1932

Abstract

High-precision gravitational measurements by orbiting spacecraft provide a means of probing the structures, fluid motions, and convective dynamos in the interiors of the rapidly rotating outer planets. Here, the classical theory of rotating homogeneous planets is briefly reviewed. Emphasis is placed on recent developments in theories and methods that relate internal structure and processes to their gravitational signatures. Whereas early theories usually treated the effects of interior density stratification and rotational distortion as perturbations to a spherical state, recent research is marked by a self-consistent perturbation approach in which the leading-order problem accounts exactly for rotational distortion, thereby determining the basic shape, internal structure, and gravitational field of the planet. The next-order problem, which is mathematically and physically coupled with the leading-order problem, describes the modifications caused by internal fluid motions. Although the theories and methods have general applicability, advances have been spurred by the need to have a basis for interpretation of the gravitational data for Jupiter and Saturn expected from the Juno and Cassini missions.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-earth-063016-020305
2017-08-30
2024-04-18
Loading full text...

Full text loading...

/deliver/fulltext/earth/45/1/annurev-earth-063016-020305.html?itemId=/content/journals/10.1146/annurev-earth-063016-020305&mimeType=html&fmt=ahah

Literature Cited

  1. Ansong M, Kleinwächter A, Reinhard M. 2003. Uniformly rotating axisymmetric fluid configurations bifurcating from highly flattened Maclaurin spheroids. MNRAS 339:515–23 [Google Scholar]
  2. Aurnou JM, Olson PL. 2001. Strong zonal winds from thermal convection in a rotating spherical shell. Geophys. Res. Lett. 28:2557–60 [Google Scholar]
  3. Bardeen JM. 1971. A reexamination of the post-Newtonian Maclaurin spheroids. Astrophys. J. 167:425–46 [Google Scholar]
  4. Burke BF, Franklin KL. 1952. Observations of a variable radio source associated with the planet Jupiter. J. Geophys. Res. 60:213–17 [Google Scholar]
  5. Busse FH. 1970. Thermal instabilities in rapidly rotating systems. J. Fluid Mech. 44:441–60 [Google Scholar]
  6. Busse FH. 1976. A simple model of convection in the Jovian atmosphere. Icarus 29:255–60 [Google Scholar]
  7. Chabrier G, Saumon D, Hubbard WB, Lunine JI. 1992. The molecular-metallic transition of hydrogen and the structure of Jupiter and Saturn. Astrophys. J. 391:817–26 [Google Scholar]
  8. Chandrasekhar S. 1933. The equilibrium of distorted polytropes. I. The rotational problem. MNRAS 93:390–406 [Google Scholar]
  9. Chandrasekhar S. 1969. Ellipsoidal Figures of Equilibrium New Haven, CT: Yale Univ. Press
  10. Chandrasekhar S. 1981. Hydrodynamic and Hydromagnetic Stability New York: Dover
  11. Connerney JEP. 1993. Magnetic fields of the outer planets. J. Geophys. Res. 98:18659–79 [Google Scholar]
  12. Cui Z, Zhang K, Liao X. 2014. On the completeness of inertial wave modes in rotating annular channels. Geophys. Astrophys. Fluid Dyn. 108:44–59 [Google Scholar]
  13. Dormy E. 2016. Strong-field spherical dynamos. J. Fluid Mech. 789:500–13 [Google Scholar]
  14. Duarte L, Gastine T, Wicht J. 2013. Anelastic dynamo models with variable electrical conductivity: an application to gas giants. Phys. Earth Planet. Int. 222:22–34 [Google Scholar]
  15. Flammer C. 1957. Spheroidal Wave Functions Stanford, CA: Stanford Univ. Press
  16. Gastine T, Wicht J. 2012. Effects of compressibility on driving zonal flow in gas giants. Icarus 219:428–42 [Google Scholar]
  17. Glatzmaier GA, Evonuk M, Rogers TM. 2009. Differential rotation in giant planets maintained by density-stratified turbulent convection. Geophys. Astrophys. Fluid Dyn. 103:31–51 [Google Scholar]
  18. Greenspan HP. 1968. The Theory of Rotating Fluids Cambridge, UK: Cambridge Univ. Press
  19. Gubbins D, Roberts PH. 1987. Magnetohydrodynamics of the Earth's core. Geomagnetism 2 JA Jacobs 1–183 London: Academic Press [Google Scholar]
  20. Guillot T. 2005. The interiors of giant planets: models and outstanding questions. Annu. Rev. Earth Planet. Sci. 33:493–530 [Google Scholar]
  21. Heimpel M, Aurnou JM, Wicht J. 2005. Simulation of equatorial and high latitude jets on Jupiter in a deep convection model. Nature 438:193–96 [Google Scholar]
  22. Helled R, Schubert G, Anderson JD. 2009. Jupiter and Saturn rotation periods. Planet. Space Sci. 57:1467–73 [Google Scholar]
  23. Hollerbach R. 1996. On the theory of the geodynamo. Phys. Earth Planet. Int. 98:163–85 [Google Scholar]
  24. Horedt GP. 2004. Polytropes: Applications in Astrophysics and Related Fields Dordrecht: Kluwer
  25. Hubbard WB. 1999. Gravitational signature of Jupiter's deep zonal flows. Icarus 137:357–59 [Google Scholar]
  26. Hubbard WB. 2012. High-precision Maclaurin-based models of rotating liquid planets. Astrophys. J. Lett. 756:L15–L17 [Google Scholar]
  27. Hubbard WB. 2013. Concentric Maclaurin spheroid models of rotating liquid planets. Astrophys. J. 768:43–50 [Google Scholar]
  28. Hubbard WB, Schubert G, Kong D, Zhang K. 2014. On the convergence of the theory of figures. Icarus 242:138–41 [Google Scholar]
  29. Ingersoll AP, Cuzzi JN. 1969. Dynamics of Jupiter's cloud band. J. Atmos. Sci. 26:981–85 [Google Scholar]
  30. Ivers DJ, Jackson A, Winch D. 2015. Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere. J. Fluid Mech. 766:468–98 [Google Scholar]
  31. James RA. 1964. The structure and stability of rotating gas masses. Astrophys. J. 140:552–82 [Google Scholar]
  32. Jones CA. 2011. Planetary magnetic fields and fluid dynamos. Annu. Rev. Fluid Mech. 43:583–614 [Google Scholar]
  33. Jones CA. 2014. A dynamo model of Jupiter's magnetic field. Icarus 241:148–59 [Google Scholar]
  34. Jones CA, Kuzanyan KM. 2012. Compressible convection in the deep atmospheres of giant planets. Icarus 204:227–38 [Google Scholar]
  35. Kaspi Y. 2013. Inferring the depth of the zonal jets on Jupiter and Saturn from odd gravity harmonics. Geophys. Res. Lett. 40:676–80 [Google Scholar]
  36. Kaspi Y, Hubbard WB, Showman AP, Flierl GR. 2010. Gravitational signature of Jupiter's internal dynamics. Geophys. Res. Lett. 37:L01204 [Google Scholar]
  37. Kaspi Y, Showman AP, Hubbard WB, Aharonson O, Helled R. 2013. Atmospheric confinement of jet streams on Uranus and Neptune.. Nature 497:344–47 [Google Scholar]
  38. Kong D, Liao X, Zhang K, Schubert G. 2013. Gravitational signature of rotationally distorted Jupiter caused by deep zonal winds. Icarus 226:1425–30 [Google Scholar]
  39. Kong D, Zhang K, Schubert G. 2015a. An exact solution for arbitrarily rotating gaseous polytropes with index unity. MNRAS 448:456–63 [Google Scholar]
  40. Kong D, Zhang K, Schubert G. 2015b. Self-consistent internal structure of a rotating gaseous planet and its comparison with an approximation by oblate spheroidal equidensity surfaces. Phys. Earth Planet. Int. 249:43–50 [Google Scholar]
  41. Kong D, Zhang K, Schubert G. 2015c. Wind-induced odd gravitational harmonics of Jupiter. MNRAS 405:L11–15 [Google Scholar]
  42. Kong D, Zhang K, Schubert G. 2016a. A fully self-consistent, multi-layered model of Jupiter. Astrophys. J. 826:127 [Google Scholar]
  43. Kong D, Zhang K, Schubert G. 2016b. Odd gravitational harmonics of Jupiter: Effects of spherical versus nonspherical geometry and mathematical smoothing of the equatorially antisymmetric zonal winds across the equatorial plane. Icarus 277:416–23 [Google Scholar]
  44. Kong D, Zhang K, Schubert G. 2016c. Using Jupiter's gravitational field to probe the Jovian convective dynamo. Sci. Rep. 6:23497 [Google Scholar]
  45. Kong D, Zhang K, Schubert G. 2017a. On the gravitational signature of zonal flows in Jupiter-like planets: an analytical solution and its numerical validation. Phys. Earth Planet. Inter. 263:1–6 [Google Scholar]
  46. Kong D, Zhang K, Schubert G. 2017b. On the interpretation of the equatorially antisymmetric Jovian gravitational field. MNRAS 469:716–20 [Google Scholar]
  47. Lagrange A-M, Bonnefoy M, Chauvin G, Apai D, Ehrenreich D. et al. 2010. A giant planet imaged in the disk of the young star β Pictoris. Science 329:57–59 [Google Scholar]
  48. Lamb H. 1932. Hydrodynamics Cambridge, UK: Cambridge Univ. Press
  49. Lian Y, Showman AP. 2010. Generation of equatorial jets by large-scale latent heating on the giant planets. Icarus 207:373–93 [Google Scholar]
  50. Liu J, Goldreich PM, Stevenson DJ. 2008. Constraints on deep-seated zonal winds inside Jupiter and Saturn. Icarus 196:653–64 [Google Scholar]
  51. Moffatt HK. 1978. Magnetic Field Generation in Electrically Conducting Fluids Cambridge, UK: Cambridge Univ. Press
  52. Monaghan JJ, Roxburgh IW. 1965. The structure of rapidly rotating polytropes. MNRAS 131:13–22 [Google Scholar]
  53. Nettelmann N, Becker A, Holst B, Redmer R. 2012. Jupiter models with improved ab initio hydrogen equation of state (H-REOS.2). Astrophys. J. 750:1388–94 [Google Scholar]
  54. Öpik EJ. 1962. Jupiter: chemical composition, structure, and origin of a giant planet. Icarus 1:200–57 [Google Scholar]
  55. Ostriker JP, Mark JW-K. 1968. Rapidly rotating stars. I. The self-consistent-field method. Astrophys. J. 151:1075–88 [Google Scholar]
  56. Porco CC, West RA, McEwen A, Del Genio AD, Ingersoll AP. et al. 2003. Cassini imaging of Jupiter's atmosphere, satellites, and rings. Science 299:1541–47 [Google Scholar]
  57. Roberts PH. 1962. On the superpotential and supermatrix of a heterogeneous ellipsoid. Astrophys. J. 136:1108–14 [Google Scholar]
  58. Roberts PH. 1963a. On highly rotating polytropes I. Astrophys. J. 137:1129–41 [Google Scholar]
  59. Roberts PH. 1963b. On highly rotating polytropes II. Astrophys. J. 138:809–19 [Google Scholar]
  60. Roberts PH, Soward AM. 1992. Dynamo theory. Annu. Rev. Fluid Mech. 24:459–512 [Google Scholar]
  61. Saumon D, Guillot T. 2004. Shock compression of deuterium and the interiors of Jupiter and Saturn. Astrophys. J. 609:1170–80 [Google Scholar]
  62. Seidelmann PK, Archinal BA, A'Hearn MF, Conrad A, Consolmagno GJ. et al. 2007. Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2006. Celestial Mech. Dyn. Astron. 98:155–80 [Google Scholar]
  63. Snellen IAG, Brandl BR, de Kok RJ, Brogi M, Birkby J. et al. 2014. Fast spin of the young extrasolar planet β Pictoris b. Nature 509:63–65 [Google Scholar]
  64. Stanley S, Glatzmaier GA. 2010. Dynamo models for planets other than Earth. Space Sci. Rev. 152:617–49 [Google Scholar]
  65. Stevenson DJ. 1982. Interiors of the giant planets. Annu. Rev. Earth Planet. Sci. 10:257–95 [Google Scholar]
  66. Stevenson DJ. 2003. Planetary magnetic fields. Earth Planet. Sci. Lett. 208:1–11 [Google Scholar]
  67. Tassoul J-L. 1978. Theory of Rotating Stars Princeton, NJ: Princeton Univ. Press
  68. Van Buren AL, Boisvert JE. 2002. Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Q. Appl. Math. 60:589–99 [Google Scholar]
  69. Van Buren AL, Boisvert JE. 2004. Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. Q. Appl. Math. 62:493–507 [Google Scholar]
  70. Zhang K. 1992. Spiralling columnar convection in rapidly rotating spherical fluid shells. J. Fluid Mech. 236:535–56 [Google Scholar]
  71. Zhang K, Busse F. 1989. Convection driven magnetohydrodynamic dynamos in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 49:97–116 [Google Scholar]
  72. Zhang K, Jones CA. 1996. On small Roberts number magnetoconvection in rapidly rotating systems. Proc. R. Soc. Lond. 452:981–95 [Google Scholar]
  73. 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. 806:270–79 [Google Scholar]
  74. Zhang K, Liao X. 2004. A new asymptotic method for the analysis of convection in a rotating sphere. J. Fluid Mech. 518:319–46 [Google Scholar]
  75. Zhang K, Liao X, Earnshaw P. 2004. On inertial waves and oscillations in a rapidly rotating fluid spheroid. J. Fluid Mech. 504:1–40 [Google Scholar]
  76. Zhang K, Schubert G. 1995. Spatial symmetry breaking in rapidly rotating convective spherical shells. Geophys. Res. Lett. 22:1265–68 [Google Scholar]
  77. Zhang K, Schubert G. 1996. Penetrative convection and zonal flow on Jupiter. Science 273:941–43 [Google Scholar]
  78. Zhang K, Schubert G. 2000.. Magnetohydrodynamics in rapidly rotating spherical systems. Annu. Rev. Fluid Mech. 32:411–45 [Google Scholar]
  79. Zharkov VN, Trubitsyn VP. 1978.. Physics of Planetary Interiors WB Hubbard. Tucson, AZ: Pachart
/content/journals/10.1146/annurev-earth-063016-020305
Loading
/content/journals/10.1146/annurev-earth-063016-020305
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error