Skip to main content
×
×
Home

Inertial wave rays in rotating spherical fluid domains

  • Anna Rabitti (a1) and Leo R. M. Maas (a1)
Abstract

The behaviour of inertial waves in a rotating spherical container, filled with homogeneous fluid, is here investigated by means of a three-dimensional ray tracing algorithm, in a linear, inviscid framework. In particular, the classical, two-dimensional association between regular modes and periodic trajectories is addressed here for the first time in a fully three-dimensional setting. Three-dimensional, repelling periodic trajectories are found and classified on the basis of the associated frequency and spatial structure, although associated frequencies are hardly reconcilable to Bryan’s (Proc. R. Soc. Lond., vol. 45, 1889, pp. 42–45) classical solutions for inertial waves in the sphere. The normalized squared frequency $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\omega ^2 = 1/2$ appears to divide the frequency range into two different trajectory regimes, where critical latitudes play a different role. Chaotic orbits are not found, as expected, while invariant, non-domain-filling orbits (whispering gallery modes) constitute the majority of the trajectories in the sphere. From ray tracing alone, the wavefield is still far from being completely reconstructed, and a study performed in such a simplified setting is clearly far from any realistic application, however, it appears that three-dimensional ray dynamics constitutes a valid approach to infer information on the spectrum and regularity properties of a system, and is then able to bring new insight in a variety of fundamental problems of geophysical and astrophysical relevance, once its power and limitations have been recognized.

Copyright
Corresponding author
Email address for correspondence: anna.rabitti@nioz.nl
References
Hide All
Aldridge, K. D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37, 307323.
Arras, P., Flanagan, E. E., Morsink, S. M., Schenk, A. K., Teukolsky, S. A. & Wasserman, I. 2003 Saturation of the -mode instability. Astrophys. J. 591, 11291151.
Baines, P. G. 1971 The reflexion of internal/inertial waves from bumpy surfaces. J. Fluid Mech. 46, 273292.
Barbosa Aguiar, A. C., Read, P. L., Wordsworth, R. D., Salter, T. & Hiro Yamazaki, Y. 2010 A laboratory model of Saturn’s North Polar Hexagon. Icarus 206 (2), 755763.
Barcilon, V. 1968 Axi-symmetric inertial oscillations of a rotating ring of fluid. Mathematika 93, 93102.
Baruteau, C. & Rieutord, M. 2013 Inertial waves in a differentially rotating spherical shell. J. Fluid Mech. 719, 4781.
Berry, M. V. 1981 Regularity and chaos in classical mechanics, illustrated by the three deformations of a circular ‘billiad’. Eur. J. Phys. 2, 91102.
Berry, M. V. 1987 Quantum chaology. Proc. R. Soc. Lond. A 413, 4245.
Broutman, D., Rottman, J. W. & Eckermann, S. D. 2004 Ray methods for internal waves in the atmosphere and ocean. Annu. Rev. Fluid Mech. 36 (1), 233253.
Bryan, G. H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Proc. R. Soc. Lond. 45, 4245.
Cartan, M. E. 1922 Sur les petites oscillations d’une masse de fluide. Bull. Sci. Math. 46, 317369.
Dintrans, B. & Ouyed, R. 2001 On Jupiter’s inertial mode oscillations. Astron. Astrophys. 375 (L47), 14.
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271297.
Drijfhout, S. & Maas, L. R. M. 2007 Impact of channel geometry and rotation on the trapping of internal tides. J. Phys. Oceanogr. 37 (11), 27402763.
Eckart, C. 1960 Hydrodynamics of Oceans and Atmospheres. Pergamon.
Eriksen, C. C. 1985 Implications of ocean bottom reflection for internal wave spectra and mixing. J. Phys. Oceanogr. (15), 11451156.
Favier, B., Barker, A. J., Baruteau, C. & Ogilvie, G. I. 2014 Nonlinear evolution of tidally forced inertial waves in rotating fluid bodies. Mon. Not. R. Astron. Soc. 439 (1), 845860.
Gilbert, D. & Garrett, C. 1989 Implications for ocean mixing of internal wave scattering off irregular topography. J. Phys. Oceanogr. (19), 17161729.
Godfrey, D. A. 1988 A hexagonal feature around Saturn’s north pole. Icarus 76 (2), 335356.
Görtler, H. 1943 Uber eine schwingungserscheinung in flussigkeiten mit stabiler dichteschichtung. Z. Angew. Math. Mech. 23, 6571.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Gutzwiller, M. C. 1990 Chaos in Classical and Quantum Mechanics. Springer.
Harlander, U. & Maas, L. R. M. 2006 Characteristics and energy rays of equatorially trapped, zonally symmetric internal waves. Meteorol. Z. 15 (4), 439450.
Harlander, U. & Maas, L. R. M. 2007 Internal boundary layers in a well-mixed equatorial atmosphere/ocean. Dyn. Atmos. Oceans 44 (1), 128.
Hazewinkel, J., Grisouard, N. & Dalziel, S. B. 2010 Comparison of laboratory and numerically observed scalar fields of an internal wave attractor. Eur. J. Mech. (B/Fluids) 30, 5156.
Hazewinkel, J., Maas, L. R. M. & Dalziel, S. B. 2011 Tomographic reconstruction of internal wave patterns in a paraboloid. Exp. Fluids 50, 247258.
Hazewinkel, J., Van Breevoort, P., Dalziel, S. B. & Maas, L. R. M. 2008 Observations on the wavenumber spectrum and evolution of an internal wave attractor. J. Fluid Mech. 598, 373382.
Heller, E. J. 1984 Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53, 15151518.
Høiland, E. 1962 Discussion of a hyperbolic equation relating to inertia and gravitational fluid oscillations. Geophys. Publ. 26, 211227.
Hollerbach, R., Futterer, B., More, T. & Egbers, C. 2004 Instabilities of the Stewartson layer part 2. Supercritical mode transitions. Theor. Comput. Fluid Dyn. 18, 197204.
Hughes, B. 1964 Effect of rotation on internal gravity waves. Nature 201 (4921), 798801.
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 (3), 16091630.
John, F. 1941 The Dirichlet problem for hyperbolic equation. Am. J. Maths 63, 141154.
Lord Kelvin, 1877 On the precessional motion of a liquid. Nature 15, 297298.
Lord Kelvin, 1880a On an experimental illustration of minimum energy. Nature 23, 6970.
Lord Kelvin, 1880b Vibrations of a columnar vortex. Phil. Mag. (10), 155168.
Koch, S., Harlander, U., Egbers, C. & Hollerbach, R. 2013 Inertial waves in a spherical shell induced by librations of the inner sphere: experimental and numerical results. Fluid Dyn. Res. 45 (3), 35504.
Kudlick, M. D.1966 On Transient Motions in a Contained Rotating Fluid. PhD thesis, Math. Dept., MIT.
Kudrolli, A., Abraham, M. C. & Gollub, J. P. 2001 Scarred patterns in surface waves. Phys. Rev. E 63 (2), 18.
Le Bars, M., Le Dizès, S. & Le Gal, P. 2007 Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers. J. Fluid Mech. 585, 323342.
Lewis, B. M. & Hawkins, H. F. 1982 Polygonal eye walls and rainbands in hurricans. Bull. Am. Meteorol. Soc. 63 (11), 12941300.
Lockitch, K. H. & Friedman, J. L. 1999 Where are the -modes of isentropic stars? Astrophys. J. 521, 764788.
Maas, L. R. M. 2003 On the amphidromic structure of inertial waves in a rectangular parallelepiped. Fluid Dyn. Res. 373, 373401.
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15 (9), 27572782.
Maas, L. R. M., Benielli, D., Sommeria, J. & Lam, F.-P. A. 1997 Observation of an internal wave attractor in a confined, stably stratified fluid. Nature 388, 557561.
Maas, L. R. M. & Lam, F.-P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 141.
Manders, A. M. M. & Maas, L. R. M. 2004 On the three-dimensional structure of the inertial wavefield in a rectangular basin with one sloping boundary. Fluid Dyn. Res. 35, 121.
Münnich, M. 1996 The influence of bottom topography on internal seiches in stratified media. Dyn. Atmos. Oceans 23, 257266.
Nöckel, J. U.1997 Resonances in nonintegrable open systems. PhD thesis, Yale University.
Nöckel, J. U., Stone, A. D., Chen, G., Grossman, H. L. & Chang, R. K. 1996 Directional emission from asymmetric resonant cavities. Opt. Lett. 21, 16091611.
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.
Nurijanyan, S., Bokhove, O. & Maas, L. R. M. 2013 A new semi-analytical solution for inertial waves in a rectangular parallelepiped. Phys. Fluids 25 (12), 126601.
Ogilvie, G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 1944.
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477509.
Phillips, O. M. 1963 Energy transfer in rotating fluids by reflection of inertial waves. Phys. Fluids 6, 513520.
Rabitti, A. & Maas, L. R. M. 2013 Meridional trapping and zonal propagation of inertial waves in a rotating fluid shell. J. Fluid Mech. 729, 445470.
Rieutord, M. 1991 Linear theory of rotating fluids using spherical harmonics part II, time-periodic flows. Geophys. Astrophys. Fluid Dyn. 59, 185208.
Rieutord, M., Georgeot, B. & Valdettaro, L. 2000 Wave attractors in rotating fluids: a paradigm for ill-posed Cauchy problems. Phys. Rev. Lett. 85 (20), 42774280.
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103144.
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.
Scolan, H., Ermanyuk, E. & Dauxois, T. 2013 Nonlinear fate of internal wave attractors. Phys. Rev. Lett. 110 (23), 234501.
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 35, 759773.
van Haren, H. & Gostiaux, L. 2012 Energy release through internal wave breaking. Oceanography 25 (2), 124131.
Whitham, G. B. 1960 A note on group velocity. J. Fluid Mech. 9, 347352.
Whitham, G. B. 1974 Linear and Non Linear Waves. John Wiley & Sons.
Wu, Y. 2005a Origin of tidal dissipation in Jupiter. I. Properties of inertial modes. Astrophys. J. 635, 674687.
Wu, Y. 2005b Origin of tidal dissipation in Jupiter. II. The value of . Astrophys. J. 13, 688710.
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36 (1), 281314.
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, 212235.
Zhang, K., Earnsahw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103119.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed