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Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry

Published online by Cambridge University Press:  07 January 2020

Jérémie Vidal Open the ORCID record for Jérémie Vidal [Opens in a new window] ,
Sylvie Su Open the ORCID record for Sylvie Su [Opens in a new window]  and
David Cébron Open the ORCID record for David Cébron [Opens in a new window]
Jérémie Vidal*
Affiliation:
Department of Applied Mathematics, University of Leeds, LeedsLS2 9JT, UK
Sylvie Su
Affiliation:
Université Grenoble Alpes, CNRS, ISTerre, Grenoble, France
David Cébron
Affiliation:
Université Grenoble Alpes, CNRS, ISTerre, Grenoble, France
*
†Email address for correspondence: [email protected]
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Abstract

Motivated by planetary-driven applications and experiments in non-spherical geometries, we study compressible fluid modes in rotating rigid ellipsoids. Such modes are also required for modal acoustic velocimetry (MAV), a promising non-invasive method to track the velocity field components in laboratory experiments. To calculate them, we develop a general spectral method in rigid triaxial ellipsoids. The description is based on an expansion onto global polynomial vector elements, satisfying the non-penetration condition on the boundary. Then, we investigate the diffusionless compressible modes in rotating (and magnetised) rigid ellipsoids. The spectral description is successfully benchmarked against three-dimensional finite-element computations and analytical predictions. A spectral convergence is obtained. Our results have direct implications for MAV in experiments, for instance in the ZoRo experiment (gas-filled rigid spheroid). So far, deformation and rotational effects have been theoretically considered separately, as small perturbations of the solutions in non-rotating spheres. We carefully compare the perturbation approach, in this illustrative geometry, to the polynomial solutions. We show that second-order ellipticity effects are often present, even in weakly deformed ellipsoids. Moreover, high-order effects due to rotation and/or ellipticity should be observed for some acoustic modes in experimental conditions. Thus, perturbation theory should be used with care in MAV. Instead, the spectral polynomial method paves the way for future MAV applications in fluid experiments with rigid ellipsoids.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 885 , 25 February 2020 , A39
Copyright
© 2020 Cambridge University Press

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References

Abramovitz, M. & Stegun, I. 1971 Handbook of Special Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Aerts, C., Christensen-Dalsgaard, J. & Kurtz, D. W. 2010 Asteroseismology. Springer.CrossRefGoogle Scholar
Backus, G. & Rieutord, M. 2017 Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid. Phys. Rev. E 95 (5), 053116.Google Scholar
Bensalah, A., Joly, P. & Mercier, J.-F. 2018 Well-posedness of a generalized time-harmonic transport equation for acoustics in flow. Math. Meth. Appl. Sci. 41 (8), 3117–3137.CrossRefGoogle Scholar
Berggren, M., Bernland, A. & Noreland, D. 2018 Acoustic boundary layers as boundary conditions. J. Comput. Phys. 371, 633–650.CrossRefGoogle Scholar
Bergliaffa, S. E. P., Hibberd, K., Stone, M. & Visser, M. 2004 Wave equation for sound in fluids with vorticity. Physica D 191 (1–2), 121–136.Google Scholar
Bernstein, I. B., Frieman, E. A., Kruskal, M. D. & Kulsrud, R. M. 1958 An energy principle for hydromagnetic stability problems. Proc. R. Soc. Lond. A 244 (1236), 17–40.Google Scholar
Bonazzola, S., Gourgoulhon, E. & Marck, J.-A. 1998 Numerical approach for high precision 3D relativistic star models. Phys. Rev. D 58 (10), 104020.Google Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Bruneau, M., Garing, C. & Leblond, H. 1986 A rate gyro based on acoustic mode coupling. J. Acoust. Soc. Am. 80 (2), 672–680.CrossRefGoogle Scholar
Campos, L. M. B. C. 1987 On waves in gases. Part II. Interaction of sound with magnetic and internal modes. Rev. Mod. Phys. 59 (2), 363–463.CrossRefGoogle Scholar
Cartan, M. E. 1922 Sur les petites oscillations d’une masse de fluide. Bull. Sci. Math. 46, 317–369.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.CrossRefGoogle Scholar
Cébron, D., Le Bars, M., Leontini, J., Maubert, P. & Le Gal, P. 2010 A systematic numerical study of the tidal instability in a rotating triaxial ellipsoid. Phys. Earth Planet. Inter. 182 (1–2), 119–128.CrossRefGoogle Scholar
Cébron, D., Le Bars, M., Maubert, P. & Le Gal, P. 2012 Magnetohydrodynamic simulations of the elliptical instability in triaxial ellipsoids. Geophys. Astrophys. Fluid Dyn. 106 (4–5), 524–546.CrossRefGoogle Scholar
Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Dover.Google Scholar
Chang, C. T. M. 1971 Natural resonant frequency of a prolate acoustical resonator. J. Acoust. Soc. Am. 49 (3A), 611–614.CrossRefGoogle Scholar
Chang, C. T. M. 1972 Natural resonant frequencies of an oblate acoustical resonator. J. Acoust. Soc. Am. 51 (1A), 1–5.CrossRefGoogle Scholar
Clausen, N. & Tilgner, A. 2014 Elliptical instability of compressible flow in ellipsoids. Astron. Astrophys. 562, A25.CrossRefGoogle Scholar
Dahlen, F. A. & Tromp, J. 1998 Theoretical Global Seismology. Princeton University Press.Google Scholar
Dassios, G. 2012 Ellipsoidal Harmonics: Theory and Applications. Cambridge University Press.CrossRefGoogle Scholar
Ecotiere, D., Tahani, N. & Bruneau, M. 2004 Inertial coupling of resonant normal modes in rotating cavities: acoustic gyrometers for high rotation rates. Acta Acust. United Ac. 90 (6), 1151–1158.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.CrossRefGoogle Scholar
Figueroa, A., Schaeffer, N., Nataf, H.-C. & Schmitt, D. 2013 Modes and instabilities in magnetized spherical Couette flow. J. Fluid Mech. 716, 445–469.CrossRefGoogle Scholar
Friedlander, S. 1987 Hydromagnetic waves in the Earth’s fluid core. Geophys. Astrophys. Fluid Dyn. 39 (4), 315–333.CrossRefGoogle Scholar
Frieman, E. & Rotenberg, M. 1960 On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys. 32 (4), 898–902.CrossRefGoogle Scholar
Gledzer, E. B. & Ponomarev, V. M. 1992 Instability of bounded flows with elliptical streamlines. J. Fluid Mech. 240, 1–30.CrossRefGoogle Scholar
Goldstein, M. E. 1978 Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech. 89 (3), 433–468.CrossRefGoogle Scholar
Gough, D. O. & Thompson, M. J. 1990 The effect of rotation and a buried magnetic field on stellar oscillations. Mon. Not. R. Astron. Soc. 242 (1), 25–55.CrossRefGoogle Scholar
Grannan, A. M., Favier, B., Le Bars, M. & Aurnou, J. M. 2016 Tidally forced turbulence in planetary interiors. Geophys. J. Intl 208 (3), 1690–1703.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Guervilly, C., Cardin, P. & Schaeffer, N. 2019 Turbulent convective length scale in planetary cores. Nature 570, 368–371.CrossRefGoogle ScholarPubMed
Guianvarch, C., Pitre, L., Bruneau, M. & Bruneau, A.-M. 2009 Acoustic field in a quasi-spherical resonator: unified perturbation model. J. Acoust. Soc. Am. 125 (3), 1416–1425.CrossRefGoogle Scholar
Horn, S. & Aurnou, J. M. 2018 Regimes of Coriolis-centrifugal convection. Phys. Rev. Lett. 120 (20), 204502.CrossRefGoogle ScholarPubMed
Horn, S. & Aurnou, J. M. 2019 Rotating convection with centrifugal buoyancy: numerical predictions for laboratory experiments. Phys. Rev. Fluids 4 (7), 073501.CrossRefGoogle Scholar
Ivers, D. 2017 Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a tri-axial ellipsoid. Geophys. Astrophys. Fluid Dyn. 111 (5), 333–354.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 (5–6), 469–487.CrossRefGoogle Scholar
Kellog, O. S. 1953 Foundations of Potential Theory. Dover.Google Scholar
Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72 (1–4), 107–144.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83–113.CrossRefGoogle Scholar
Koulakis, J. P., Pree, S. & Putterman, S. 2018 Acoustic resonances in gas-filled spherical bulb with parabolic temperature profile. J. Acoust. Soc. Am. 144 (5), 2847–2851.CrossRefGoogle ScholarPubMed
Labbé, F., Jault, D. & Gillet, N. 2015 On magnetostrophic inertia-less waves in quasi-geostrophic models of planetary cores. Geophys. Astrophys. Fluid Dyn. 109 (6), 587–610.CrossRefGoogle Scholar
Le Bars, M., Cébron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163–193.CrossRefGoogle Scholar
Le Reun, T., Favier, B. & Le Bars, M. 2019 Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence. J. Fluid Mech. 879, 296–326.CrossRefGoogle Scholar
Lebovitz, N. R. 1982 Perturbation expansions on perturbed domains. SIAM Rev. 24 (4), 381–400.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Lignières, F., Rieutord, M. & Reese, D. 2006 Acoustic oscillations of rapidly rotating polytropic stars. Part I. Effects of the centrifugal distortion. Astron. Astrophys. 455 (2), 607–620.CrossRefGoogle Scholar
Lynden-Bell, D. & Ostriker, J. P. 1967 On the stability of differentially rotating bodies. Mon. Not. R. Astron. Soc. 136 (3), 293–310.CrossRefGoogle Scholar
Malkus, W. V. R. 1967 Hydromagnetic planetary waves. J. Fluid Mech. 28 (4), 793–802.CrossRefGoogle Scholar
Mehl, J. B. 2007 Acoustic eigenvalues of a quasispherical resonator: second order shape perturbation theory for arbitrary modes. J. Res. Natl Inst. Stand. 112 (3), 163–173.CrossRefGoogle ScholarPubMed
Menaut, R., Corre, Y., Huguet, L., Le Reun, T., Alboussière, T., Bergman, M., Deguen, R., Labrosse, S. & Moulin, M. 2019 Experimental study of convection in the compressible regime. Phys. Rev. Fluids 4 (3), 033502.CrossRefGoogle Scholar
Moldover, M. R., Mehl, J. B. & Greenspan, M. 1986 Gas-filled spherical resonators: theory and experiment. J. Acoust. Soc. Am. 79 (2), 253–272.CrossRefGoogle Scholar
Morse, P. M. & Ingard, K. U. 1986 Theoretical Acoustics. Princeton University Press.Google Scholar
Nduka, A. 1971 The Roche problem in an eccentric orbit. Astrophys. J. 170, 131–142.CrossRefGoogle Scholar
Noir, J., Brito, D., Aldridge, K. & Cardin, P. 2001 Experimental evidence of inertial waves in a precessing spheroidal cavity. Geophys. Res. Lett. 28 (19), 3785–3788.CrossRefGoogle 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.CrossRefGoogle Scholar
Pierce, A. D. 1990 Wave equation for sound in fluids with unsteady inhomogeneous flow. J. Acoust. Soc. Am. 87 (6), 2292–2299.CrossRefGoogle Scholar
Reese, D., Lignières, F. & Rieutord, M. 2006 Acoustic oscillations of rapidly rotating polytropic stars. Part II. Effects of the Coriolis and centrifugal accelerations. Astron. Astrophys. 455 (2), 621–637.CrossRefGoogle Scholar
Rieutord, M., Espinosa Lara, F. & Putigny, B. 2016 An algorithm for computing the 2D structure of fast rotating stars. J. Comput. Phys. 318, 277–304.CrossRefGoogle Scholar
Rieutord, M., Triana, S. A., Zimmerman, D. S. & Lathrop, D. P. 2012 Excitation of inertial modes in an experimental spherical Couette flow. Phys. Rev. E 86 (2), 026304.Google Scholar
Rieutord, M. & Valdettaro, L. 2018 Axisymmetric inertial modes in a spherical shell at low Ekman numbers. J. Fluid Mech. 844, 597–634.CrossRefGoogle Scholar
Roberts, P. H. & Loper, D. E. 1979 On the diffusive instability of some simple steady magnetohydrodynamic flows. J. Fluid Mech. 90 (4), 641–668.CrossRefGoogle Scholar
Saviot, L. & Murray, D. B. 2009 Acoustic vibrations of anisotropic nanoparticles. Phys. Rev. B 79 (21), 214101.CrossRefGoogle Scholar
Schaeffer, N., Jault, D., Nataf, H.-C. & Fournier, A. 2017 Turbulent geodynamo simulations: a leap towards Earth’s core. Geophys. J. Intl 211 (1), 1–29.CrossRefGoogle Scholar
Schmitt, D., Alboussiere, T., Brito, D., Cardin, P., Gagnière, N., Jault, D. & Nataf, H.-C. 2008 Rotating spherical Couette flow in a dipolar magnetic field: experimental study of magneto-inertial waves. J. Fluid Mech. 604, 175–197.CrossRefGoogle Scholar
Seyranian, A. P. 1993 Sensitivity analysis of multiple eigenvalues. J. Struct. Mech. 21 (2), 261–284.Google Scholar
Su, S., Cébron, D., Nataf, H.-C., Cardin, P., Vidal, J., Solazzo, M. & Do, Y.2020 Acoustic spectra of a spheroidal cavity. Preprint (available at https://hal.archives-ouvertes.fr/hal-02404242).Google Scholar
Tigrine, Z., Nataf, H.-C., Schaeffer, N., Cardin, P. & Plunian, F. 2019 Torsional Alfvén waves in a dipolar magnetic field: experiments and simulations. Geophys. J. Int. 219 (Supplement_1), S83–S100.CrossRefGoogle Scholar
Tisseur, F. & Meerbergen, K. 2001 The quadratic eigenvalue problem. SIAM Rev. 43 (2), 235–286.CrossRefGoogle Scholar
Triana, S. A., Zimmerman, D. S., Nataf, H.-C., Thorette, A., Lekic, V. & Lathrop, D. P. 2014 Helioseismology in a bottle: modal acoustic velocimetry. New J. Phys. 16 (11), 113005.CrossRefGoogle Scholar
Valdettaro, L., Rieutord, M., Braconnier, T. & Frayssé, V. 2007 Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi–Chebyshev algorithm. J. Comput. Appl. Math. 205 (1), 382–393.CrossRefGoogle Scholar
Valette, B. 1989a Etude d’une classe de problèmes spectraux. C. R. Acad. Sci. Paris 309 (Série I), 785–788.Google Scholar
Valette, B. 1989b Spectre des vibrations propres dun corps élastique, auto-gravitant, en rotation uniforme et contenant une partie fluide. C. R. Acad. Sci. Paris 309 (Série I), 419–422.Google Scholar
Vantieghem, S. 2014 Inertial modes in a rotating triaxial ellipsoid. Proc. R. Soc. Lond. A 470 (2168), 20140093.CrossRefGoogle Scholar
Vantieghem, S., Cébron, D. & Noir, J. 2015 Latitudinal libration driven flows in triaxial ellipsoids. J. Fluid Mech. 771, 193–228.CrossRefGoogle Scholar
Vidal, J. & Cébron, D. 2017 Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits. J. Fluid Mech. 833, 469–511.CrossRefGoogle Scholar
Vidal, J., Cébron, D., ud Doula, A. & Alecian, E. 2019a Fossil field decay due to nonlinear tides in massive binaries. Astron. Astrophys. 629, A142.CrossRefGoogle Scholar
Vidal, J., Cébron, D., Schaeffer, N. & Hollerbach, R. 2018 Magnetic fields driven by tidal mixing in radiative stars. Mon. Not. R. Astron. Soc. 475 (4), 4579–4594.CrossRefGoogle Scholar
Vidal, J., Su, S. & Cébron, D. 2019b Polynomial description of acoustic modes in fluid ellipsoids. In Comptes-Rendus de la 22e Rencontre du Non-Linéaire.Google Scholar
Visscher, W. M., Migliori, A., Bell, T. M. & Reinert, R. A. 1991 On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects. J. Acoust. Soc. Am. 90 (4), 2154–2162.CrossRefGoogle Scholar
Willatzen, M. & Lew Yan Voon, L. C. 2004 Eigenmodes of triaxial ellipsoidal acoustical cavities with mixed boundary conditions. J. Acoust. Soc. Am. 116 (6), 3279–3283.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Zhang, K. & Liao, X. 2017 Theory and Modelling of Rotating Fluids: Convection, Inertial Waves and Precession. Cambridge University Press.CrossRefGoogle Scholar
Zhang, K., Liao, X. & Schubert, G. 2003 Nonaxisymmetric instabilities of a toroidal magnetic field in a rotating sphere. Astrophys. J. 585 (2), 1124–1137.CrossRefGoogle Scholar
Zimmerman, D. S., Triana, S. A., Nataf, H.-C. & Lathrop, D. P. 2014 A turbulent, high magnetic Reynolds number experimental model of Earth’s core. J. Geophys. Res. 119 (6), 4538–4557.CrossRefGoogle Scholar