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Long nonlinear waves in stratified shear flows

Published online by Cambridge University Press:  19 April 2006

S. A. Maslowe  and
L. G. Redekopp
S. A. Maslowe
Affiliation:
Mathematics Department, McGill University, Montreal, P.Q. H3A 2K6
L. G. Redekopp
Affiliation:
Department of Aerospace Engineering, University of Southern California, Los Angeles, California 90007
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Abstract

The propagation of finite-amplitude internal waves in a shear flow is considered for wavelengths that are long compared to the shear-layer thickness. Both singular and regular modes are investigated, and the equation governing the amplitude evolution is derived. The theory is generalized to allow for a radiation condition when the region outside the stratified shear layer is unbounded and weakly stratified. In this case, the evolution equation contains a damping term describing energy loss by radiation which can be used to estimate the persistence of solitary waves or nonlinear wave packets in realistic environments. A continuous three-layer model is studied in detail and closed-form expressions are obtained for the phase speed and the coefficients of the nonlinear and dispersive terms in the amplitude equation as a function of Richardson number.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 101 , Issue 2 , 27 November 1980 , pp. 321 - 348
Copyright
© 1980 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1967 Handbook of Mathematical Functions. Washington: National Bureau of Standards.
Apel, J. R., Byrne, H. M., Proni, J. R. & Charnell, R. L. 1975 Observations of oceanic internal and surface waves from the Earth Resources Technology Satellite. J. Geophys. Res. 80, 865–881.Google Scholar
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form. J. Fluid Mech. 29, 241–270.Google Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.Google Scholar
Benney, D. J. 1966 Long nonlinear waves in fluid flows. J. Math. Phys. 45, 52–63.Google Scholar
Chen, H. H., Lee, Y. C. & Pereira, N. R. 1979 Algebraic internal wave solitons and the integrable Calogero-Moser-Sutherland N-body problem. Phys. Fluids 22, 187–188.Google Scholar
Christie, D. R., Muirhead, K. J. & Hales, A. L. 1978 On solitary waves in the atmosphere. J. Atmos. Sci. 35, 805–825.Google Scholar
Davis, R. E. & Acrivos, A. 1967 Solitary internal waves in deep water. J. Fluid Mech. 29, 593–607.Google Scholar
Farmer, D. M. & Smith, J. D. 1978 Nonlinear internal waves in a fjord. Hydrodynamics of Estuaries and Fjords (ed. J. Nihoul). Elsevier.
Hadamard, J. 1923 Lectures on Cauchy's Problem in Linear Partial Differential Equations. Yale University Press.
Hoiland, E. 1953 On the dynamic effect of variation in density on two-dimensional perturbations of flow with constant shear. Geophys. Publ. 18, no. 10.Google Scholar
Joseph, R. I. 1977 Solitary waves in a finite depth fluid. J. Phys. A, Math. Gen. 10, L225–L227.Google Scholar
Keulegan, G. H. 1953 Characteristics of internal solitary waves. J. Res. Nat. Bur. Stand. 51, 133–140.Google Scholar
Kubota, T., Ko, D. R. S. & Dobbs, L. 1978 Weakly-nonlinear, long internal gravity waves in stratified fluids of finite depth. J. Hydronautics 12, 157–165.Google Scholar
Lee, C. & Beardsley, R. 1974 The generation of long nonlinear internal waves in a weakly stratified shear flow. J. Geophys. Res. 79, 453–462.Google Scholar
Long, R. 1965 On the Boussinesq approximation and its role in the theory of internal waves. Tellus 17, 46–52.Google Scholar
Lyra, G. 1943 Theorie der stationären Leewellenströmung in freier Atmosphäre. Z. angew. Math. Mech. 23, 1–28.Google Scholar
Maslowe, S. A. 1972 The generation of clear air turbulence by nonlinear waves. Stud. Appl. Math. 51, 1–16.Google Scholar
Maslowe, S. A. 1973 Finite-amplitude Kelvin-Helmholtz billows. Boundary-Layer Met. 5, 43–52.Google Scholar
Matsuno, Y. 1979 Exact multi-soliton solution of the Benjamin-Ono equation. J. Phys. A, Math. Gen. 12, 619–621.Google Scholar
Miles, J. W. 1968 Waves and wave drag in stratified flows. Proc. 12th Int. Cong. Appl. Mech., Stanford, pp. 50–75.
Ono, H. 1975 Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39, 1082–1091.Google Scholar
Peters, A. S. & Stoker, J. J. 1960 Solitary waves in liquids having non-constant density. Comm. Pure Appl. Math. 13, 115–164.Google Scholar
Queney, P., Corby, G., Gerbier, N., Koschmieder, H. & Zierep, J. 1960 The airflow over mountains. World Meteor. Organization, Tech. Note 34.Google Scholar
Redekopp, L. G. 1977 On the theory of solitary Rossby waves. J. Fluid Mech. 82, 725–745.Google Scholar
Satsuma, J., Ablowitz, M. J. & Kodama, Y. 1980 On an internal wave equation describing a stratified fluid with finite depth. Phys. Lett. A (to appear).Google Scholar