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On the stability of Kelvin waves

Published online by Cambridge University Press:  26 April 2006

W. K. Melville ,
G. G. Tomasson  and
D. P. Renouard
W. K. Melville
Affiliation:
R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
G. G. Tomasson
Affiliation:
R. M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
D. P. Renouard
Affiliation:
Institut de Mecanique de Grenoble, Saint-Martin d'Heres, France
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Abstract

We consider the evolution of weakly nonlinear dispersive long waves in a rotating channel. The governing equations are derived and approximate solutions obtained for the initial data corresponding to a Kelvin wave. In consequence of the small nonlinear speed correction it is shown that weakly nonlinear Kelvin waves are unstable to a direct nonlinear resonance with the linear Poincaré modes of the channel. Numerical solutions of the governing equations are computed and found to give good agreement with the approximate analytical solutions. It is shown that the curvature of the wavefront and the decay of the leading wave amplitude along the channel are attributable to the Poincaré waves generated by the resonance. These results appear to give a qualitative explanation of the experimental results of Maxworthy (1983), and Renouard, Chabert d'Hières & Zhang (1987).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 206 , September 1989 , pp. 1 - 23
Copyright
© 1989 Cambridge University Press

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References

Armi, L. & Farmer, D. 1984 The internal hydraulies of the Strait of Gibraltar and associated sills and narrows. Oceanol. Acta 8, 37–46.Google Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic.
Grimshaw, R. 1985 Evolution equations for weakly nonlinear, long internal waves in a rotating fluid. Stud. Appl. Maths 73, 1–33.Google Scholar
Grimshaw, R. & Melville, W. K. 1989 On the derivation of the modified Kadomtsev-Petviashvili equation. Stud. App. Maths (In press).Google Scholar
Katsis, C. & Akylas, T. R. 1987 Solitary internal waves in a rotating channel: A numerical study. Phys. Fluids 30, 297–301.Google Scholar
Macomb, E. S. 1986 The interaction of nonlinear waves and currents with coastal topography. M.S. thesis, Department of Civil Engineering, MIT.
Macomb, E. S. & Melville, W. K. 1987 On the generation of long nonlinear waves in a channel. Unpublished manuscript.
Maxworthy, T. 1983 Experiments on solitary internal Kelvin waves. J. Fluid Mech. 129, 365–383.Google Scholar
Melville, W. K., Renouard, D. & Zhang, X. 1988 On the generation of nonlinear internal Kelvin waves in a rotating channel. J. Phys. Oceanogr. (Sub judice).Google Scholar
Renouard, D. P., Chabert d'Hières, G. & Zhang, X. 1987 An experimental study of strongly nonlinear waves in a rotating system. J. Fluid Mech. 177, 381–394.Google Scholar
Tomasson, G. G. 1988 On the stability of long nonlinear Kelvin waves. M.S. thesis, Department of Civil Engineering, MIT.
Whitham, G. B. 1974 Linear and nonlinear waves. Wiley.
Winther, R. 1985 Model equations for long, almost plane waves in nonlinear dispersive systems. Unpublished manuscript.
Zhang, X. 1986 Contribution a l'étude des ondes internes non linéaires en présence d'une topographie ou de rotation. Doctoral thesis, University of Grenoble.