Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-25T17:19:21.182Z Has data issue: false hasContentIssue false
  • English
  • Français

Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits

Published online by Cambridge University Press:  06 November 2017

Jérémie Vidal Open the ORCID record for Jérémie Vidal [Opens in a new window]  and
David Cébron Open the ORCID record for David Cébron [Opens in a new window]
Jérémie Vidal
Affiliation:
Université Grenoble Alpes, CNRS, ISTerre, F-38000 Grenoble, France
David Cébron
Affiliation:
Université Grenoble Alpes, CNRS, ISTerre, F-38000 Grenoble, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the hydrodynamic stability of homogeneous, incompressible and rotating ellipsoidal fluid masses. The latter are the simplest models of fluid celestial bodies with internal rotation and subjected to tidal forces. The classical problem is the stability of Roche–Riemann ellipsoids moving on circular Kepler orbits. However, previous stability studies have to be reassessed. Indeed, they only consider global perturbations of large wavelength or local perturbations of short wavelength. Moreover many planets and stars undergo orbital motions on eccentric Kepler orbits, implying time-dependent ellipsoidal semi-axes. This time dependence has never been taken into account in hydrodynamic stability studies. In this work we overcome these stringent assumptions. We extend the hydrodynamic stability analysis of rotating ellipsoids to the case of eccentric orbits. We have developed two open-source and versatile numerical codes to perform global and local inviscid stability analyses. They give sufficient conditions for instability. The global method, based on an exact and closed Galerkin basis, handles rigorously global ellipsoidal perturbations of unprecedented complexity. Tidally driven and libration-driven elliptical instabilities are first recovered and unified within a single framework. Then we show that new global fluid instabilities can be triggered in ellipsoids by tidal effects due to eccentric Kepler orbits. Their existence is confirmed by a local analysis and direct numerical simulations of the fully nonlinear and viscous problem. Thus a non-zero orbital eccentricity may have a strong destabilising effect in celestial fluid bodies, which may lead to space-filling turbulence in most of the parameters range.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Parametric instability Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 833 , 25 December 2017 , pp. 469 - 511
Copyright
© 2017 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

Aizenman, M. L. 1968 The equilibrium and the stability of the Roche–Riemann ellipsoids. Astrophys. J. 153, 511–544.Google Scholar
Backus, G., Parker, R. L. & Constable, C. 1996 Foundations of Geomagnetism. Cambridge University Press.Google Scholar
Backus, G. & Rieutord, M. 2017a Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid. Phys. Rev. E 95 (5), 053116.Google Scholar
Backus, G. & Rieutord, M. 2017b Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid. Phys. Rev. E 95 (5), 053116.Google Scholar
Barker, A. J. 2016a Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability. Mon. Not. R. Astron. Soc. 459 (1), 939–956.Google Scholar
Barker, A. J. 2016b On turbulence driven by axial precession and tidal evolution of the spin–orbit angle of close-in giant planets. Mon. Not. R. Astron. Soc. 460 (3), 2339–2350.Google Scholar
Barker, A. J., Braviner, H. J. & Ogilvie, G. I. 2016 Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability. Mon. Not. R. Astron. Soc. 459 (1), 924–938.Google Scholar
Barker, A. J. & Lithwick, Y. 2013 Non-linear evolution of the elliptical instability in the presence of weak magnetic fields. Mon. Not. R. Astron. Soc. 437 (1), 305–315.CrossRefGoogle Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57 (17), 2160.CrossRefGoogle ScholarPubMed
Bondi, H. & Lyttleton, R. A. 1953 On the dynamical theory of the rotation of the Earth. II. The effect of precession on the motion of the liquid core. In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 49, pp. 498–515. Cambridge University Press.Google Scholar
Broadbent, E. G. & Moore, D. W. 1979 Acoustic destabilization of vortices. Phil. Trans. R. Soc. Lond. A 290 (1372), 353–371.Google Scholar
Busse, F. H. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33 (04), 739–751.CrossRefGoogle Scholar
Cébron, D. & Hollerbach, R. 2014 Tidally driven dynamos in a rotating sphere. Astrophys. J. Lett. 789 (1), L25.Google Scholar
Cébron, D., Le Bars, M., Le Gal, P., Moutou, C., Leconte, J. & Sauret, A. 2013 Elliptical instability in hot jupiter systems. Icarus 226 (2), 1642–1653.Google Scholar
Cébron, D., Le Bars, M., Leontini, J., Maubert, P. & Le Gal, P. 2010a A systematic numerical study of the tidal instability in a rotating triaxial ellipsoid. Phys. Earth Planet. Inter. 182 (1), 119–128.Google Scholar
Cébron, D., Le Bars, M. & Meunier, P. 2010b Tilt-over mode in a precessing triaxial ellipsoid. Phys. Fluids 22 (11), 116601.CrossRefGoogle Scholar
Cébron, D., Le Bars, M., Moutou, C. & Le Gal, P. 2012a Elliptical instability in terrestrial planets and moons. Astron. Astrophys. 539, A78.Google Scholar
Cébron, D., Le Bars, M., Noir, J. & Aurnou, J. M. 2012b Libration driven elliptical instability. Phys. Fluids 24 (6), 061703.Google Scholar
Cébron, D., Maubert, P. & Le Bars, M. 2010c Tidal instability in a rotating and differentially heated ellipsoidal shell. Geophys. J. Intl 182 (3), 1311–1318.Google Scholar
Cébron, D., Vantieghem, S. & Herreman, W. 2014 Libration-driven multipolar instabilities. J. Fluid Mech. 739, 502–543.CrossRefGoogle Scholar
Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium, vol. 10. Yale University Press.Google Scholar
Clausen, N. & Tilgner, A. 2014 Elliptical instability of compressible flow in ellipsoids. Astron. Astrophys. 562, A25.Google Scholar
Craik, A. D. D. 1989 The stability of unbounded two-and three-dimensional flows subject to body forces: some exact solutions. J. Fluid Mech. 198, 275–292.Google Scholar
Craik, A. D. D. & Criminale, W. O. 1986 Evolution of wavelike disturbances in shear flows: a class of exact solutions of the Navier–Stokes equations. Proc. R. Soc. Lond. A 406, 13–26.Google Scholar
Dassios, G. 2012 Ellipsoidal Harmonics: Theory and Applications. Cambridge University Press.Google Scholar
Dehant, V. & Mathews, P. M. 2015 Precession, Nutation and Wobble of the Earth. Cambridge University Press.CrossRefGoogle Scholar
Eckhardt, B. & Yao, D. 1995 Local stability analysis along Lagrangian paths. Chaos, Solitons Fractals 5 (11), 2073–2088.Google Scholar
Eloy, C., Le Gal, P. & Le Dizès, S. 2003 Elliptic and triangular instabilities in rotating cylinders. J. Fluid Mech. 476, 357–388.Google Scholar
Favier, B., Grannan, A. M., Le Bars, M. & Aurnou, J. M. 2015 Generation and maintenance of bulk turbulence by libration-driven elliptical instability. Phys. Fluids 27 (6), 066601.Google Scholar
Friedlander, S. & Lipton-Lifschitz, A. 2003 Localized instabilities in fluids. In Handbook of Mathematical Fluid Dynamics (ed. Friedlander, S. J. & Serre, D.), vol. 2, pp. 289–354. Elsevier.Google Scholar
Friedlander, S. & Vishik, M. M. 1991 Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Lett. 66 (17), 2204.Google Scholar
Giuricin, G., Mardirossian, F. & Mezzetti, M. 1984 Synchronization in eclipsing binary stars. Astron. Astrophys. 131, 152–158.Google Scholar
Gledzer, E. B. & Ponomarev, V. M. 1978 Finite-dimensional approximation of the motions of an incompressible fluid in an ellipsoidal cavity. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 820–827.Google Scholar
Gledzer, E. B. & Ponomarev, V. M. 1992 Instability of bounded flows with elliptical streamlines. J. Fluid Mech. 240, 1–30.Google Scholar
Goepfert, O. & Tilgner, A. 2016 Dynamos in precessing cubes. New J. Phys. 18, 103019.Google Scholar
Goodman, J. & Lackner, C. 2009 Dynamical tides in rotating planets and stars. Astrophys. J. 696 (2), 2054.Google Scholar
Grannan, A. M., Favier, B., Le Bars, M. & Aurnou, J. M. 2017 Tidally forced turbulence in planetary interiors. Geophys. J. Intl 208 (3), 1690.Google Scholar
Grannan, A. M., Le Bars, M., Cébron, D. & Aurnou, J. M. 2014 Experimental study of global-scale turbulence in a librating ellipsoid. Phys. Fluids 26 (12), 126601.Google Scholar
Greenspan, H. 1968 The theory of rotating fluids. Cambridge University Press.Google Scholar
Greff-Lefftz, M., Métivier, L. & Legros, H. 2005 Analytical solutions of love numbers for a hydrostatic ellipsoidal incompressible homogeneous Earth. Celestial Mech. Dyn. Astron. 93 (1), 113–146.Google Scholar
Herreman, W., Le Bars, M. & Le Gal, P. 2009 On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids. Phys. Fluids 21 (4), 046602.CrossRefGoogle Scholar
Hindmarsh, A. C., Brown, P. N., Grant, K. E., Lee, S. L., Serban, R., Shumaker, D. E. & Woodward, C. S. 2005 SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. (TOMS) 31 (3), 363–396.Google Scholar
Hough, S. S. 1895 The oscillations of a rotating ellipsoidal shell containing fluid. Phil. Trans. R. Soc. Lond. A 469–506.Google Scholar
Hut, P. 1981 Tidal evolution in close binary systems. Astron. Astrophys. 99, 126–140.Google Scholar
Hut, P. 1982 Tidal evolution in close binary systems for high eccentricity. Astron. Astrophys. 110, 37–42.Google Scholar
Ivers, D. 2017 Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a triaxial ellipsoid. Geophys. Astrophys. Fluid Dyn. 111 (5), 333–354.Google Scholar
Kerswell, R. R. 1993a Elliptical instabilities of stratified, hydromagnetic waves. Geophys. Astrophys. Fluid Dyn. 71 (1–4), 105–143.Google Scholar
Kerswell, R. R. 1993b The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1–4), 107–144.CrossRefGoogle Scholar
Kerswell, R. R. 1996 Upper bounds on the energy dissipation in turbulent precession. J. Fluid Mech. 321, 335–370.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83–113.Google Scholar
Kerswell, R. R. & Malkus, W. V. R. 1998 Tidal instability as the source for Io’s magnetic signature. Geophys. Res. Lett. 25 (5), 603–606.Google Scholar
Lacaze, L., Le Gal, P. & Le Dizès, S. 2004 Elliptical instability in a rotating spheroid. J. Fluid Mech. 505, 1–22.Google Scholar
Le Bars, M. 2016 Flows driven by libration, precession, and tides in planetary cores. Phys. Rev. Fluids 1 (6), 060505.Google Scholar
Le Bars, M., Cébron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193.Google Scholar
Le Bars, M., Lacaze, L., Le Dizès, S., Le Gal, P. & Rieutord, M. 2010 Tidal instability in stellar and planetary binary systems. Phys. Earth Planet. Inter. 178 (1), 48–55.Google Scholar
Le Bars, M. & Le Dizés, S. 2006 Thermo-elliptical instability in a rotating cylindrical shell. J. Fluid Mech. 563, 189–198.Google Scholar
Le Dizès, S. 2000 Three-dimensional instability of a multipolar vortex in a rotating flow. Phys. Fluids 12 (11), 2762–2774.Google Scholar
Le Duc, A.2001 Etude d’écoulements faiblement compressibles, de giration, puis d’impact sur paroi, par théorie linéaire et simulation numérique directe. PhD thesis.Google Scholar
Le Reun, T., Favier, B., Barker, A. J. & Le Bars, M. 2017 Inertial wave turbulence driven by elliptical instability. Phys. Rev. Lett. 119 (3), 034502.Google Scholar
Lebovitz, N. R. 1989 The stability equations for rotating, inviscid fluids: Galerkin methods and orthogonal bases. Geophys. Astrophys. Fluid Dyn. 46 (4), 221–243.Google Scholar
Lebovitz, N. R. & Lifschitz, A. 1996a New global instabilities of the Riemann ellipsoids. Astrophys. J. 458, 699.Google Scholar
Lebovitz, N. R. & Lifschitz, A. 1996b Short-wavelength instabilities of Riemann ellipsoids. Phil. Trans. R. Soc. Lond. A 354 (1709), 927–950.Google Scholar
Lemasquerier, D., Grannan, A. M., Vidal, J., Cébron, D., Favier, B., Le Bars, M. & Aurnou, J. M. 2017 Libration-driven flows in ellipsoidal shells. J. Geophys. Res. 122 (9), 1926–1950.Google Scholar
Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3 (11), 2644–2651.CrossRefGoogle Scholar
Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111–128.Google Scholar
Lorenzani, S. & Tilgner, A. 2003 Inertial instabilities of fluid flow in precessing spheroidal shells. J. Fluid Mech. 492, 363–379.CrossRefGoogle Scholar
Malkus, W. V. R. 1963 Precessional torques as the cause of geomagnetism. J. Geophys. Res. 68 (10), 2871–2886.Google Scholar
Malkus, W. V. R. 1968 Precession of the Earth as the cause of geomagnetism. Science 160 (3825), 259–264.Google Scholar
Malkus, W. V. R. 1989 An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48 (1–3), 123–134.Google Scholar
Miyazaki, T. 1993 Elliptical instability in a stably stratified rotating fluid. Phys. Fluids A 5 (11), 2702–2709.Google Scholar
Murray, C. D. & Dermott, S. F. 1999 Solar System Dynamics. Cambridge University Press.Google Scholar
Naing, M. M. & Fukumoto, Y. 2009 Local instability of an elliptical flow subjected to a Coriolis force. J. Phys. Soc. Japan 78 (12), 124401.Google Scholar
Nduka, A. 1971 The Roche problem in an eccentric orbit. Astrophys. J. 170, 131.Google Scholar
Noir, J., Cardin, P., Jault, D. & Masson, J.-P. 2003 Experimental evidence of non-linear resonance effects between retrograde precession and the tilt-over mode within a spheroid. Geophys. J. Intl 154 (2), 407–416.Google Scholar
Noir, J. & Cébron, D. 2013 Precession-driven flows in non-axisymmetric ellipsoids. J. Fluid Mech. 737, 412–439.Google Scholar
Noir, J., Cébron, D., Le Bars, M., Sauret, A. & Aurnou, J. M. 2012 Experimental study of libration-driven zonal flows in non-axisymmetric containers. Phys. Earth Planet. Inter. 204, 1–10.Google Scholar
Noir, J., Hemmerlin, F., Wicht, J., Baca, S. M. & Aurnou, J. M. 2009 An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans. Phys. Earth Planet. Inter. 173 (1), 141–152.Google Scholar
Ogilvie, G. I. 2009 Tidal dissipation in rotating fluid bodies: a simplified model. Mon. Not. R. Astron. Soc. 396 (2), 794–806.Google Scholar
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610 (1), 477.Google Scholar
Ogilvie, G. I. & Lin, D. N. C. 2007 Tidal dissipation in rotating solar-type stars. Astrophys. J. 661 (2), 1180.Google Scholar
Pais, M. A. & Le Mouël, J.-L. 2001 Precession-induced flows in liquid-filled containers and in the Earth’s core. Geophys. J. Intl 144 (3), 539–554.Google Scholar
Peterson, P. 2009 F2PY: a tool for connecting Fortran and Python programs. Intl J. Comput. Sci. Engng 4 (4), 296–305.Google Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57 (17), 2157.Google Scholar
Poincaré, H. 1910 Sur la précession des corps déformables. Bulletin Astronomique I 27, 321–356.Google Scholar
Rambaux, N. & Williams, J. G. 2011 The Moon’s physical librations and determination of their free modes. Celestial Mech. Dyn. Astronomy 109 (1), 85–100.Google Scholar
Riemann, B. 1860 Untersuchungen uber die bewegung eines flussigen gleich-artigen ellipsoides. Abh. Konigl. Gesell. Wiss. Gottingen 9, 3–36.Google Scholar
Rieutord, M. 2004 Evolution of rotation in binaries: physical processes. In Symposium-International Astronomical Union, vol. 215, pp. 394–403. Cambridge University Press.Google Scholar
Rieutord, M. 2008 The dynamics of rotating fluids and binary stars. Eur. Astron. Soc. Publ. Ser. 29, 127–147.Google Scholar
Rieutord, M. & Valdettaro, L. 2010 Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J. Fluid Mech. 643, 363–394.Google Scholar
Roberts, P. H. & Stewartson, K. 1965 On the motion of a liquid in a spheroidal cavity of a precessing rigid body. II. Math. Proc. Camb. Phil. Soc. 61 (01), 279–288.Google Scholar
Roberts, P. H. & Wu, C.-C. 2011 On flows having constant vorticity. Physica D 240 (20), 1615–1628.Google Scholar
Sauret, A., Cébron, D., Le Bars, M. & Le Dizès, S. 2012 Fluid flows in a librating cylinder. Phys. Fluids 24 (2), 026603.Google Scholar
Schaeffer, N. 2013 Efficient spherical harmonic transforms aimed at pseudospectral numerical simulations. Geochem. Geophys. Geosyst. 14 (3), 751–758.Google Scholar
Schenk, O. & Gärtner, K. 2004 Solving unsymmetric sparse systems of linear equations with PARDISO. Future Generation Comput. Sys. 20 (3), 475–487.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Sieber, M., Paschereit, C. O. & Oberleithner, K. 2016 Spectral proper orthogonal decomposition. J. Fluid Mech. 792, 798–828.Google Scholar
Sloudsky, T. 1895 De la rotation de la Terre supposée fluide à son intérieur. Bull. Soc. Imp. Natur. Mosc. IX, 285–318.Google Scholar
Stewartson, K. & Roberts, P. H. 1963 On the motion of liquid in a spheroidal cavity of a precessing rigid body. J. Fluid Mech. 17 (01), 1–20.CrossRefGoogle Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.Google Scholar
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17 (3), 034104.Google Scholar
Tilgner, A. 2007 Kinematic dynamos with precession driven flow in a sphere. Geophys. Astrophys. Fluid Dyn. 101 (1), 1–9.Google Scholar
Tilgner, A. 2015 Rotational dynamics of the core. In Treatise on Geophysics, vol. 8, pp. 183–212. Elsevier.Google Scholar
Tilgner, A. & Busse, F. H. 2001 Fluid flows in precessing spherical shells. J. Fluid Mech. 426, 387–396.Google Scholar
Van Der Walt, S., Colbert, S. C. & Varoquaux, G. 2011 The NumPy array: a structure for efficient numerical computation. Comput. Sci. Engng 13 (2), 22–30.Google Scholar
Van Hoolst, T., Rambaux, N., Karatekin, Ö., Dehant, V. & Rivoldini, A. 2008 The librations, shape, and icy shell of Europa. Icarus 195 (1), 386–399.Google Scholar
Vantieghem, S. 2014 Inertial modes in a rotating triaxial ellipsoid. Proc. R. Soc. Lond. A 470 (2168), 20140093.Google Scholar
Vantieghem, S., Cébron, D. & Noir, J. 2015 Latitudinal libration driven flows in triaxial ellipsoids. J. Fluid Mech. 771, 193–228.Google Scholar
Vidal, J., Cébron, D. & Schaeffer, N. 2016 Diffusionless hydromagnetic modes in rotating ellipsoids: A road to weakly nonlinear models? In Comptes-rendus de la 19e Rencontre du Non-Linéaire, pp. 121–126. Université Paris Diderot.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2 (1), 76–80.Google Scholar
Wu, C.-C. & Roberts, P. H. 2009 On a dynamo driven by topographic precession. Geophys. Astrophys. Fluid Dyn. 103 (6), 467–501.Google Scholar
Wu, C.-C. & Roberts, P. H. 2011 High order instabilities of the Poincaré solution for precessionally driven flow. Geophys. Astrophys. Fluid Dyn. 105 (2–3), 287–303.Google Scholar
Wu, C.-C. & Roberts, P. H. 2013 On a dynamo driven topographically by longitudinal libration. Geophys. Astrophys. Fluid Dyn. 107 (1–2), 20–44.Google Scholar
Wu, Y. 2005a Origin of tidal dissipation in jupiter. I. Properties of inertial modes. Astrophys. J. 635 (1), 674.Google Scholar
Wu, Y. 2005b Origin of tidal dissipation in Jupiter. II. The value of Q. Astrophys. J. 635 (1), 688.Google Scholar
Zahn, J.-P. 1966 Les marées dans une étoile double serrée. In Annales d’Astrophysique, vol. 29, p. 313.Google Scholar
Zahn, J.-P. 1975 The dynamical tide in close binaries. Astron. Astrophys. 41, 329–344.Google Scholar
Zahn, J.-P. 2008 Tidal dissipation in binary systems. Europ. Astron. Soc. Publ. Ser. 29, 67–90.Google Scholar
Zhang, K., Chan, K. H. & Liao, X. 2010 On fluid flows in precessing spheres in the mantle frame of reference. Phys. Fluids 22 (11), 116604.Google Scholar
Zhang, K., Chan, K. H. & Liao, X. 2012 Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration. J. Fluid Mech. 692, 420–445.Google Scholar
Zhang, K., Chan, K. H. & Liao, X. 2014 On precessing flow in an oblate spheroid of arbitrary eccentricity. J. Fluid Mech. 743, 358.Google Scholar
Zhang, K., Chan, K. H., Liao, X. & Aurnou, J. M. 2013 The non-resonant response of fluid in a rapidly rotating sphere undergoing longitudinal libration. J. Fluid Mech. 720, 212.Google Scholar
Zhang, K., Liao, X. & Earnshaw, P. 2004 On inertial waves and oscillations in a rapidly rotating spheroid. J. Fluid Mech. 504, 1–40.Google Scholar