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Equatorial wave attractors and inertial oscillations

Published online by Cambridge University Press:  14 October 2021

Leo R. M. Maas  and
Uwe Harlander
Leo R. M. Maas
Affiliation:
Royal Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Texel, The Netherlands
Uwe Harlander
Affiliation:
Royal Netherlands Institute for Sea Research, P.O. Box 59, 1790 AB Texel, The Netherlands
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Abstract

Three different approximations to the axisymmetric small-disturbance dynamics of a uniformly rotating thin spherical shell are studied for the equatorial region assuming time-harmonic motion. The first is the standard β-plane model. The second is Stern's (Tellus, vol. 15, 1963, p. 246) homogeneous, equatorial β-plane model of inertial waves (that includes all Coriolis terms). The third is a version of Stern's equation extended to include uniform stratification. It is recalled that the boundary value problem (BVP) that governs the streamfunction of zonally symmetric waves in the meridional plane becomes separable only for special geometries. These separable BVPs allow us to make a connection between the streamfunction field and the underlying geometry of characteristics of the governing equation. In these cases characteristics are each seen to trace a purely periodic path. For most geometries, however, the BVP is non-separable and characteristics and therefore wave energy converge towards a limit cycle, referred to as an equatorial wave attractor. For Stern's model we compute exact solutions for wave attractor regimes. These solutions show that wave attractors correspond to singularities in the velocity field, indicating an infinite magnification of kinetic energy density along the attractor. The instability that arises occurs without the necessity of any ambient shear flow and is referred to as geometric instability.

For application to ocean and atmosphere, Stern's model is extended to include uniform stratification. Owing to the stratification, characteristics are trapped near the equator by turning surfaces. Characteristics approach either equatorial wave attractors, or point attractors situated at the intersections of turning surfaces and the bottom. At these locations, trapped inertia–gravity waves are perceived as near-inertial oscillations. It is shown that trapping of inertia–gravity waves occurs for any monochromatic frequency within the allowed range, while equatorial wave attractors exist in a denumerable, infinite set of finite-sized continuous frequency intervals. It is also shown that the separable Stern equation, obtained as an approximate equation for waves in a homogeneous fluid confined to the equatorial part of a spherical shell, gives an exact description for buoyancy waves in uniformly but radially stratified fluids in such shells.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 570 , 10 January 2007 , pp. 47 - 67
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

van Aken, H. M., Maas, L. R. M. & van Haren, H. 2005 Observations of inertial wave events near the continental slope off Goban Spur. J. Phys. Oceanogr. 35, 1329–1340.Google Scholar
Bretherton, F. P. 1964 Low frequency oscillations trapped near the equator. Tellus 16, 181–185.10.3402/tellusa.v16i2.8922CrossRefGoogle 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.CrossRefGoogle Scholar
Farrell, B. 1982 The initial growth of disturbances in a baroclinic flow. J. Atmos. Sci. 39, 1663–1686.2.0.CO;2>CrossRefGoogle Scholar
Friedlander, S. & Siegmann, W. L. 1982 Internal waves in a rotating stratified fluid in an arbitrary gravitational field. Geophys. Astrophys. Fluid Dyn. 19, 267–291.CrossRefGoogle Scholar
Fu, L. L. 1981 Observations and models of inertial waves in the deep ocean. Rev. Geophys. Space Phys. 19, 141–170.CrossRefGoogle Scholar
Galperin, B., Nakano, H., Huang, H.-P. & Sukoriansky, S. 2004 The ubiquitous zonal jets in the atmospheres of giant planets and Earth's oceans. Geophys. Res. Lett. 31, L13303, doi:10.1029/2004GL019691.CrossRefGoogle Scholar
Gerkema, T. & Shrira, V. I. 2005a Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195–219.CrossRefGoogle Scholar
Gerkema, T. & Shrira, V. I. 2005b Near-inertial waves on the ‘non-traditional’ β plane. J. Geophys. Res. 110, C01003, doi:10.1029/2004JC002519.Google Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic.Google Scholar
Gille, S. T., Smith, S. G. L. & Statom, N. M. 2005 Global observations of the land breeze. Geophys. Res. Lett. 32, L05605, doi:10.1029/2004GL022139.CrossRefGoogle Scholar
Greenspan, H. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
van Haren, H. 2005 Sharp near-equatorial transitions in inertial motions and deep-ocean step-formation. Geophys. Res. Lett. 32, L01605, doi:10.1029/2004GL021630.Google Scholar
van Haren, H., Maas, L. R. M. & van Aken, H. 2002 On the nature of internal wave spectra near a continental slope. Geophys. Res. Lett. 29 (12), doi:10.1029/2001GL014341.CrossRefGoogle Scholar
Harlander, U. & Maas, L. R. M. 2006a Characteristics and energy rays of equatorially trapped, zonally symmetric internal waves. Met. Z. 15, 439–450.CrossRefGoogle Scholar
Harlander, U. & Maas, L. R. M. 2006b On inertial boundary layers in a well mixed equatorial atmosphere. Dyn. Atmos. Oceans (submitted).Google Scholar
Høiland, E. 1962 Discussion of a hyperbolic equation relating to inertia and gravitational fluid oscillations. Geophys. Publ. XXIV, 211–227.Google Scholar
Hua, B. L., Moore, D. W. & Gentil, S. L. 1997 Inertial nonlinear equilibration of equatorial flows. J. Fluid Mech. 331, 345–371.CrossRefGoogle Scholar
Israeli, M. 1972 On trapped oscillations of rotating fluids in spherical shells. Stud. Appl. Maths L1, 219–237.CrossRefGoogle Scholar
John, F. 1975 Partial Differential Equations, 2nd edn. Springer.CrossRefGoogle 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.CrossRefGoogle Scholar
Maas, L. R. M. 2003 On the amphidromic structure of inertial waves in a rectangular parallelepiped. Fluid Dyn. Res. 33, 373–401.CrossRefGoogle Scholar
Maas, L. R. M. 2005 Wave attractors: linear yet nonlinear. Intl J. Bifurcation Chaos 15, 2757–2782.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. & Lam, F.-P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.CrossRefGoogle 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.CrossRefGoogle 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
Matsuno, T. 1966 Quasi-geostrophic motions in the equatorial area. J. Met. Soc. Japan II 44, 25–43.CrossRefGoogle Scholar
McEwan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–640.CrossRefGoogle Scholar
Merzbacher, E. 1970 Quantum Mechanics, 2nd edn. Wiley & Sons.Google Scholar
Munk, W. 1980 Internal wave spectra at the buoyant and inertial frequencies. J. Phys. Oceanogr. 10, 1718–1728.2.0.CO;2>CrossRefGoogle Scholar
Munk, W. & Phillips, N. A. 1968 Coherence and band structure of inertial motion in the sea. Rev. Geophys. 4, 447–472.CrossRefGoogle Scholar
Münnich, W. 1993 On the influence of bottom topography on the vertical structure of internal seiches. PhD thesis ETA Zürich.Google Scholar
Nedelec, J.-C. 2001 Acoustic and Electromagnetic Equations. Springer.10.1007/978-1-4757-4393-7CrossRefGoogle Scholar
Ogilvie, G. I. 2005 Wave attractors and the asymptotic dissipation rate of tidal disturbances. J. Fluid Mech. 543, 19–44.CrossRefGoogle Scholar
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610, 477–509.CrossRefGoogle Scholar
Ooyama, K. 1966 On the stability of baroclinic circular vortex: a sufficient crieterion for stability. J. Atmos. Sci. 23, 43–53.2.0.CO;2>CrossRefGoogle Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2000 Wave attractors in rotating fluids: a paradigm for ill-posed Cauchy problems. Phys. Rev. Lett. 85, 4277–4280.CrossRefGoogle ScholarPubMed
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum. J. Fluid Mech. 435, 103–144.10.1017/S0022112001003718CrossRefGoogle Scholar
Rieutord, M. & Valdettaro, L. 1997 Inertial waves in a rotating spherical shell. J. Fluid Mech. 341, 77–99.CrossRefGoogle Scholar
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, 345–360.CrossRefGoogle Scholar
Stern, M. E. 1963 Trapping of low frequency oscillations in an equatorial ‘boundary layer’. Tellus 15, 246–250.CrossRefGoogle Scholar
Stewartson, K. 1971 On trapped oscillations of a rotating fluid in a thin spherical shell. Tellus 23, 506–510.CrossRefGoogle Scholar
Stewartson, K. 1972 On trapped oscillations of a rotating fluid in a thin spherical shell II. Tellus 24, 283–287.CrossRefGoogle Scholar
Stewartson, K. & Rickard, J. A. 1969 Pathological oscillations of a rotating fluid. J. Fluid Mech. 5, 577–592.CrossRefGoogle Scholar
Stewartson, K. & Walton, I. C. 1976 On inertial oscillations in the oceans. Tellus 28, 71–73.CrossRefGoogle Scholar
Tilgner, A. 1999 Driven inertial oscillations in spherical shells. Phys. Rev. E 59, 1789–1794.CrossRefGoogle Scholar
Yavneh, I., McWilliams, J. C. & Molemaker, M. J. 2001 Non-axisymmetyric instability of centrifugally stable stratified Taylor—Couette flow. J. Fluid Mech. 448, 1–21.CrossRefGoogle Scholar