Skip to main content
×
×
Home

Meridional trapping and zonal propagation of inertial waves in a rotating fluid shell

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

Inertial waves propagate in homogeneous rotating fluids, and constitute a challenging and simplified case study for the broader class of inertio-gravity waves, present in all geophysical and astrophysical media, and responsible for energetically costly processes such as diapycnal and angular momentum mixing. However, a complete analytical description and understanding of internal waves in arbitrarily shaped enclosed domains, such as the ocean or a planet liquid core, is still missing. In this work, the inviscid, linear inertial wave field is investigated by means of three-dimensional ray tracing in spherical shell domains, having in mind possible oceanographic applications. Rays are here classically interpreted as representative of energy paths, but in contrast to previous studies, they are now launched with a non-zero initial zonal component allowing for a more realistic, localized forcing and the development of azimuthal inhomogeneities. We find that meridional planes generally act in the shell geometry as attractors for ray trajectories. In addition, the existence of trajectories that are not subject to meridional trapping is here observed for the first time. Their dynamics was not captured by the previous purely meridional studies and unveils a new class of possible solutions for inertial motion in the spherical shell. Both observed behaviours shed some new light on possible mechanisms of energy localization, a key process that still deserves further investigation in our ocean, as well as in other stratified, rotating media.

Copyright
Corresponding author
Email address for correspondence: anna.rabitti@nioz.nl
References
Hide All
Baines, P. G. 1971 The reflexion of internal/inertial waves from bumpy surfaces. J. Fluid Mech. 46, 273292.
Balona, L. A., Bohm, T., Foing, B. H., Ghosh, K. K., Lagrange, A.-M., Lawson, W. A., James, S. D. & Baudrand, J. 1996 Line profile variations in Doradus. Mon. Not. R. Astron. Soc. 281, 13151325.
Brandt, P., Funk, A., Hormann, V., Dengler, M., Greatbatch, R. J. & Toole, J. M. 2011 Interannual atmospheric variability forced by the deep equatorial Atlantic Ocean. Nature 473 (7348), 497500.
Bretherton, F. P. 1964 Low frequency oscillations trapped near the equator. Tellus 16, 181185.
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.
Calkins, M. A., Noir, J., Eldredge, J. D. & Aurnou, J. M. 2010 Axisymmetric simulations of libration-driven fluid dynamics in a spherical shell geometry. Phys. Fluids 22 (086602).
Cartan, M. E. 1922 Sur les petites oscillations dune masse de fluide. Bull. Sci. Math. 46, 317369.
Dauxois, T. & Young, W. R. 2000 Near-critical reflection of internal waves. J. Fluid Mech. 390, 271295.
Dengler, M. & Quadfasel, D. 2002 Equatorial deep jets and abyssal mixing in the Indian Ocean. J. Phys. Oceanogr. 32 (4), 11651180.
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), 2740.
Eriksen, C. C. 1982 Observations of internal wave reflection off sloping bottoms. J. Geophys. Res. 87, 525538.
Eriksen, C. C. 1985 Implications of ocean bottom reflection for internal wave spectra and mixing. J. Phys. Oceanogr. (15), 11451156.
Firing, E. 1987 Deep zonal currents in the central equatorial Pacific. J. Mar. Res. 45 (4), 791812.
Friedlander, S. 1982 Internal waves in a rotating stratified fluid in arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. (19), 267291.
Friedlander, S. & Siegmann, W. L. 1982 Internal waves in a contained rotating stratified fluid. J. Fluid Mech. (114), 123156.
Galperin, B. 2004 The ubiquitous zonal jets in the atmospheres of giant planets and Earth’s oceans. Geophys. Res. Lett. 31 (13), 15.
Gerkema, T. & Shrira, V. I. 2005 Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195219.
Gilbert, D. & Garrett, C. 1989 Implications for ocean mixing of internal wave scattering off irregular topography. J. Phys. Oceanogr. (19), 17161729.
Görtler, H. 1943 Uber eine schwingungserscheinung in flussigkeiten mit stabiler dichteschichtung. Z. Angew. Math. Mech. 23 (1), 6571.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press.
Grisouard, N. & Bühler, O. 2012 Forcing of oceanic mean flows by dissipating internal tides. J. Fluid Mech. 708, 250278.
Harlander, U. & Maas, L. R. M. 2006 Characteristics and energy rays of equatorially trapped, zonally symmetric internal waves. Meteorologische 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. 2010a Comparison of laboratory and numerically observed scalar fields of an internal wave attractor. Eur. J. Mech. (B/Fluids) 30 (1), 5156.
Hazewinkel, J., Maas, L. R. M. & Dalziel, S. B. 2010b Tomographic reconstruction of internal wave patterns in a paraboloid. Exp. Fluids 50 (2), 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.
Horn, W. & Meincke, J. 1976 Note on the tidal current field in the continental slope area of Northwest Africa. Memo. Soc. Roy. Sci. Liege 6 (9), 3142.
Hughes, B. 1964 Effect of rotation on internal gravity waves. Nature 201 (4921), 798801.
John, F. 1941 The Dirichlet problem for hyperbolic equation. Am. J. Maths 63, 141154.
Koch, S., Harlander, U., Egbers, C. & Hollerbach, R. 2012 Inertial oscillations in a spherical shell induced by librations of the inner sphere. J. Fluid Mech. (under review).
Maas, L. R. M. 2001 Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids. J. Fluid Mech. 437, 1328.
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15 (9), 27572782.
Maas, L. R. M. 2009 Exact analytic self-similar solution of a wave attractor field. Physica D: Nonlinear Phenomena 238 (5), 502505.
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. & Harlander, U. 2007 Equatorial wave attractors and inertial oscillations. J. Fluid Mech. 570, 4767.
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. 2003 Observations of inertial waves in a rectangular basin with one sloping boundary. J. Fluid Mech. 493, 5988.
Manders, A. M. M. & Maas, L. R. M. 2004 On the three-dimensional structure of the inertial wave field in a rectangular basin with one sloping boundary. Fluid Dyn. Res. 35, 121.
Morize, C., Le Bars, M., Le Gal, P. & Tilgner, A. 2010 Experimental determination of zonal winds driven by tides. Phys. Rev. Lett. 104 (21), 2831.
Münnich, M. 1996 The influence of bottom topography on internal seiches in stratified media. Dyn. Atmos. Oceans 23, 257266.
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.
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge.
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. 2010 Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J. Fluid Mech. 643, 363.
Sauret, A., Cebron, D., Morize, C. & Le Bars, M. 2010 Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 662, 260268.
Send, U, Eden, C & Schott, F 2002 Atlantic equatorial deep jets: space–time structure and cross-equatorial fluxes. J. Phys. Oceanogr. 32, 891902.
Shen, M. & Keller, J. 1975 Uniform ray theory of surface, internal and acoustic wave propagation in a rotating ocean or atmosphere. SIAM J. Appl. Maths 28 (4), 857875.
Stern, M. E. 1963 Trapping of low frequency oscillations in an equatorial ‘boundary layer’. Tellus 15, 246250.
Stewartson, K. 1971 On trapped oscillations of a rotating fluid in a thin spherical shell I. Tellus 23, 506510.
Stewartson, K. 1972 On trapped oscillations of a rotating fluid in a thin spherical shell II. Tellus 24, 283287.
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 35, 759773.
Swart, A., Manders, A. M. M., Harlander, U. & Maas, L. R. M. 2010 Experimental observation of strong mixing due to internal wave focusing over sloping terrain. Dyn. Atmos. Oceans 50 (1), 1634.
Thorpe, S. A. 1997 On the interactions of internal waves reflecting from slopes. J. Phys. Oceanogr. 27, 20722078.
Thorpe, S. A. 2001 On the reflection of internal wave groups from sloping topography. J. Phys. Oceanogr. 31 (1999), 31213126.
Tilgner, A. 2007 Zonal wind driven by inertial modes. Phys. Rev. Lett. 99 (19), 14.
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.
Wunsch, C. 1968 On the propagation of internal waves up a slope. Deep-Sea Res. 15, 251258.
Wunsch, C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131141.
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36 (1), 281314.
Zhang, K., Earnsahw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103119.
Zhao, Z., Alford, M. H., MacKinnon, J. A. & Pinkel, R. 2010 Long-range propagation of the semidiurnal internal tide from the Hawaiian ridge. J. Phys. Oceanogr. 40 (4), 713736.
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