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Tilted incompressible Coriolis modes in spheroids

Published online by Cambridge University Press:  02 November 2017

D. J. Ivers Open the ORCID record for D. J. Ivers [Opens in a new window]
D. J. Ivers*
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
*
†Email address for correspondence: david.ivers@sydney.edu.au
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Abstract

The incompressible flow of a uniform fluid, which fills a rigid spheroid rotating about an arbitrary axis fixed in an inertial frame, is dominated at small Rossby and Ekman numbers by the rotation through the Coriolis force. The effects of rotation on the flow can be found by treating the Coriolis force modified by a pressure gradient as a skew-symmetric bounded linear operator $\boldsymbol{{\mathcal{C}}}$ acting on smooth inviscid incompressible flows in the spheroid. It is shown that the space of incompressible polynomial flows of degree $N$ or less in the spheroid is invariant under $\boldsymbol{{\mathcal{C}}}$ for any $N$. The skew symmetry of $\boldsymbol{{\mathcal{C}}}$ implies the Coriolis operator $\boldsymbol{{\mathcal{C}}}$ is non-defective for such flows with an orthogonal set of eigenmodes (inertial and geostrophic modes) which form a basis for the finite-dimensional space of spheroidal polynomial flows. The eigenmodes are tilted if the rotation axis is not aligned with the symmetry axis of the spheroid. The non-defective property of $\boldsymbol{{\mathcal{C}}}$ enables enumeration of the modes and proof of their completeness using the Weierstrass polynomial approximation theorem. The fundamental tool, which is required to establish invariance of spheroidal polynomial flows under $\boldsymbol{{\mathcal{C}}}$ and completeness of the Coriolis modes, is that the solution of the polynomial Poisson–Neumann problem, i.e. Poisson’s equation with Neumann boundary condition and polynomial data, in a spheroid is a polynomial. The Coriolis modes of degree one and all geostrophic modes are explicitly constructed. Only the modes of degree one have non-zero angular momentum in the boundary frame.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 833 , 25 December 2017 , pp. 131 - 163
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© 2017 Cambridge University Press 

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References

Aldridge, K. D. & Lumb, L. I. 1987 Inertial waves identified in the Earth’s fluid outer core. Nature 325, 421–423.CrossRefGoogle Scholar
Aldridge, K. D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37, 307–323.Google Scholar
Backus, G. & Rieutord, M. 2017 Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid. Phys. Rev. E 95, 053116.Google Scholar
Balick, B. & Frank, A. 2002 Shapes and shaping of planetary nebulae. Annu. Rev. Astron. Astrophys. 40, 439–486.Google Scholar
Bryan, G. H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. A 180, 187–219.Google Scholar
Chan, K. H., Liao, X. & Zhang, K. 2011 Simulations of fluid motion in spheroidal planetary cores driven by latitudinal libration. Phys. Earth Planet. Inter. 187, 404–415.Google Scholar
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
Dermott, S. F. 1979 Shapes and gravitational moments of satellites and asteroids. Icarus 37, 575–586.CrossRefGoogle Scholar
Dyson, J. & Schutz, P. F. 1979 Perturbations and stability of rotating stars. I. Completeness of normal modes. Proc. R. Soc. Lond. A 368, 389–410.Google Scholar
Gans, R. F. 1970 On the hydromagnetic precession in a cylinder. J. Fluid Mech. 45, 111–130.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hough, S. S. 1895 The oscillations of a rotating ellipsoidal shell containing fluid. Phil. Trans. R. Soc. Lond. A 186, 469–506.Google Scholar
Ivanov, P. B. & Papaloizou, J. C. B. 2010 Inertial waves in rotating bodies: a WKBJ formalism for inertial modes and a comparison with numerical results. Mon. Not. R. Astron. Soc. 407, 1609–1630.Google Scholar
Ivers, D. J. 2017a Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a tri-axial ellipsoid. Geophys. Astrophys. Fluid Dyn. 111, 333–354.Google Scholar
Ivers, D. J. 2017b Kinematic dynamos in spheroidal geometries. Proc. R. Soc. Lond. A 473, 20170432.Google Scholar
Ivers, D. J., Jackson, A. & Winch, D. 2015 Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere. J. Fluid Mech. 766, 468–498.CrossRefGoogle Scholar
Kelley, D. H., Triana, S. A., Zimmerman, D. S., Tilgner, A. & Lathrop, D. P. 2007 Inertial waves driven by differential rotation in a planetary geometry. Geophys. Astrophys. Fluid Dyn. 101, 469–487.CrossRefGoogle Scholar
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72, 107–144.Google Scholar
Kerswell, R. R. 1994 Tidal excitation of hydromagnetic waves and their damping in the earth. J. Fluid Mech. 214, 219–241.CrossRefGoogle Scholar
Kudlick, M. D.1966 On transient motions in a contained rotating fluid. PhD thesis, MIT.Google Scholar
Le Bars, M., Dizès, S. L. & Le Gal, P. 2007 Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers. J. Fluid Mech. 585, 323–342.CrossRefGoogle Scholar
Lebovitz, N. R. 1989 The stability equations for rotating, inviscid fluids: Galerkin methods and orthogonal bases. Geophys. Astrophys. Fluid Dyn. 46, 221–243.Google Scholar
Liao, X. & Zhang, K. 2009 A new integral property of inertial waves in rotating fluid spheres. Proc. R. Soc. Lond. A 465, 1075–1091.Google Scholar
Liao, X. & Zhang, K. 2010a Asymptotic and numerical solutions of the initial value problem in rotating planetary fluid cores. Geophys. J. Intl 180, 181–192.CrossRefGoogle Scholar
Liao, X. & Zhang, K. 2010b A new Legendre-type polynomial and its application to geostrophic flow in rotating fluid spheres. Proc. R. Soc. Lond. A 466, 2203–2217.Google Scholar
Lorenzani, S. & Tilgner, A. 2003 Inertial instabilities of fluid flow in precessing spheroidal shells. J. Fluid Mech. 492, 363–379.Google Scholar
MacRobert, T. M. 1947 Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications, 2nd edn. Methuen.Google Scholar
Maffei, S., Jackson, A. & Livermore, P. W. 2017 Characterization of columnar inertial modes in rapidly rotating spheres and spheroids. Proc. R. Soc. Lond. A 473, 20170181.Google ScholarPubMed
Malkus, W. V. R. 1967 Hydromagnetic planetary waves. J. Fluid Mech. 28, 793–802.CrossRefGoogle 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, 141–152.Google Scholar
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477–509.CrossRefGoogle Scholar
Poincaré, H. 1885 Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7, 259–380.CrossRefGoogle Scholar
Schmitt, D. 2006 Numerical study of viscous modes in a rotating spheroid. J. Fluid Mech. 567, 399–414.Google Scholar
Schmitt, D. & Jault, D. 2004 Numerical study of a rotating fluid in a spheroidal container. J. Comput. Phys. 197, 671–685.Google Scholar
Stacey, F. D. & Davis, P. M. 2008 Physics of the Earth, 4th edn. Cambridge University Press.Google Scholar
Tassoul, J. L. 1978 Theory of Rotating Stars. Princeton University Press.Google Scholar
Thomas, P. C. 2010 Sizes, shapes, and derived properties of the saturnian satellites after the Cassini nominal mission. Nature 208, 395–401.Google Scholar
Thomson, W. 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Tilgner, A. 2007 Zonal wind driven by inertial modes. Phys. Rev. Lett. 99, 194501.Google Scholar
Vantieghem, S. 2014 Inertial modes in a rotating triaxial ellipsoid. Proc. R. Soc. Lond. A 470, 20140093.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, 287–303.Google Scholar
Zhang, K. 1993 On equatorially trapped boundary inertial waves. J. Fluid Mech. 248, 203–217.CrossRefGoogle Scholar
Zhang, K. 1994 On coupling between the Poincaré equation and the heat equation. J. Fluid Mech. 268, 211–229.Google Scholar
Zhang, K. 1995 On coupling between the Poincaré equation and the heat equation: non-slip boundary condition. J. Fluid Mech. 284, 239–256.Google Scholar
Zhang, K., Chan, K. H. & Liao, X. 2014 On precessing flow in an oblate spheroid of arbitrary eccentricity. J. Fluid Mech. 743, 358–384.Google Scholar
Zhang, K., Earnshaw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103–119.Google Scholar
Zhang, K. & Liao, X. 2004 A new asymptotic method for the analysis of convection in a rapidly rotating sphere. J. Fluid Mech. 518, 319–346.Google Scholar
Zhang, K. & Liao, X. 2017 Theory and Modeling of Rotating Fluids: Convection, Inertial Waves and Precession. Cambridge University Press.Google Scholar
Zhang, K., Liao, X. & Earnshaw, P. 2004a On inertial waves and oscillations in a rapidly rotating spheroid. J. Fluid Mech. 504, 1–40.Google Scholar
Zhang, K., Liao, X. & Earnshaw, P. 2004b The Poincaré equation: a new polynomial and its unusual properties. J. Math. Phys. 45, 4777–4790.Google Scholar