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Axisymmetric inertial modes in a spherical shell at low Ekman numbers

  • M. Rieutord (a1) (a2) and L. Valdettaro (a3)


We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$ , $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin (\unicode[STIX]{x03C0}/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.


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Backus, G. & Rieutord, M. 2017 Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid. Phys. Rev. E 95 (5), 053116.
Baruteau, C. & Rieutord, M. 2013 Inertial waves in a differentially rotating spherical shell – I. Free modes of oscillation. J. Fluid Mech. 719, 4781.10.1017/jfm.2012.605
Bryan, G. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Phil. Trans. R. Soc. Lond. 180, 187219.
Chatelin, F. 2012 Eigenvalues of Matrices, Revised edn. SIAM Classics in Applied Mathematics.10.1137/1.9781611972467
Cui, Z., Zhang, K. & Liao, X. 2014 On the completeness of inertial wave modes in rotating annular channels. Geophys. Astrophys. Fluid Dyn. 108, 4459.10.1080/03091929.2013.821117
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271297.10.1017/S0022112099006308
Fotheringham, P. & Hollerbach, R. 1998 Inertial oscillations in a spherical shell. Geophys. Astrophys. Fluid Dyn. 89, 2343.10.1080/03091929808213647
Friedlander, S. 1982 Turning surface behaviour for internal waves subject to general gravitational fields. Geophys. Astrophys. Fluid Dyn. 21, 189200.10.1080/03091928208209012
Friedlander, S. & Siegmann, W. 1982 Internal waves in a contained rotating stratified fluid. J. Fluid Mech. 114, 123156.10.1017/S002211208200007X
Gerkema, T., Zimmerman, J. T. F., Maas, L. R. M. & van Haren, H. 2008 Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev. Geophys. 46, RG2004.10.1029/2006RG000220
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Hollerbach, R. & Kerswell, R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327339.10.1017/S0022112095003338
Ivers, D. J., Jackson, A. & Winch, D. 2015 Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a sphere. J. Fluid Mech. 766, 468498.10.1017/jfm.2015.27
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155168.
Kerswell, R. 1995 On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary layers. J. Fluid Mech. 298, 311325.10.1017/S0022112095003326
Le Dizès, S. 2015 Wave field and zonal flow of a librating disk. J. Fluid Mech. 782, 178208.10.1017/jfm.2015.530
Maas, L., Benielli, D., Sommeria, J. & Lam, F.-P. 1997 Observation of an internal wave attractor in a confined, stably stratified fluid. Nature 388, 557561.10.1038/41509
Maas, L. & Lam, F.-P. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 141.10.1017/S0022112095003582
Manders, A. M. M. & Maas, L. R. M. 2003 Observations of inertial waves in a rectangular basin with one sloping boundary. J. Fluid Mech. 493, 5988.10.1017/S0022112003005998
Mirouh, G. M., Baruteau, C., Rieutord, M. & Ballot, 2016 Gravito-inertial waves in a differentially rotating spherical shell. J. Fluid Mech. 800, 213247.10.1017/jfm.2016.382
Noir, J., Brito, D., Aldridge, K. & Cardin, P. 2001 Experimental evidence of inertial waves in a precessing spheroidal cavity. Geophys. Res. Lett. 28, 37853788.10.1029/2001GL012956
Ogilvie, G. 2009 Tidal dissipation in rotating fluid bodies: a simplified model. Mon. Not. R. Astron. Soc. 396, 794806.10.1111/j.1365-2966.2009.14814.x
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477509.10.1086/421454
Poincaré, H. 1885 Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Mathematica 7, 259380.10.1007/BF02402204
Rieutord, M. 1987 Linear theory of rotating fluids using spherical harmonics. I. Steady flows. Geophys. Astrophys. Fluid Dyn. 39, 163182.10.1080/03091928708208811
Rieutord, M. 2015 Fluid Dynamics: An Introduction. Springer.
Rieutord, M., Georgeot, B. & Valdettaro, L. 2000 Waves attractors in rotating fluids: a paradigm for ill-posed cauchy problems. Phys. Rev. Lett. 85, 42774280.10.1103/PhysRevLett.85.4277
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103144.10.1017/S0022112001003718
Rieutord, M. & Valdettaro, L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 7799.10.1017/S0022112097005491
Rieutord, M. & Valdettaro, L. 2010 Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J. Fluid Mech. 643, 363394.10.1017/S002211200999214X
Rieutord, M., Valdettaro, L. & Georgeot, B. 2002 Analysis of singular inertial modes in a spherical shell: the slender toroidal shell model. J. Fluid Mech. 463, 345360.10.1017/S0022112002008881
Sauret, A. & Le Dizès, S. 2013 Libration-induced mean flow in a spherical shell. J. Fluid Mech. 718, 181209.10.1017/jfm.2012.604
Stewartson, K. & Rickard, J. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 35, 759773.10.1017/S002211206900142X
Valdettaro, L., Rieutord, M., Braconnier, T. & Fraysse, V. 2007 Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi–Chebyshev algorithm. J. Comput. Appl. Maths 205, 382393.10.1016/
Zhang, K.-K., Earnshaw, P., Liao, X. & Busse, F. 2001 On inertial waves in a rotating sphere. J. Fluid Mech. 437, 2001.10.1017/S0022112001004049
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Axisymmetric inertial modes in a spherical shell at low Ekman numbers

  • M. Rieutord (a1) (a2) and L. Valdettaro (a3)


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