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

Published online by Cambridge University Press:  24 July 2013

Anna Rabitti  and
Leo R. M. Maas
Anna Rabitti*
Affiliation:
Department of Physical Oceanography, NIOZ Royal Netherlands Institute for Sea Research, PO Box 59, 1790 AB Texel, The Netherlands
Leo R. M. Maas
Affiliation:
Department of Physical Oceanography, NIOZ Royal Netherlands Institute for Sea Research, PO Box 59, 1790 AB Texel, The Netherlands
*
†Email address for correspondence: anna.rabitti@nioz.nl
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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.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 445 - 470
Copyright
©2013 Cambridge University Press 

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References

Baines, P. G. 1971 The reflexion of internal/inertial waves from bumpy surfaces. J. Fluid Mech. 46, 273–292.Google Scholar
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 $\gamma $ Doradus. Mon. Not. R. Astron. Soc. 281, 1315–1325.Google Scholar
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), 497–500.CrossRefGoogle ScholarPubMed
Bretherton, F. P. 1964 Low frequency oscillations trapped near the equator. Tellus 16, 181–185.CrossRefGoogle Scholar
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), 233–253.Google Scholar
Bryan, G. H. 1889 The waves on a rotating liquid spheroid of finite ellipticity. Proc. R. Soc. Lond. 45, 42–45.Google Scholar
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).CrossRefGoogle Scholar
Cartan, M. E. 1922 Sur les petites oscillations dune masse de fluide. Bull. Sci. Math. 46, 317–369.Google Scholar
Dauxois, T. & Young, W. R. 2000 Near-critical reflection of internal waves. J. Fluid Mech. 390, 271–295.Google Scholar
Dengler, M. & Quadfasel, D. 2002 Equatorial deep jets and abyssal mixing in the Indian Ocean. J. Phys. Oceanogr. 32 (4), 1165–1180.Google Scholar
Dintrans, B., Rieutord, M. & Valdettaro, L. 1999 Gravito-inertial waves in a rotating stratified sphere or spherical shell. J. Fluid Mech. 398, 271–297.Google Scholar
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.CrossRefGoogle Scholar
Eriksen, C. C. 1982 Observations of internal wave reflection off sloping bottoms. J. Geophys. Res. 87, 525–538.Google Scholar
Eriksen, C. C. 1985 Implications of ocean bottom reflection for internal wave spectra and mixing. J. Phys. Oceanogr. (15), 1145–1156.Google Scholar
Firing, E. 1987 Deep zonal currents in the central equatorial Pacific. J. Mar. Res. 45 (4), 791–812.Google Scholar
Friedlander, S. 1982 Internal waves in a rotating stratified fluid in arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. (19), 267–291.Google Scholar
Friedlander, S. & Siegmann, W. L. 1982 Internal waves in a contained rotating stratified fluid. J. Fluid Mech. (114), 123–156.CrossRefGoogle Scholar
Galperin, B. 2004 The ubiquitous zonal jets in the atmospheres of giant planets and Earth’s oceans. Geophys. Res. Lett. 31 (13), 1–5.Google Scholar
Gerkema, T. & Shrira, V. I. 2005 Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195–219.Google Scholar
Gilbert, D. & Garrett, C. 1989 Implications for ocean mixing of internal wave scattering off irregular topography. J. Phys. Oceanogr. (19), 1716–1729.2.0.CO;2>CrossRefGoogle Scholar
Görtler, H. 1943 Uber eine schwingungserscheinung in flussigkeiten mit stabiler dichteschichtung. Z. Angew. Math. Mech. 23 (1), 65–71.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press.Google Scholar
Grisouard, N. & Bühler, O. 2012 Forcing of oceanic mean flows by dissipating internal tides. J. Fluid Mech. 708, 250–278.Google Scholar
Harlander, U. & Maas, L. R. M. 2006 Characteristics and energy rays of equatorially trapped, zonally symmetric internal waves. Meteorologische Z. 15 (4), 439–450.Google Scholar
Harlander, U. & Maas, L. R. M. 2007 Internal boundary layers in a well-mixed equatorial atmosphere/ocean. Dyn. Atmos. Oceans 44 (1), 1–28.Google Scholar
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), 51–56.CrossRefGoogle Scholar
Hazewinkel, J., Maas, L. R. M. & Dalziel, S. B. 2010b Tomographic reconstruction of internal wave patterns in a paraboloid. Exp. Fluids 50 (2), 247–258.CrossRefGoogle Scholar
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, 373–382.CrossRefGoogle Scholar
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), 31–42.Google Scholar
Hughes, B. 1964 Effect of rotation on internal gravity waves. Nature 201 (4921), 798–801.CrossRefGoogle Scholar
John, F. 1941 The Dirichlet problem for hyperbolic equation. Am. J. Maths 63, 141–154.Google Scholar
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).Google Scholar
Maas, L. R. M. 2001 Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids. J. Fluid Mech. 437, 13–28.Google Scholar
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15 (9), 2757–2782.Google Scholar
Maas, L. R. M. 2009 Exact analytic self-similar solution of a wave attractor field. Physica D: Nonlinear Phenomena 238 (5), 502–505.Google Scholar
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, 557–561.CrossRefGoogle Scholar
Maas, L. R. M. & Harlander, U. 2007 Equatorial wave attractors and inertial oscillations. J. Fluid Mech. 570, 47–67.Google Scholar
Maas, L. R. M. & Lam, F.-P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.Google Scholar
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, 59–88.Google Scholar
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, 1–21.CrossRefGoogle Scholar
Morize, C., Le Bars, M., Le Gal, P. & Tilgner, A. 2010 Experimental determination of zonal winds driven by tides. Phys. Rev. Lett. 104 (21), 28–31.Google Scholar
Münnich, M. 1996 The influence of bottom topography on internal seiches in stratified media. Dyn. Atmos. Oceans 23, 257–266.CrossRefGoogle Scholar
Ogilvie, G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 19–44.Google Scholar
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477–509.CrossRefGoogle Scholar
Phillips, O. M. 1963 Energy transfer in rotating fluids by reflection of inertial waves. Phys. Fluids (6), 513–520.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge.Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2000 Wave attractors in rotating fluids: a paradigm for ill-posed Cauchy problems. Phys. Rev. Lett. 85 (20), 4277–4280.Google Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103–144.Google Scholar
Rieutord, M. & Valdettaro, L. 2010 Viscous dissipation by tidally forced inertial modes in a rotating spherical shell. J. Fluid Mech. 643, 363.CrossRefGoogle Scholar
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, 260–268.CrossRefGoogle Scholar
Send, U, Eden, C & Schott, F 2002 Atlantic equatorial deep jets: space–time structure and cross-equatorial fluxes. J. Phys. Oceanogr. 32, 891–902.Google Scholar
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), 857–875.Google Scholar
Stern, M. E. 1963 Trapping of low frequency oscillations in an equatorial ‘boundary layer’. Tellus 15, 246–250.Google Scholar
Stewartson, K. 1971 On trapped oscillations of a rotating fluid in a thin spherical shell I. Tellus 23, 506–510.Google Scholar
Stewartson, K. 1972 On trapped oscillations of a rotating fluid in a thin spherical shell II. Tellus 24, 283–287.Google Scholar
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 35, 759–773.CrossRefGoogle Scholar
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), 16–34.Google Scholar
Thorpe, S. A. 1997 On the interactions of internal waves reflecting from slopes. J. Phys. Oceanogr. 27, 2072–2078.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S. A. 2001 On the reflection of internal wave groups from sloping topography. J. Phys. Oceanogr. 31 (1999), 3121–3126.Google Scholar
Tilgner, A. 2007 Zonal wind driven by inertial modes. Phys. Rev. Lett. 99 (19), 1–4.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar
Wunsch, C. 1968 On the propagation of internal waves up a slope. Deep-Sea Res. 15, 251–258.Google Scholar
Wunsch, C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131–141.Google Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36 (1), 281–314.Google Scholar
Zhang, K., Earnsahw, P., Liao, X. & Busse, F. H. 2001 On inertial waves in a rotating fluid sphere. J. Fluid Mech. 437, 103–119.CrossRefGoogle Scholar
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), 713–736.Google Scholar