Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-09T21:49:26.736Z Has data issue: false hasContentIssue false
  • English
  • Français

Decay of an internal tide due to random topography in the ocean

Published online by Cambridge University Press:  18 April 2011

OLIVER BÜHLER  and
MIRANDA HOLMES-CERFON
OLIVER BÜHLER*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
MIRANDA HOLMES-CERFON
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Topographic effects Waves/Free-surface Flows: Wave scattering
Type
Papers
Information
Journal of Fluid Mechanics , Volume 678 , 10 July 2011 , pp. 271 - 293
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

‡

Current address: School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.

References

REFERENCES

Alford, M. H. 2003 Redistribution of energy available for ocean mixing by long-range propagation of internal waves. Nature 423, 159–162.CrossRefGoogle ScholarPubMed
Arnold, V. I. & Khesin, B. A. 1998 Topological Method in Hydrodynamics. Springer.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
Bell, T. H. 1975 Statistical features of sea-floor topography. Deep-Sea Res. 22 (12), 883–892.Google Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge Monographs on Mechanics. Cambridge University Press.CrossRefGoogle Scholar
Bühler, O. & Holmes-Cerfon, M. 2009 Particle dispersion by random waves in rotating shallow water. J. Fluid Mech. 638, 5–26.CrossRefGoogle Scholar
Bühler, O. & Muller, C. M. 2007 Instability and focusing of internal tides in the deep ocean. J. Fluid Mech. 588, 1–28.CrossRefGoogle Scholar
Chen, E. 2009 Degradation of the internal tide over long bumpy topography. In Proc. Annual Summer Study in Geophysical Fluid Dynamics, pp. 248–268. Woods Hole Oceanographic Institution.Google Scholar
Echeverri, P. & Peacock, T. 2010 Internal tide generation by arbitrary two-dimensional topography. J. Fluid Mech. 659, 247–266.CrossRefGoogle Scholar
Garrett, C. 2003 Internal tides and ocean mixing. Science 301, 1858–1859.CrossRefGoogle ScholarPubMed
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Goff, J. & Jordan, T. 1988 Stochastic modelling of seafloor morphology: inversion of sea beam data for second-order statistics. J. Geophys. Res. 93 (B11), 13589–13608.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
Harlander, U. & Maas, L. R. M. 2007 Two alternatives for solving hyperbolic boundary value problems of geophysical fluid dynamics. J. Fluid Mech. 588, 331–351.CrossRefGoogle Scholar
Khatiwala, S. 2003 Generation of internal tides in an ocean of finite depth: analytical and numerical calculations. Deep-Sea Res. I. Oceanogr. Res. Papers 50 (1), 3–21.CrossRefGoogle Scholar
Kunze, E. & LlewellynSmith, S. G. Smith, S. G. 2004 The role of small-scale topography in turbulent mixing of the global ocean. Oceanography 17, 55–64.CrossRefGoogle Scholar
LlewellynSmith, S. G. Smith, S. G. & Young, W. R. 2003 Tidal conversion at a very steep ridge. J. Fluid Mech. 495, 175–191.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1969 On the reflexion of wave characteristics from rough surfaces. J. Fluid Mech. 37, 231–250.CrossRefGoogle Scholar
Maas, L. R. & Lam, F. P. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.CrossRefGoogle Scholar
Muller, C. M. & Bühler, O. 2009 Saturation of the internal tides and induced mixing in the abyssal ocean. J. Phys. Oceanogr. 39, 2077–2096.CrossRefGoogle Scholar
Muller, P. & Liu, X. B. 2000 a 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
Muller, P. & Liu, X. B. 2000 b Scattering of internal waves at finite topography in two dimensions. Part II. Spectral calculations and boundary mixing. J. Phys. Oceanogr. 30 (3), 550–563.2.0.CO;2>CrossRefGoogle Scholar
Muller, 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
Nachbin, A. & Papanicolaou, G. C. 1992 Water waves in shallow channels of rapidly varying depth. J. Fluid Mech. 241, 311–332.CrossRefGoogle Scholar
Nikurashin, M. & Ferrari, R. 2010 Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: application to the southern ocean. J. Phys. Oceanogr. 40, 2025–2042.CrossRefGoogle Scholar
Papanicolaou, G. C. & Kohler, W. 1974 Asymptotic theory of mixing stochastic ordinary differential equations. Commun. Pure Appl. Math. 27, 641–668.CrossRefGoogle Scholar
Petrelis, F., Smith, S. L. & Young, W. R. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36 (6), 1053–1071.CrossRefGoogle Scholar
Sobolev, S. L. 1954 On a new problem in mathematical physics. Izvestia Akad. Nauk SSSR 18, 3–50.Google Scholar
St Laurent, L. & 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
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar
Yaglom, A. M. 1962 An Introduction to the Theory of Stationary Random Functions. Dover.Google Scholar
Zhao, Z., Alford, M. H., MacKinnon, J. A. & Pinkel, R. 2010 Long-range propagation of the semidiurnal tide from the Hawaiian ridge. J. Phys. Oceanogr. 40, 713–736.CrossRefGoogle Scholar