Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-27T06:09:07.457Z Has data issue: false hasContentIssue false
  • English
  • Français

Inviscid critical and near-critical reflection of internal waves in the time domain

Published online by Cambridge University Press:  14 March 2011

ALBERTO SCOTTI
ALBERTO SCOTTI*
Affiliation:
Department of Marine Sciences, University of North Carolina, Chapel Hill, NC 27599, USA
*
†Email address for correspondence: ascotti@unc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A solution that describes the inviscid reflection of internal waves off a sloping bottom in time is derived under conditions of linearity and uniform stratification. The solution is well behaved even under critical conditions. In the region ky < Nt, where k is the along-slope wavenumber of the incoming wave, y is the slope-normal direction, N is the Brünt–Väisälä frequency and t is time, an approximation can be written in terms of Lommel's function of two variables. The analysis can easily be extended to the case of a beam of finite width. In the non-critical case, the streamfunction relaxes to the classical Phillips steady-state solution in the region y < cgt, where cg is the slope-normal component of the group velocity for waves at the forcing frequency. However, it is found that the region where the along-boundary component of the velocity relaxes to the Phillips solution is also bounded from below, leaving a region very close to the wall where the classical solution misses important elements of the reflectionprocess. This leads to interesting properties near the boundary, especially relatively to the formation of shear-driven unstable conditions.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 464 - 488
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.)

References

REFERENCES

Arnold, V. I. & Khesin, B. A. 1998 Topological Methods in Hydrodynamics. Springer.CrossRefGoogle Scholar
Barenblatt, G. I. 1996 Scaling, Self-Similarity and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
Cacchione, D.A. & Drake, D. E. 1986 Nepheloid layers and internal waves over continental shelves and slopes. Geo. Mar. Lett. 16, 147–152.CrossRefGoogle Scholar
Cacchione, D. A., Pratson, L. F. & Ogston, A. S. 2002 The shaping of continental slopes by internal tides. Science 296, 724–727.CrossRefGoogle ScholarPubMed
Cacchione, D. A. & Wunsch, C. 1974 Experimental study of internal waves on a slope. J. Fluid Mech. 66, 223–239.CrossRefGoogle Scholar
Dauxois, T. & Young, W. R. 1999 Near-critical reflection of internal waves. J. Fluid Mech. 390, 271–295.CrossRefGoogle Scholar
De Silva, I. P. D., Imberger, J. & Ivey, G. N. 1997 Localized mixing due to a breaking internal wave ray at a sloping bed. J. Fluid Mech. 350, 1–27.CrossRefGoogle Scholar
Dickson, R. R. & McCave, I. N. 2002 Nepheloid layers on the continental slope west of Porcupine Bank. Deep-Sea Res. 33, 797–818.Google Scholar
Erdélyi, A. 1956 Asymptotic Expansions. Dover.Google Scholar
Gostiaux, L., Dauxois, T., Didelle, H., Sommeria, J. & Viboud, S. 2006 Quantitative laboratory observations of internal wave reflection on ascending slopes. Phys. Fluids 18, 056602, doi:10.1063/1.2197528.CrossRefGoogle Scholar
Ivey, G. N. & Nokes, R. I. 1989 Vertical mixing due to the breaking of critical internal waves on sloping boundaries. J. Fluid Mech. 204, 479–500.CrossRefGoogle Scholar
Kistovich, Yu. V. & Chashechkin, Yu. D. 1995 The reflection of beams of internal gravity waves at a flat rigid surface. J. Appl. Math. Mech. 59 (4), 579–585.CrossRefGoogle Scholar
Legg, S. & Adcroft, A. 2003 Internal wave breaking at concave and convex continental slopes. J. Phys. Oceanogr. 33, 2224–2246.2.0.CO;2>CrossRefGoogle Scholar
von Lommel, E. C. J. 1884–1886 Die beugungserscheinungen einer kreisrunden oeffnung und eines kreisrunden schirmchens theoretisch und experimentell bearbeitet. Abh. der Math. Phys. Classe der k. b. Akad. der Wiss. (Mnchen) 15, 229–328.Google Scholar
MacIntyre, S. 1998 Turbulent mixing and resource supply to phytoplankton. In Physical Processes in Lakes and Oceans, Coastal Estuarine Studies (ed. Imberger, J.), vol. 54, pp. 561–590. AGU.Google Scholar
McPhee-Shaw, E. E. 2006 Boundary-interior exchange: reviewing the idea that internal-wave mixing enhances lateral dispersal near continental margins. Deep-Sea Res. 43, 42–59.Google Scholar
McPhee-Shaw, E. E. & Kunze, E. 2002 Boundary layer intrusions from a sloping bottom: a mechanism for generating intermediate nepheloid layers. J. Geophys. Res. 107 (C6), 3050.Google Scholar
Mercier, M., Martinand, D., Mathur, M., Gostiuax, L., Peacock, T. & Dauxois, T. 2010 New wave generation. J. Fluid Mech. 657 308–334.CrossRefGoogle Scholar
Moum, J. N., Caldwell, D. R., Nash, J. D. & Gunderson, G. D. 2002 Observations of boundary mixing over the continental slope. J. Phys. Ocean. 32, 2113–2130.2.0.CO;2>CrossRefGoogle Scholar
Nash, J. D., Kunze, E., Toole, J. & Schmitt, R. 2004 Internal tide reflection and turbulent mixing on the continental slope. J. Phys. Oceanogr. 34, 117–1134.2.0.CO;2>CrossRefGoogle Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Pocklington, H. C. 1905 Growth of a wave-group when the group velocity is negative. Nature 71, 607–608.CrossRefGoogle Scholar
Slinn, D. N. & Riley, J. J. 1998 Turbulent dynamics of a critically reflecting internal gravity wave. Theor. Comput. Fluid Dyn. 11, 281–303.CrossRefGoogle Scholar
Sobolev, S. L. 1954 On a new problem of mathematical physics. Izvestiya USSR Acad. Sci., Ser. Mat. 18, 3–50.Google Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559–593.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2003 Nonlinear internal gravity beams. J. Fluid Mech. 482, 141–161.CrossRefGoogle Scholar
Tabaei, A., Akylas, T. R. & Lamb, K. G. 2005 Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech. 526, 217–243.CrossRefGoogle Scholar
Thorpe, S. A. 1987 On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279–302.CrossRefGoogle Scholar
Venayagamoorthy, S. K. & Fringer, O. B. 2006 Numerical simulations of the interaction of internal waves with a shelf break. Phys. Fluids 18, 076603.CrossRefGoogle Scholar
Watson, G. N. 1995 A Treatise on the Theory of Bessel Functions. Cambridge University Press.Google Scholar
Wunsch, C. 1968 On the propagation of internal waves up a slope. Deep-Sea Res. 25, 251–258.Google Scholar
Wunsch, C. 1969 Progressive internal waves on slopes. J. Fluid Mech. 35, 131–144.CrossRefGoogle Scholar