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Long nonlinear internal waves in channels of arbitrary cross-section

Published online by Cambridge University Press:  12 April 2006

R. H. J. Grimshaw
R. H. J. Grimshaw
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
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Abstract

Using a two-layer fluid model, equations are developed which describe long internal waves propagating along a channel of arbitrary cross-section. Expressions for the phase speed of these waves are derived in terms of geometric properties of the cross-section. When weakly nonlinear effects are balanced by weak dispersion, a Korteweg-de Vries equation is derived to describe the waves. The effects of a slowly varying cross-section are included. Applications of the theory are made to recent observations of internal surges in lakes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 86 , Issue 3 , 14 June 1978 , pp. 415 - 431
Copyright
© 1978 Cambridge University Press

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References

Farmer, D. M. 1978 Observations of long nonlinear internal waves in a lake. J. Phys. Ocean. 8 (to appear).Google Scholar
Grimshaw, R. 1970 The solitary wave in water of variable depth. J. Fluid Mech. 42, 639–656.Google Scholar
Halpern, D. 1971 Observations of short-period internal waves in Massachusetts Bay. J. Mar. Res. 29, 116–132.Google Scholar
Hunkins, K. & Fliegel, M. 1973 Internal undular surges in Seneca Lake: a natural occurrence of solitons. J. Geophys. Res. 78, 539–548.Google Scholar
Johnson, R. S. 1973a On the development of a solitary wave moving over an uneven bottom. Proc. Camb. Phil. Soc. 73, 183–203.Google Scholar
Johnson, R. S. 1973b On an asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients. J. Fluid Mech. 60, 813–824.Google Scholar
Keulegan, G. H. 1953 Characteristics of internal solitary waves. J. Res. Nat. Bur. Stand. 51, 133–140.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lee, C.-V. & Beardsley, R. C. 1974 The generation of long nonlinear internal waves in a weakly stratified shear flow. J. Geophys. Res. 79, 453–462.Google Scholar
Long, R. R. 1956 Solitary waves in one- and two-fluid systems. Tellus 8, 460–471.Google Scholar
Long, R. R. 1972 The steepening of long internal waves. Tellus 24, 88–99.Google Scholar
Miles, J. W. 1977 Note on a solitary wave in a slowly varying channel. J. Fluid Mech. 80, 149–152.Google Scholar
Peregrine, D. H. 1968 Long waves in a uniform channel of arbitrary cross-section. J. Fluid Mech. 32, 353–365.Google Scholar
Shen, M. C. 1968 Long waves in a stratified fluid over a channel of arbitrary cross-section. Phys. Fluids 11, 1853–1862.Google Scholar
Shen, M. C. & Keller, J. B. 1973 Ray method for nonlinear wave propagation in a rotating fluid of variable depth. Phys. Fluids 16, 1565–1572.Google Scholar
Thorpe, S. A. 1971 Asymmetry of the internal seiche in Loch Ness. Nature 231, 306–308.Google Scholar
Thorpe, S. A. 1974 Near-resonant forcing in a shallow two-layer fluid: a model for the internal surge in Loch Ness. J. Fluid Mech. 69, 509–527.Google Scholar
Thorpe, S. A., Hall, A. & Crofts, I. 1972 The internal surge in Loch Ness. Nature 237, 96–98.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Academic Press.
Winant, C. D. 1974 Internal surges in coastal waters. J. Geophys. Res. 79, 4523–4526.Google Scholar
Ziegenbein, J. 1969 Short internal waves in the Strait of Gibraltar. Deep-Sea Res. 16, 479–487.Google Scholar