- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Rapidly rotating self-gravitating Boussinesq fluid. II. Onset of thermal inertial convection in oblate spheroidal cavities

Wenbo Li

Dali Kong*

  • State Key Laboratory of Lunar and Planetary Sciences, Macau University of Science and Technology, Taipa, Macao 999078, China; CNSA Macau Center for Space Exploration and Science, Macao 999078, China; and CAS Key Laboratory of Planetary Sciences, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, 200030, China

  • CAS Key Laboratory of Planetary Sciences, Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, 200030, China

  • *dkong@shao.ac.cn
PDF

Phys. Rev. Fluids 7, 103502 – Published 27 October, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.103502

Abstract

The problem of thermal instability in rapidly rotating, self-gravitating fluid bodies has been widely modeled in spheres or spherical shells, which implicitly neglects the flattening effect due to the centrifugal force. In our previous paper [Kong, Phys. Rev. Fluids 7, 074803 (2022)], by self-consistently taking into account the centrifugal force, rapidly rotating stably stratified Boussinesq fluid was modeled in oblate spheroidal cavities whose geometric shapes are determined by the theory of figure. A closed-form solution was obtained for gravity and temperature. The stable stratification was demonstrated to be motionless in the corotating frame of reference. Based on this nonspherical model of the conduction state in a rapidly rotating spheroidal cavity, the problem of thermal instability is formulated and discussed by this paper in the regime of inertial convection, which is marked by asymptotically small Ekman number and sufficiently small Prandtl number. The critical properties of inertial modes are explicitly derived. The dependence of the onset of thermal inertial convection on the oblateness of spheroid is systematically explored.

View figure in article

Physics Subject Headings (PhySH)

See Also

Rapidly rotating self-gravitating Boussinesq fluid. III. A previously unknown zonal oscillation at the onset of rotating convection

Wenbo Li and Dali Kong
Phys. Rev. Fluids 8, L011501 (2023)

Article Text

References (73)

  1. D. Gubbins, T. G. Masters, and J. A. Jacobs, Thermal evolution of the Earth's core, Geophys. J. Int. 59, 57 (1979).
  2. S. Müller, R. Helled, and A. Cumming, The challenge of forming a fuzzy core in Jupiter, Astron. Astrophys. 638, A121 (2020).
  3. M. Bouffard, G. Choblet, S. Labrosse, and J. Wicht, Chemical convection and stratification in the Earth's outer core, Front. Earth Sci. 7, 99 (2019).
  4. C. A. Jones, Planetary magnetic fields and fluid dynamos, Annu. Rev. Fluid Mech. 43, 583 (2011).
  5. N. Schaeffer, D. Jault, H.-C. Nataf, and A. Fournier, Turbulent geodynamo simulations: A leap towards Earth's core, Geophys. J. Inte. 211, 1 (2017).
  6. J. Wicht and S. Sanchez, Advances in geodynamo modelling, Geophys. Astrophys. Fluid Dyn. 113, 2 (2019).
  7. K. Zhang and G. Schubert, Magnetohydrodynamics in rapidly rotating spherical systems, Annu. Rev. Fluid Mech. 32, 409 (2000).
  8. J. Sánchez Umbría and M. Net, Continuation of double Hopf points in thermal convection of rotating fluid spheres, SIAM J. Appl. Dyn. Syst. 20, 208 (2021).
  9. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, UK, 1962)
  10. P. H. Roberts, On the thermal instability of a rotating-fluid sphere containing heat sources, Phil. Trans. R. Soc. Lond. A 263, 93 (1968).
  11. F. Busse, Thermal instabilities in rapidly rotating systems, J. Fluid Mech. 44, 441 (1970).
  12. A. M. Soward, On the finite amplitude thermal instability of a rapidly rotating fluid sphere, Geophys. Astrophys. Fluid Dyn. 9, 19 (1977).
  13. J.-I. Yano, Asymptotic theory of thermal convection in rapidly rotating systems, J. Fluid Mech. 243, 103 (1992).
  14. K. Zhang, On coupling between the Poincaré equation and the heat equation, J. Fluid Mech. 268, 211 (1994).
  15. K. Zhang, On coupling between the Poincaré equation and the heat equation: Nonslip boundary condition, J. Fluid Mech. 284, 239 (1995).
  16. C. A. Jones, A. W. Soward, and A. I. Mussa, The onset of thermal convection in a rapidly rotating sphere, J. Fluid Mech. 405, 157 (2000).
  17. K. Zhang and X. Liao, A new asymptotic method for the analysis of convection in a rapidly rotating sphere, J. Fluid Mech. 518, 319 (2004).
  18. E. Dormy, A. M. Soward, C. A. Jones, D. Jault, and P. Cardin, The onset of thermal convection in rotating spherical shells, J. Fluid Mech. 501, 43 (2004).
  19. P. Olson, 8.01-core dynamics: An introduction and overview, in Treatise on Geophysics, 2nd ed., edited by G. Schubert (Elsevier, Oxford, UK, 2015), pp. 1–25.
  20. F. Busse, A simple model of convection in the Jovian atmosphere, Icarus 29, 255 (1976).
  21. A. P. Ingersoll and D. Pollard, Motion in the interiors and atmospheres of Jupiter and Saturn: Scale analysis, anelastic equations, barotropic stability criterion, Icarus 52, 62 (1982).
  22. K. Zhang, Spiralling columnar convection in rapidly rotating spherical fluid shells, J. Fluid Mech. 236, 535 (1992).
  23. M. Heimpel, J. Aurnou, and J. Wicht, Simulation of equatorial and high-latitude jets on Jupiter in a deep convection model, Nature (London) 438, 193 (2005).
  24. T. Gastine, J. Wicht, and J. Aubert, Scaling regimes in spherical shell rotating convection, J. Fluid Mech. 808, 690 (2016).
  25. S. Maffei, A. Jackson, and P. W. Livermore, Characterization of columnar inertial modes in rapidly rotating spheres and spheroids, Proc. R. Soc. A. 473, 20170181 (2017).
  26. C. Guervilly, P. Cardin, and N. Schaeffer, Turbulent convective length scale in planetary cores, Nature (London) 570, 368 (2019).
  27. Y. Lin and A. Jackson, Large-scale vortices and zonal flows in spherical rotating convection, J. Fluid Mech. 912, A46 (2020).
  28. S. Chandrasekhar, Ellipsoidal figures of equilibrium-an historical account, Comm. Pure Appl. Math. 20, 251 (1967).
  29. H. Lamb, Hydrodynamics, Dover Books on Physics (Dover Publications, Oxford, UK, 1945).
  30. W. B. Hubbard, G. Schubert, D. Kong, and K. Zhang, On the convergence of the theory of figures, Icarus 242, 138 (2014).
  31. N. Nettelmann, N. Movshovitz, D. Ni et al., Theory of figures to the seventh order and the interiors of Jupiter and Saturn, Planet. Sci. J. 2, 241 (2021).
  32. A. Jackson, A. Sheyko, P. Marti et al., A spherical shell numerical dynamo benchmark with pseudo-vacuum magnetic boundary conditions, Geophys. J. Int. 196, 712 (2014).
  33. H. Matsui, E. Heien, J. Aubert et al., Performance benchmarks for a next generation numerical dynamo model, Geochem. Geophys. Geosyst. 17, 1586 (2016).
  34. K. H. Chan, K. Zhang, L. Li, and X. Liao, A new generation of convection-driven spherical dynamos using EBE finite element method, Phys. Earth Planet. Inter. 163, 251 (2007).
  35. K. H. Chan, Y. He, K. Zhang, and J. Zou, A finite element analysis on fluid motion in librating triaxial ellipsoids, Numer. Methods Partial Differ. Equat. 30, 1518 (2014).
  36. E. H. von Zeipel, The radiative equilibrium of a rotating system of gaseous masses, Mon. Not. R. Astron. Soc. 84, 665 (1924).
  37. A. Eddington, Circulating currents in rotating stars, Observatory 48, 73 (1925).
  38. J. H. Jeans, On a theorem of v. zeipel on radiative equilibrium, Mon. Not. R. Astron. Soc. 85, 526 (1925).
  39. M. Schwarzschild, On stellar rotation, Astrophys. J. 95, 441 (1942).
  40. E. J. Öpik, Rotational currents, Mon. Not. R. Astron. Soc. 111, 278 (1951).
  41. F. Busse, Do eddington-sweet circulations exist? Geophys. Astrophys. Fluid Dyn. 17, 215 (1981).
  42. J.-L. Tassoul and M. Tassoul, Meridional circulation in rotating stars. I. A boundary layer analysis of mean steady motions in early-type stars, Astrophys. J. Suppl. Series 49, 317 (1982).
  43. H. C. Spruit and E. Knobloch, Baroclinic instability in stars, Astron. Astrophys. 132, 89 (1984).
  44. U. Schneider, Baroclinic zonal currents in rotating stars, Astron. Astrophys. 238, 142 (1990).
  45. J. P. Zahn, Circulation and turbulence in rotating stars, Astron. Astrophys. 265, 115 (1992).
  46. M. Rieutord, The dynamics of the radiative envelope of rapidly rotating stars, I. A spherical boussinesq model, Astron. Astrophys. 451, 1025 (2006).
  47. R. D. Simitev and F. H. Busse, Baroclinially-driven flows and dynamo action in rotating spherical fluid shells, Geophys. Astrophys. Fluid Dyn. 111, 369 (2017).
  48. D. Kong, Rapidly rotating self-gravitating Boussinesq fluid: A nonspherical model of motionless stable stratification, Phys. Rev. Fluids 7, 074803 (2022).
  49. A. Meilland, P. Stee, M. Vannier et al., First direct detection of a Keplerian rotating disk around the be star α Arae using Amber/VLTI, Astron. Astrophys. 464, 59 (2007).
  50. P. Kervella, A. Domiciano de Souza, S. Kanaan et al., The environment of the fast rotating star Achernar, II. Thermal infrared interferometry with VLTI/MIDI, Astron. Astrophys. 493, L53 (2009).
  51. A. Maeder, C. Georgy, and D. Meynet, Convective envelopes in rotating OB stars, Astron. Astrophys. 479, L37 (2008).
  52. S. M. Wahl, W. B. Hubbard, B. Militzer et al., Comparing Jupiter interior structure models to Juno gravity measurements and the role of a dilute core, Geophys. Res. Lett. 44, 4649 (2017).
  53. L. Iess, W. M. Folkner, D. Durante et al., Measurement of Jupiter's asymmetric gravity field, Nature (London) 555, 220 (2018).
  54. D. Kong, K. Zhang, G. Schubert, and J. D. Anderson, Origin of Jupiter's cloud-level zonal winds remains a puzzle even after Juno, Proc. Natl. Acad. Sci. USA 115, 8499 (2018).
  55. L. Iess, B. Militzer, Y. Kaspi et al., Measurement and implications of Saturn's gravity field and ring mass, Science 364, eaat2965 (2019).
  56. D. Kong, K. Zhang, and G. Schubert, A fully self-consistent multi-layered model of Jupiter, Astrophys. J. 826, 127 (2016).
  57. D. Kong, K. Zhang, and G. Schubert, Depth of the dynamo region and zonal circulation of the molecular layer in Saturn inferred from its equatorially symmetric gravitational field, Month. Not. Roy. Astron. Soc. 488, 5633 (2019).
  58. D. Ni, Understanding Saturn's interior from the Cassini grand finale gravity measurements, Astron. Astrophys. 639, A10 (2020).
  59. K. Zhang, X. Liao, and P. Earnshaw, On inertial waves and oscillations in a rapidly rotating spheroid, J. Fluid Mech. 504, 1 (1999).
  60. K. Zhang, K. Lam, and D. Kong, Asymptotic theory for torsional convection in rotating fluid spheres, J. Fluid Mech. 813, R2 (2017).
  61. K.-K. Zhang and F. H. Busse, On the onset of convection in rotating spherical shells, Geophys. Astrophys. Fluid Dynam. 39, 119 (1987).
  62. K. Zhang, X. Liao, and F. H. Busse, Asymptotic solutions of convection in rapidly rotating nonslip spheres, J. Fluid Mech. 578, 371 (2007).
  63. K. Zhang and X. Liao, Theory and Modeling of Rotating Fluids: Convection, Inertial Waves and Precession (Cambridge University Press, Cambridge, UK, 2017).
  64. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, CA, 1957).
  65. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Government Printing Office, Washington, D.C., 1964) Chap. 21.
  66. K. Chan, K. Zhang, and X. Liao, An EBE finite element method for simulating nonlinearflows in rotating spheroidal cavities, Int. J. Numer. Methods Fluids 63, 395 (2010).
  67. U. M. Ascher and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (SIAM, Philadelphia, PA, 1998).
  68. V. John, Finite Element Spaces for Linear Saddle Point Problems (Springer, Cham, 2016).
  69. G. L. G. Sleijpen and D. R. Fokkema, Bicgstab(l) for linear equations involving unsymmetric matrices with complex spectrum, Electr. Trans. Numer. Anal. 1, 11 (1993).
  70. D. Kong, X. Liao, and K. Zhang, The sidewall-localized mode in a resonant precessing cylinder, Phys. Fluids 26, 051703 (2014).
  71. E. J. Kaplan, N. Schaeffer, J. Vidal, and P. Cardin, Subcritical Thermal Convection of Liquid Metals in a Rapidly Rotating Sphere, Phys. Rev. Lett. 119, 094501 (2017).
  72. D. Kong, K. Zhang, K. Lam, and A. P. Willis, Axially symmetric and latitudinally propagating nonlinear patterns in rotating spherical convection, Phys. Rev. E 98, 031101(R) (2018).
  73. P. H. Roberts, On the thermal instability of a highly rotating fluid sphere, Astrophys. J. 141, 240 (1965).

Outline

Information

Phys. Rev. Fluids 7, 103502– Published 27 October, 2022
  • Received 9 July 2022
  • Accepted 12 September 2022
Crossmark - Check for updates button

©2022 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation