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Precession-driven flows in non-axisymmetric ellipsoids

Published online by Cambridge University Press:  26 November 2013

J. Noir  and
D. Cébron
J. Noir*
Affiliation:
Institut für Geophysik, ETH Zürich, Sonneggstrasse 5, Zürich CH-8092, Switzerland
D. Cébron
Affiliation:
Institut für Geophysik, ETH Zürich, Sonneggstrasse 5, Zürich CH-8092, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

We study the flow forced by precession in rigid non-axisymmetric ellipsoidal containers. To do so, we revisit the inviscid and viscous analytical models that have been previously developed for the spheroidal geometry by, respectively, Poincaré (Bull. Astronomique, vol. XXVIII, 1910, pp. 1–36) and Busse (J. Fluid Mech., vol. 33, 1968, pp. 739–751), and we report the first numerical simulations of flows in such a geometry. In strong contrast with axisymmetric spheroids, where the forced flow is systematically stationary in the precessing frame, we show that the forced flow is unsteady and periodic. Comparisons of the numerical simulations with the proposed theoretical model show excellent agreement for both axisymmetric and non-axisymmetric containers. Finally, since the studied configuration corresponds to a tidally locked celestial body such as the Earth’s Moon, we use our model to investigate the challenging but planetary-relevant limit of very small Ekman numbers and the particular case of our Moon.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 737 , 25 December 2013 , pp. 412 - 439
Copyright
©2013 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. 1972 Handbook of Mathematical Functions. Dover.Google Scholar
Agrawal, B. N. 1993 Dynamic characteristics of liquid motion in partially filled tanks of a spinning spacecraft. J. Guid. Control Dyn. 16 (4), 636–640.Google Scholar
Bao, G. W. & Pascal, M. 1997 Stability of a spinning liquid-filled spacecraft. Arch. Appl. Mech. 67 (6), 407–421.Google Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57 (17), 2160–2163.Google Scholar
Boisson, J., Cébron, D., Moisy, F. & Cortet, P. P. 2012 Earth rotation prevents exact solid body rotation of fluids in the laboratory. Europhys. Lett. 98, 59002.Google Scholar
Bondi, H. & Lyttleton, R. A. 1953 On the dynamic theory of the rotation of the Earth: the effect of precession on the motion of the liquid core. Proc. Camb. Phil. Soc 49, 498–515.Google Scholar
Bullard, E. C. 1949 The magnetic flux within the Earth. Proc. R. Soc. Lond. A 197, 433–453.Google Scholar
Busse, F. H. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33 (4), 739–751.CrossRefGoogle 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, 119–128.Google Scholar
Cébron, D., Le Bars, M. & Meunier, P. 2010b Tilt-over mode in a precessing triaxial ellipsoid. Phys. Fluids 22, 116601.CrossRefGoogle Scholar
Dwyer, C. A., Stevenson, D. J. & Nimmo, F. 2011 A long-lived lunar dynamo driven by continuous mechanical stirring. Nature 479, 212–214.Google Scholar
Faller, A. J. 1991 Instability and transition of disturbed flow over a rotating disk. J. Fluid Mech. 230, 245–269.Google Scholar
Gans, R. F. 1984 Dynamics of a near-resonant fluid-filled gyroscope. AIAA J. 22 (10), 1465–1471.CrossRefGoogle Scholar
Garg, S. C., Furumoto, N. & Vanyo, J. P. 1986 Spacecraft nutational instability prediction by energy-dissipation measurements. J. Guid. Control Dyn. 9, 357–362.Google Scholar
Glampedakis, K., Andersson, N. & Jones, D. I. 2009 Do superfluid instabilities prevent neutron star precession? Mon. Not. R. Astron. Soc. 394, 1908–1924.CrossRefGoogle Scholar
Goldreich, P. 1967 Precession of the moon’s core. J. Geophys. Res. 72 (12), 3135–3137.Google Scholar
Goto, S., Ishii, N., Kida, S. & Nishioka, M. 2007 Turbulence generator using a precessing sphere. Phys. Fluids 19, 061705.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17 (3), 385–404.Google Scholar
Greff-Lefftz, M. & Legros, H. 1999 Core rotational dynamics and geological events. Science 286 (5445), 1707–1709.Google 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. 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 186, 469–506.Google Scholar
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1), 107–144.Google Scholar
Kerswell, R. R. 1996 Upper bounds on the energy dissipation in turbulent precession. J. Fluid Mech. 321, 335–370.Google Scholar
Kida, S. 2011 Steady flow in a rapidly rotating sphere with weak precession. J. Fluid Mech. 680, 150–193.Google Scholar
Kida, S. & Nakazawa, N. 2010 Super-rotation flow in a precessing sphere. Theor. Comput. Fluid Dyn. 24 (1–4), 259–263.CrossRefGoogle Scholar
Lacaze, L., Le Gal, P. & Le Dizès, S. 2004 Elliptical instability in a rotating spheroid. J. Fluid Mech. 505, 1–22.Google Scholar
Lagrange, R., Eloy, C., Nadal, F. & Meunier, P. 2008 Instability of a fluid inside a precessing cylinder. Phys. Fluids 20, 081701.Google Scholar
Lagrange, R., Meunier, P., Nadal, F. & Eloy, C. 2011 Precessional instability of a fluid cylinder. J. Fluid Mech. 666, 104–145.CrossRefGoogle 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 (1–2), 48–55.Google Scholar
Le Bars, M., Wieczorek, M. A., Karatekin, Ö., Cébron, D. & Laneuville, M. 2011 An impact-driven dynamo for the early moon. Nature 479, 215–218.CrossRefGoogle ScholarPubMed
Le Dizès, S. 2000 Three-dimensional instability of a multipolar vortex in a rotating flow. Phys. Fluids 12, 2762–2774.Google Scholar
Lingwood, R. J. 1997 Absolute instability of the Ekman layer and related rotating flows. J. Fluid Mech. 331, 405–428.Google Scholar
Lorenzani, S. 2001 Fluid instabilities in precessing ellipsoidal shells. PhD thesis, Niedersächsische Staats- und Universitätsbibliothek Göttingen.Google Scholar
Lorenzani, S. & Tilgner, A. 2001 Fluid instabilities in precessing spheroidal cavities. J. Fluid Mech. 447, 111–128.CrossRefGoogle 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. 1968 Precession of the Earth as the cause of geomagnetism: experiments lend support to the proposal that precessional torques drive the Earth’s dynamo. Science 160 (3825), 259–264.Google Scholar
Meunier, P., Eloy, C., Lagrange, R. & Nadal, F. C. O. 2008 A rotating fluid cylinder subject to weak precession. J. Fluid Mech. 599, 405–440.Google Scholar
Meyer, J. & Wisdom, J. 2011 Precession of the lunar core. Icarus 211, 921–924.Google Scholar
Noir, J., Cardin, P., Jault, D. & Masson, J.-P. 2003 Experimental evidence of nonlinear resonance effects between retrograde precession and the tilt-over mode within a spheroid. Geophys. J. Intl 154 (2), 407–416.Google Scholar
Noir, J., Jault, D. & Cardin, P. 2001 Numerical study of the motions within a slowly precessing sphere at low Ekman number. J. Fluid Mech. 437, 283–299.Google Scholar
Nore, C., Léorat, J., Guermond, J. L. & Luddens, F. 2011 Nonlinear dynamo action in a precessing cylindrical container. Phys. Rev. E 84 (1), 016317.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
Poincaré, H 1910 Sur la précéssion des corps déformables. Bull. Astronomique XXVIII, 1–36.Google Scholar
Sloudsky, T. 1895 De la rotation de la Terre supposée fluide à son intérieur. Bulletin de la Société Impériale des Naturalistes de Moscou, Moscou, 1895, Vol. 2.Google Scholar
Stewartson, K. 1959 On the stability of a spinning top containing liquid. J. Fluid Mech. 5 (4), 577–592.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 (1), 1–20.Google Scholar
Tilgner, A 1998 On models of precession driven core flow. Stud. Geophys. Geod. 42 (3), 232–238.Google Scholar
Tilgner, A. 1999 Magnetohydrodynamic flow in precessing spherical shells. J. Fluid Mech. 379, 303–318.Google Scholar
Tilgner, A. 2005 Precession driven dynamos. Phys. Fluids 17, 034104.Google Scholar
Tilgner, A. 2007 Kinematic dynamos with precession driven flow in a sphere. Geophys. Astrophys. Fluid Dyn. 101 (1), 1–9.CrossRefGoogle Scholar
Tilgner, A. & Busse, F. H. 2001 Fluid flows in precessing spherical shells. J. Fluid Mech. 426, 387–396.Google Scholar
Triana, S. A., Zimmerman, D. S. & Lathrop, D. P. 2012 Precessional states in a laboratory model of the Earth’s core. J. Geophys. Res. 117 (B4), XXX–XXX.Google Scholar
Vanyo, J. P. 1991 A geodynamo powered by luni-solar precession. Geophys. Astrophys. Fluid Dyn. 59 (1), 209–234.CrossRefGoogle Scholar
Vanyo, J., Wilde, P., Cardin, P. & Olson, P. 1995 Experiments on precessing flows in the Earth’s liquid core. Geophys. J. Intl 121 (1), 136–142.Google Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A: Fluid Dyn. 2, 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
Zhang, K., Chan, K. H. & Liao, X. 2010 On fluid flows in precessing spheres in the mantle frame of reference. Phys. Fluids 22, 116604.CrossRefGoogle 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