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Internal wave attractors over random, small-amplitude topography

Published online by Cambridge University Press:  09 December 2015

Yuan Guo  and
Miranda Holmes-Cerfon
Yuan Guo
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY, USA
Miranda Holmes-Cerfon*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, NY, USA
*
†Email address for correspondence: holmes@cims.nyu.edu
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Abstract

We consider whether small-amplitude topography in a two-dimensional ocean may contain internal wave attractors. These are closed orbits formed by the characteristics (or wave beam paths) of the linear, inviscid, steady-state Boussinesq equations, and their existence may imply enhanced scattering and energy decay for the internal tide when dissipation is present. We develop a numerical code to detect attractors over arbitrary topography, and apply this to random, Gaussian topography with different covariance functions. The rate of attractors per length of topography increases with the fraction of supercritical topography, but surprisingly, it also increases as the amplitude of the topography is decreased, while the supercritical fraction is held constant. This can partly be understood by appealing to Rice’s formula for the rate of zero crossings of a stochastic process. We compute the rate of attractors for a covariance function typical of ocean bathymetry away from large features and find it is about 10 attractors per 1000 km. This could have implications for the overall energy budget of the ocean.

JFM classification

Geophysical and Geological Flows: Internal waves Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Geophysical and Geological Flows: Topographic effects

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Papers
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Journal of Fluid Mechanics , Volume 787 , 25 January 2016 , pp. 148 - 174
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© 2015 Cambridge University Press 

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References

Adler, R. J. & Taylor, J. E. 2007 Random Fields and Geometry. Springer.Google Scholar
Alford, M. H., Peacock, T. et al. 2015 The formation and fate of internal waves in the South China Sea. Nature 521, 65–71.CrossRefGoogle ScholarPubMed
Bajars, J., Frank, J. & Maas, L. R. M. 2013 On the appearance of internal wave attractors due to an initial or parametrically excited disturbance. J. Fluid Mech. 714, 283–311.CrossRefGoogle Scholar
Balmforth, N. J., Ierley, G. R. & Young, W. R. 2002 Tidal conversion by subcritical topography. J. Phys. Oceanogr. 32, 2900–2914.2.0.CO;2>CrossRefGoogle Scholar
Balmforth, N. J. & Peacock, T. 2009 Tidal conversion by supercritical topography. J. Phys. Oceanogr. 39 (8), 1965–1974.CrossRefGoogle Scholar
Balmforth, N. J., Spiegel, E. A. & Tresser, C. 1995 Checkerboard maps. Chaos 5, 216–226.CrossRefGoogle ScholarPubMed
Bell, T. H. 1975a Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech. 67, 705–722.CrossRefGoogle Scholar
Bell, T. H. 1975b Statistical features of sea-floor topography. Deep-Sea Res. 22, 883–892.Google Scholar
Bühler, O. & Holmes-Cerfon, M. 2011 Decay of an internal tide due to random topography in the ocean. J. Fluid Mech. 678, 271–293.CrossRefGoogle Scholar
Drijfhout, S. & Maas, L. R. M. 2007 Impact of channel geometry and rotation on the trapping on internal tides. J. Phys. Oceanogr. 37, 2740–2763.CrossRefGoogle Scholar
Echeverri, P., Yokossi, T., Balmforth, N. J. & Peacock, T. 2011 Tidally generated internal-wave attractors between double ridges. J. Fluid Mech. 669, 354–374.CrossRefGoogle Scholar
Egbert, G. D. & Ray, R. D. 2000 Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature 405, 775–778.CrossRefGoogle ScholarPubMed
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253–282.CrossRefGoogle Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Grimshaw, R., Pelinovsky, E. & Talipova, T. 2010 Nonreflecting internal wave beam propagation in the deep ocean. J. Phys. Oceanogr. 40 (4), 802–813.CrossRefGoogle Scholar
Hazewinkel, J., Breevoort, P., Van 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
Hida, T. & Hitsuda, M. 1993 Gaussian Process. American Mathematical Society.Google Scholar
John, F. 1941 The Dirichlet problem for a hyperbolic equation. Am. J. Maths 63, 141–154.CrossRefGoogle Scholar
Kelly, S. M., Jones, N. L., Nash, J. D. & Waterhouse, A. F. 2013 The geography of semidurnal mode-1 internal-tide energy loss. Geophys. Res. Lett. 40, 4689–4693.CrossRefGoogle Scholar
Khatiwala, S. 2003 Generation of internal tides in an ocean of finite depth: analytical and numerical calculations. Deep-Sea Res. 50, 3–21.CrossRefGoogle Scholar
Klymak, J. M., Buijsman, M., Legg, S. & Pinkel, R. 2013 Parameterizing surface and internal tide scattering and breaking on supercritical topography: the one- and two-ridge cases. J. Phys. Oceanogr. 43, 1380–1397.CrossRefGoogle Scholar
Laurent, L. St & Garrett, C. 2002 The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32 (10), 2882–2899.2.0.CO;2>CrossRefGoogle Scholar
Legg, S. 2014 Scattering of low-mode internal waves at finite isolated topography. J. Phys. Oceanogr. 44, 359–383.CrossRefGoogle Scholar
Li, Y. & Mei, C. C. 2014 Scattering of internal tides by irregular bathymetry of large extent. J. Fluid Mech. 747, 481–505.CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Young, W. R. 2002 Conversion of the barotropic tide. J. Phys. Oceanogr. 32 (5), 1554–1566.2.0.CO;2>CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Young, W. R. 2003 Tidal conversion at a very steep ridge. J. Fluid Mech. 495, 175–191.CrossRefGoogle 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. 2005 Wave attractors – linear yet nonlinear. Intl J. Bifurcation Chaos 15, 2757–2782.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. 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
Mathur, M., Carter, G. S. & Peacock, T. 2014 Topographic scattering of the low-mode internal tide in the deep ocean. J. Geophys. Res. Oceans 119, 2165–2182.CrossRefGoogle Scholar
Müller, P. & Liu, X. 2000 Scattering of internal waves at finite topography in two dimensions part i: theory and case studies. J. Phys. Oceanogr. 30 (3), 532–549.2.0.CO;2>CrossRefGoogle Scholar
Müller, P. & Xu, N. 1992 Scattering of oceanic internal gravity waves off random bottom topography. J. Phys. Oceanogr. 22 (5), 474–488.2.0.CO;2>CrossRefGoogle Scholar
Nycander, J. 2005 Generation of internal waves in the deep ocean by tides. J. Geophys. Res. 110, C10028.Google 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
Petrelis, F., Llewellyn Smith, S. G. & Young, W. R. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36 (6), 1053–1071.CrossRefGoogle Scholar
Rabitti, A. & Maas, L. R. M. 2013 Meridional trapping and zonal propagation of inertial waves in a rotating fluid shell. J. Fluid Mech. 729, 445–470.CrossRefGoogle Scholar
Rabitti, A. & Maas, L. R. M. 2014 Inertial wave rays in rotating spherical fluid domains. J. Fluid Mech. 758, 621–654.CrossRefGoogle Scholar
Ray, R. D. & Mitchum, G. T. 1996 Surface manifestation of internal tides generated near Hawaii. Geophys. Res. Lett. 23, 2101–2104.CrossRefGoogle Scholar
Rice, S. O. 1945 Mathematical analysis of random noise. Bell Syst. Tech. J. 24, 46–156.CrossRefGoogle Scholar
Rieutord, M., Georgeot, B. & Valdettaro, L. 2001 Internal waves in a rotating spherical shell: attractors and symptomatic spectrum. J. Fluid Mech. 435, 103–144.CrossRefGoogle Scholar
Schwarz, D.2010 http://www.mathworks.com/matlabcentral/fileexchange/11837-fast-and-robust-curve-intersections.Google Scholar
Sobolev, S. L. 1954 On a new problem in mathematical physics. Izv. Akad. Nauk SSSR 18, 3–50.Google Scholar
Swart, A., Sleijpen, G. L. G., Maas, L. R. M. & Brandts, J. 2007 Numerical solution of the two-dimensional Poincaré equation. J. Comput. Appl. Maths 200, 317–341.CrossRefGoogle Scholar
Tilgner, A. 1999 Driven inertial oscillations in spherical shells. Phys. Rev. E 59, 1789–1794.CrossRefGoogle Scholar
Wunsch, C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131–141.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the ocean. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar
Yaglom, A. M. 1962 An Introduction to the Theory of Stationary Random Functions. Oxford University Press.Google Scholar
Zhao, Z., Alford, M. & MacKinnon, J. 2010 Long-range propagation of the semidiurnal tide from the Hawaiian ridge. J. Phys. Oceanogr. 40, 713–736.CrossRefGoogle Scholar
Zhao, Z., Alford, M. H., Girton, J., Johnston, T. M. S. & Carter, G. 2011 Internal tides around the Hawaiian Ridge estimated from multisatellite altimetry. J. Geophys. Res. 116, C12039.Google Scholar