphere lies between 0.11 and 0.21, higher than the Voyager determination. Based on the discrepancy between the Galileo and Voyager results for Jupiter, and our models, we predict that revised observational results for Saturn will yield a higher atmospheric helium mass fraction relative to the Voyager value.     

The structure of the planets Jupiter and Saturn

Planetary models for Jupiter and Saturn are computed using a fourth-order theory and a new molecular equation of state. The equation of state for the molecular hydrogen and helium planetary envelopes is taken from the Monte Carlo calculations of Slattery and Hubbard [Icarus 29, 187–192 (1976)]. Models for Jupiter are found that have a small amount of heavy elements either mixed with hydrogen and helium throughout the interior of the planet or concentrated in a small dense core. Saturn is modeled with a solar-composition hydrogen and helium envelope and a small derse core. We conclude that the molecular equation of state linked with suitable interior equations of state can produce Jovian models which satisfy the observational data. The planetary models show that the enrichment of heavy elements (relative to solar composition) is approximately 3 times for Jupiter and 10 times for Saturn.     

Time-dependent flexure of the lithospheres on the icy satellites of Jupiter and Saturn

Previous analyses into flexural deformation on the icy satellites of Jupiter and Saturn have assumed static, elastic lithospheres. Viscous creep within the lithosphere, however, can cause evolution over time. Here, we apply a finite-element model that employs a time-dependent elastic–viscous-plastic rheology in order to investigate flexure on icy satellites. Factors that affect this time-dependent response are those that control creep rates; surface temperature, heat flow, and grain size. Our results show that surface temperature is by far the dominant factor. At higher surface temperatures (100–130 K), the evolution of the deformation is such that the thickness of a modeled elastic lithosphere could vary by up to an order of magnitude, depending on the time scale over which the deformation occurred. Because the flexure observed on icy satellites generally indicates transient high heat flow events, our results indicate that the duration of the heat pulse is an important factor. For the icy worlds of Jupiter and Saturn, static models of lithospheric flexure should be used with caution.     

Near-infrared polarization studies of Saturn and Jupiter

Polarization measurements of Jupiter, Saturn, and Saturn's rings from 1 to 3.5 μm are presented. At 1.6 μm on the discs of the two planets, the radially directed limb polarizations observed in the visible undergo, in some cases, a surprising 90° rotation to a tangential direction, particularly on the poles. The only immediate explanation for this effect is double Mie scattering, due to aerosols which must be of the order of a micrometer in size. On Jupiter the patterns are not uniform and are not stable, reflecting variable aerosol concentrations on the two poles. The ring polarization is uniformly negative (E vector parallel to the ecliptic plane) from the visible through 3.5 μm, and is inversely proportional to the albedo. This is as expected from Wolff's model for scattering from rough solid surfaces; but the degree of polarization seems uncommonly high, exceeding 2% at 3.5 μm.     

The effects of vigorous mixing in a convective model of zonal flow on the ice giants

Previous studies have used models of three-dimensional (3D) Boussinesq convection in a rotating spherical shell to explain the zonal flows on the gas giants, Jupiter and Saturn. In this paper we demonstrate that this approach can also generate flow patterns similar to those observed on the ice giants, Uranus and Neptune. The equatorial jets of Uranus and Neptune are often assumed to result from baroclinic cloud layer processes and have been simulated with shallow layer models. Here we show that vigorous, 3D convection in a spherical shell can produce the retrograde (westward) equatorial flows that occur on the ice giants as well as the prograde (eastward) equatorial flows of the gas giants. In our models, the direction of the equatorial jet depends on the ratio of buoyancy to Coriolis forces in the system. In cases where Coriolis forces dominate buoyancy, cylindrical Reynolds stresses drive prograde equatorial jets. However, as buoyancy forces approach and exceed Coriolis forces, the cylindrical nature of the flow is lost and 3D mixing homogenizes the fluid's angular momentum; the equatorial jet reverses direction, while strong prograde jets form in the polar regions. Although the results suggest that conditions involving strong atmospheric mixing are responsible for generating the zonal flows on the ice giants, our present models require roughly 100 and 10 times the internal heat fluxes observed on Uranus and Neptune, respectively.     

Hadley circulations and Kelvin wave-driven equatorial jets in the atmospheres of Jupiter and Saturn

We propose a dynamical mechanism that can plausibly explain the origin of the broad prograde equatorial winds observed on Jupiter and Saturn, and examine the feasibility of this mechanism using two- (2D) and three-dimensional (3D) numerical simulation models. The idea is based on combining a narrow Gaussian jet peaking at the equator, which is induced by the momentum transfer from an upward propagating equatorial Kelvin-wave, and a pair of off-equatorial jets due to a meridional-vertical circulation similar to the tropical Hadley circulation on Earth. We employ for this feasibility study a 2D mechanistic mean-flow model which incorporates the influence of prescribed waves, and a 3D general circulation model, based on the generalised primitive equations of atmospheric motion. We then confirm that the dynamical models of both kinds can successfully reproduce theoretically expected flows of a reasonable magnitude, and that when two mechanisms are combined, a broad super-rotating jet is produced with off-equatorial maxima in zonal velocity for both Jupiter and Saturn, approximately in accordance with observations.     

Instability of Zonal Flows in Rotating Spherical Shells: An Application to Jupiter

Measurements from the Galileo probe suggest that the zonal winds are deep rooted. Jupiter's high rotation rate makes it likely that the whole outer molecular H/He layer is involved in these long-lived jet flows. Assuming that the primary flows are geostrophic, and that the banded surface structure stretches right through the molecular H/He layer, we examine the conditions for such flows to be stable. As a first step, the linear stability of some prescribed banded zonal flows in a rotating spherical shell is explored. Incompressibility is assumed for simplicity, and the boundary condition is stress-free. We compare solutions for two aspect ratios, appropriate for the molecular H/He layers of Jupiter and Saturn, and two Ekman numbers (E=10⁻² and E=10⁻⁴). Convective and shear flow instabilities compete in our system. The convective instabilities are of the well-known columnar structure. Shear flow instabilities for the larger Ekman number are similar to the Taylor-Couette instability in rotating annuli. At the lower Ekman number, shear flow instabilities adopt a geostrophic character, assuming the form of rotating columns, similar to the convective instabilities. While the convective instability always sets in outside the tangent cylinder, shear instability can become unstable inside the tangent cylinder. If even a weak zonal flow is present inside the tangent cylinder, the flow is unstable to shear instability. This offers an explanation why the jovian zonal jet structure is much weaker at the higher latitudes that correspond to locations inside the tangent cylinder.     

Retrievals of atmospheric variables on the gas giants from ground-based mid-infrared imaging

Thermal-infrared imaging of Jupiter and Saturn using the NASA/IRTF and Subaru observatories are quantitatively analyzed to assess the capabilities for reproducing and extending the zonal mean atmospheric results of the Cassini/CIRS experiment. We describe the development of a robust, systematic and reproducible approach to the acquisition and reduction of planetary images in the mid-infrared (7-25 μm), and perform an adaptation and validation of the optimal estimation, correlated-k retrieval algorithm described by Irwin et al. [Irwin, P., Teanby, N., de Kok, R., Fletcher, L., Howett, C., Tsang, C., Wilson, C., Calcutt, S., Nixon, C., Parrish, P., 2008. J. Quant. Spectrosc. Radiat. Trans. 109 (6), 1136-1150] for channel-integrated radiances. Synthetic spectral analyses and a comparison to Cassini results are used to verify our abilities to retrieve temperatures, haze opacities and gaseous abundances from filtered imaging. We find that ground-based imaging with a sufficiently high spatial resolution is able to reproduce the three-dimensional temperature and para-H₂ fields measured by spacecraft visiting Jupiter and Saturn, allowing us to investigate vertical wind shear, pressure and, with measured cloud-top winds, Ertel potential vorticity on potential temperature surfaces. Furthermore, by scaling vertical profiles of NH₃, PH₃, haze opacity and hydrocarbons as free parameters during thermal retrievals, we can produce meridional results comparable with CIRS spectroscopic investigations. This paper demonstrates that mid-IR imaging instruments operating at ground-based observatories have access to several dynamical and chemical diagnostics of the atmospheric state of the gas giants, offering the prospect for quantitative studies over much longer baselines and often covering much wider areas than is possible from spaceborne platforms.     

Stability of jets on Jupiter and Saturn

The linear stability of a zonal jet that decays with depth is investigated under the assumption that the thermal stratification is very small. A westerly cosine jet is found to be more stable than it is in a thin fluid shell with two-dimensional flow. This is in agreement with observations of Jupiter and Saturn, where jet curvature exceeds the barotropic stability criterion. This result constitutes an alternative hypothesis to that of Ingersoll and Pollard [Icarus 52 (1982) 62], who showed that deep jets extending through the interior are also more stable than thin shell jets. The flow regime assumed in the present work requires that a small stratification can exist and persist even in the presence of horizontal temperature gradients. Further work will be needed to test whether this is realistic.