On the convergence of the theory of figures

Hubbard, W. B.; Schubert, G.; Kong, Dali; Zhang, K.

doi:10.1016/j.icarus.2014.08.014

Cited by 30 publications

(25 citation statements)

References 5 publications

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“…In summary, deciphering the effect of the atmospheric and internal flows from the measured gravity spectrum of Jupiter and Saturn provides a major challenge. The methods suggested to date have been either limited to barotropic cases (e.g., Hubbard, 1982Hubbard, , 1999Hubbard, , 2012Kong et al, 2012;Hubbard et al, 2014), or approximations limited to spherical symmetry or partial solutions (e.g., Kaspi et al, 2010;Zhang et al, 2015;Cao and Stevenson, 2015). Here, we have developed a self-consistent perturbation approach to the thermal wind balance that incorporates all physical effects, including the effects of oblateness on the dynamics and the gravity perturbation induced by the flow itself.…”

Section: Discussionmentioning

confidence: 99%

A full, self-consistent treatment of thermal wind balance on oblate fluid planets

2016

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The nature of the flow below the cloud level on Jupiter and Saturn is still unknown. Relating the flow on these planets to perturbations in their density field is key to the analysis of the gravity measurements expected from both the Juno (Jupiter) and Cassini (Saturn) spacecrafts during 2016-18. Both missions will provide latitude-dependent gravity fields, which in principle could be inverted to calculate the vertical structure of the observed cloud-level zonal flow on these planets. Theories to date connecting the gravity field and the flow structure have been limited to potential theories under a barotropic assumption, or estimates based on thermal wind balance that allow analyzing baroclinic wind structures, but have made simplifying assumptions that neglected several physical effects. These include the effects of the deviations from spherical symmetry, the centrifugal force due to density perturbations, and self-gravitational effects of the density perturbations. Recent studies attempted to include some of these neglected terms, but lacked an overall approach that is able to include all effects in a self-consistent manner. The present study introduces such a self-consistent perturbation approach to the thermal wind balance that incorporates all physical effects, and applies it to several example wind structures, both barotropic and baroclinic. The contribution of each term is analyzed, and the results are compared in the barotropic limit to those of potential theory. It is found that the dominant balance involves the original simplified thermal wind approach. This balance produces a good order-of-magnitude estimate of the gravitational moments, and is able, therefore, to address the order one question of how deep the flows are given measurements of gravitational moments. The additional terms are significantly smaller yet can affect the gravitational moments to some degree. However, none of these terms is dominant so any approximation attempting to improve over the simplified thermal wind approach needs to include all other terms.

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Section: Discussionmentioning

confidence: 99%

A full, self-consistent treatment of thermal wind balance on oblate fluid planets

2016

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“…While the spheroidal-shape approximation has only small effects on the lower-order gravitational coefficients such as J 2 , the small-scale density variation resulting from the non-spheroidal shape can make a substantial contribution to the high-order gravitational coefficients J n with  n 10 ( Kong et al 2015). Hubbard (2013) developed a radially discontinuous numerical method, which, because of the convergence radius of the expansion (Hubbard et al 2014), is valid only for moderate angular velocity Ω but applicable to all giant planets in the solar system. By dividing a rotating body into a number of concentric incompressible layers within which the density is assumed constant, Hubbard (2013) iterated over the shapes of the concentric layers until all the interfaces became equipotential surfaces.…”

Section: Introductionmentioning

confidence: 99%

A Fully Self-Consistent Multi-Layered Model of Jupiter

Kong

Zhang

Schubert

2016

ApJ

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We construct a three-dimensional, fully self-consistent, multi-layered, non-spheroidal model of Jupiter consisting of an inner core, a metallic electrically conducting dynamo region, and an outer molecular electrically insulating envelope. We assume that the Jovian zonal winds are on cylinders parallel to the rotation axis but, due to the effect of magnetic braking, are confined within the outer molecular envelope. We also assume that the location of the molecular-metallic interface is characterized by its equatorial radius HR e , where R e is the equatorial radius of Jupiter at the 1 bar pressure level and H is treated as a parameter of the model. We solve the relevant mathematical problem via a perturbation approach. The leading-order problem determines the density, size, and shape of the inner core, the irregular shape of the 1 bar pressure level, and the internal structure of Jupiter that accounts for the full effect of rotational distortion, but without the influence of the zonal winds; the next-order problem determines the variation of the gravitational field solely caused by the effect of the zonal winds on the rotationally distorted non-spheroidal Jupiter. The leading-order solution produces the known mass, the known equatorial and polar radii, and the known zonal gravitational coefficient J 2 of Jupiter within their error bars; it also yields the coefficients J 4 and J 6 within about 5% accuracy, the core equatorial radius R 0.09 e and the core density r =´-2.0 10 kg m corresponding to 3.73 Earth masses; the next-order solution yields the wind-induced variation of the zonal gravitational coefficients of Jupiter.

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“…using spherical coordinates so that φ is longitude and µ = cos θ. Note that unlike the dynamical gravity moments, the static gravity moments are dominated by the oblate shape of the planet, and therefore needs to be calculated by other methods (e.g., Zharkov and Trubitsyn 1978;Kong et al 2012;Hubbard 2012;Hubbard et al 2014;Wisdom and Hubbard 2016).…”

Section: Methodsmentioning

confidence: 99%

An Adjoint-Based Method for the Inversion of the Juno and Cassini Gravity Measurements Into Wind Fields

Galanti

Kaspi

2016

ApJ

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During 2016-17 the Juno and Cassini spacecraft will both perform close eccentric orbits of Jupiter and Saturn, respectively, obtaining high-precision gravity measurements for these planets. This data will be used to estimate the depth of the observed surface flows on these planets. All models to date, relating the winds to the gravity field, have been in the forward direction, thus allowing only calculation of the gravity field from given wind models. However, there is a need to do the inverse problem since the new observations will be of the gravity field. Here, an inverse dynamical model is developed to relate the expected measurable gravity field, to perturbations of the density and wind fields, and therefore to the observed cloud-level winds . In order to invert the gravity field into the 3D circulation, an adjoint model is constructed for the dynamical model, thus allowing backward integration. This tool is used for examination of various scenarios, simulating cases in which the depth of the wind depends on latitude. We show that it is possible to use the gravity measurements to derive the depth of the winds, both on Jupiter and Saturn, taking into account also measurement errors. Calculating the solution uncertainties, we show that the wind depth can be determined more precisely in the low-to-midlatitudes. In addition, the gravitational moments are found to be particularly sensitive to flows at the equatorial intermediate depths. Therefore we expect that if deep winds exist on these planets they will have a measurable signature by Juno and Cassini.

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On the convergence of the theory of figures

Cited by 30 publications

References 5 publications

A full, self-consistent treatment of thermal wind balance on oblate fluid planets

A full, self-consistent treatment of thermal wind balance on oblate fluid planets

A Fully Self-Consistent Multi-Layered Model of Jupiter

An Adjoint-Based Method for the Inversion of the Juno and Cassini Gravity Measurements Into Wind Fields

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