Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-01T03:22:31.860Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  24 November 2016

Eli Galanti ,
Yohai Kaspi  and
Eli Tziperman
Eli Galanti*
Affiliation:
Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot 76100, Israel
Yohai Kaspi
Affiliation:
Department of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot 76100, Israel
Eli Tziperman
Affiliation:
Department of Earth and Planetary Sciences and School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
*
†Email address for correspondence: eli.galanti@weizmann.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

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 2016–2018. 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 baroclinic wind structures to be analysed, 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 analysed, and the results are compared in the barotropic limit with 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.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 810 , 10 January 2017 , pp. 175 - 195
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Busse, F. H. 1976 A simple model of convection in the Jovian atmosphere. Icarus 29, 255–260.CrossRefGoogle Scholar
Cao, H. & Stevenson, D. J.2016 Gravity and zonal flows of giant planets: from the Euler equation to the thermal wind equation. arXiv:1508.02764.CrossRefGoogle Scholar
Cho, J. & Polvani, L. M. 1996 The formation of jets and vortices from freely-evolving shallow water turbulence on the surface of a sphere. Phys. Fluids 8, 1531–1552.CrossRefGoogle Scholar
Galanti, E. & Kaspi, Y. 2016a An adjoint based method for the inversion of the Juno and Cassini gravity measurements into wind fields. Astrophys. J. 820 (2), 91.CrossRefGoogle Scholar
Galanti, E. & Kaspi, Y. 2016b Deciphering Jupiter’s deep flow dynamics using the upcoming Juno gravity measurements and an adjoint based dynamical model. Icarus (submitted).Google Scholar
Guillot, T. 2005 The interiors of giant planets: models and outstanding questions. Annu. Rev. Earth Planet. Sci. 33, 493–530.CrossRefGoogle Scholar
Heimpel, M., Gastine, T. & Wicht, J. 2016 Simulation of deep-seated zonal jets and shallow vortices in gas giant atmospheres. Nature Geosci. 9, 19–23.CrossRefGoogle Scholar
Helled, R. & Guillot, T. 2013 Interior models of Saturn: including the uncertainties in shape and rotation. Astrophys. J. 767, 113.CrossRefGoogle Scholar
Hubbard, W. B. 1982 Effects of differential rotation on the gravitational figures of Jupiter and Saturn. Icarus 52, 509–515.CrossRefGoogle Scholar
Hubbard, W. B. 1999 Note: gravitational signature of Jupiter’s deep zonal flows. Icarus 137, 357–359.CrossRefGoogle Scholar
Hubbard, W. B. 2012 High-precision Maclaurin-based models of rotating liquid planets. Astrophys. J. Lett. 756, L15.CrossRefGoogle Scholar
Hubbard, W. B. 2013 Concentric maclaurian spheroid models of rotating liquid planets. Astrophys. J. 768 (1).CrossRefGoogle Scholar
Hubbard, W. B., Schubert, G., Kong, D. & Zhang, K. 2014 On the convergence of the theory of figures. Icarus 242, 138–141.CrossRefGoogle Scholar
Ingersoll, A. P. & Pollard, D. 1982 Motion in the interiors and atmospheres of Jupiter and Saturn: scale analysis, anelastic equations, barotropic stability criterion. Icarus 52, 62–80.CrossRefGoogle Scholar
Kaspi, Y. 2013 Inferring the depth of the zonal jets on Jupiter and Saturn from odd gravity harmonics. Geophys. Res. Lett. 40, 676–680.CrossRefGoogle Scholar
Kaspi, Y., Davighi, J. E., Galanti, E. & Hubbard, W. B. 2016 The gravitational signature of internal flows in giant planets: comparing the thermal wind approach with barotropic potential-surface methods. Icarus 276, 170–181.CrossRefGoogle Scholar
Kaspi, Y. & Flierl, G. R. 2007 Formation of jets by baroclinic instability on gas planet atmospheres. J. Atmos. Sci. 64, 3177–3194.CrossRefGoogle Scholar
Kaspi, Y., Hubbard, W. B., Showman, A. P. & Flierl, G. R. 2010 Gravitational signature of Jupiter’s internal dynamics. Geophys. Res. Lett. 37, L01204.CrossRefGoogle Scholar
Kaspi, Y., Showman, A. P., Hubbard, W. B., Aharonson, O. & Helled, R. 2013 Atmospheric confinement of jet-streams on Uranus and Neptune. Nature 497, 344–347.CrossRefGoogle ScholarPubMed
Kong, D., Zhang, K. & Schubert, G. 2012 On the variation of zonal gravity coefficients of a giant planet caused by its deep zonal flows. Astrophys. J. 748 (2), 143.CrossRefGoogle Scholar
Lian, Y. & Showman, A. P. 2010 Generation of equatorial jets by large-scale latent heating on the giant planets. Icarus 207, 373–393.CrossRefGoogle Scholar
Liu, J. & Schneider, T. 2010 Mechanisms of jet formation on the giant planets. J. Atmos. Sci. 67, 3652–3672.CrossRefGoogle Scholar
Liu, J., Schneider, T. & Fletcher, L. N. 2014 Constraining the depth of Saturn’s zonal winds by measuring thermal and gravitational signals. Icarus 239, 260–272.CrossRefGoogle Scholar
Liu, J., Schneider, T. & Kaspi, Y. 2013 Predictions of thermal and gravitational signals of Jupiter’s deep zonal winds. Icarus 224, 114–125.CrossRefGoogle Scholar
Militzer, B., Hubbard, W. B., Vorberger, J., Tamblyn, I. & Bonev, S. A. 2008 A massive core in Jupiter predicted from first-principles simulations. Astrophys. J. 688, L45–L48.CrossRefGoogle Scholar
Nettelmann, N., Becker, A., Holst, B. & Redmer, R. 2012 Jupiter models with improved ab initio hydrogen equation of state (H-REOS.2). Astrophys. J. 750, 52.CrossRefGoogle Scholar
Porco, C. C., West, R. A., McEwen, A., Del Genio, A. D., Ingersoll, A. P., Thomas, P., Squyres, S., Dones, L., Murray, C. D., Johnson, T. V. et al. 2003 Cassini imaging of Jupiter’s atmosphere, satellites and rings. Science 299, 1541–1547.CrossRefGoogle ScholarPubMed
Scott, R. K. & Polvani, L. M. 2007 Forced-dissipative shallow-water turbulence on the sphere and the atmospheric circulation of the giant planets. J. Atmos. Sci. 64, 3158–3176.CrossRefGoogle Scholar
Showman, A. P., Gierasch, P. J. & Lian, Y. 2006 Deep zonal winds can result from shallow driving in a giant-planet atmosphere. Icarus 182, 513–526.CrossRefGoogle Scholar
Showman, A. P., Kaspi, Y., Achterberg, R. & Ingersoll, A. P. 2016 The global atmospheric circulation of Saturn. In Saturn in the 21st Century. Cambridge University Press.Google Scholar
Strang, G. 2006 Linear Algebra and its Applications, 4th edn. Thomson, Brooks/Cole.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics, p. 770. Cambridge University Press.CrossRefGoogle Scholar
Vasavada, A. R. & Showman, A. P. 2005 Jovian atmospheric dynamics: an update after Galileo and Cassini. Rep. Prog. Phys. 68, 1935–1996.CrossRefGoogle Scholar
Williams, G. P. 1978 Planetary circulations: 1. Barotropic representation of the Jovian and terrestrial turbulence. J. Atmos. Sci. 35, 1399–1426.2.0.CO;2>CrossRefGoogle Scholar
Wisdom, J. & Hubbard, W. B. 2016 Differential rotation in Jupiter: a comparison of methods. Icarus 267, 315–322.CrossRefGoogle Scholar
Zhang, K., Kong, D. & Schubert, G. 2015 Thermal-gravitational wind equation for the wind-induced gravitational signature of giant gaseous planets: mathematical derivation, numerical method and illustrative solutions. Astrophys. J. 806, 270–279.CrossRefGoogle Scholar