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A weakly nonlinear theory of continental shelf waves

Published online by Cambridge University Press:  20 April 2006

Stefano Pierini
Stefano Pierini
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge Present address: Instituto Universitario Navale, Via A. Acton, 38, Napoli, Italy
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Abstract

In this paper three systems of evolution equations are presented which describe the free propagation of long continental shelf waves in the linear and weakly nonlinear regime. Two different degrees of nonlinearity are considered: for the first the Korteweg-de Vries equation is found to govern the dynamics of the system in the case of a single energy-containing mode (theories by Smith 1972; Grimshaw 1977), whereas, for the second nonlinear range, a nonlinear hyperbolic equation is derived. The nonlinear interactions among shelf-wave modes are also considered: they are modelled through nonlinear coupling terms in the evolution equations. This theory allows the timescale for the development of dispersive and nonlinear effects to be determined for each parameter range. The amplitude ranges corresponding to linear and nonlinear shelf waves are evaluated for the Oregon and the East Australian shelves, and some qualitative conclusions on the importance of nonlinear effects are derived. Finally the case of a shelf with longshore variation in topography is analysed and coupling terms in the evolution equations appear. They account for the scattering of energy between the various modes due to the linear and nonlinear interactions of the wave with the topographic changes.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 138 , January 1984 , pp. 197 - 208
Copyright
© 1984 Cambridge University Press

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References

Adams, J. K. & Buchwald, V. T. 1969 The generation of continental shelf waves J. Fluid Mech. 35, 815–826.Google Scholar
Allen, J. S. 1976 Continental shelf waves and alongshore variations in bottom topography and coastline J. Phys. Oceanogr. 6, 864–878.Google Scholar
Buchwald, V. T. & Adams, J. K. 1968 The propagation of continental shelf waves.Proc. R. Soc. Lond A 305, 235–250.
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1974 Korteweg-de Vries equation and generalizations. VI. Methods for exact solution Commun. Pure Appl. Maths. 27, 97–133.Google Scholar
Gill, A. E. & Schumann, E. H. 1974 The generation of long shelf waves by the wind J. Phys. Oceanogr. 4, 83–90.Google Scholar
Grimshaw, R. 1977 a The stability of continental shelf waves, I. Side band instability and long wave resonance. J. Austral. Math. Soc. B 20, 13–30.
Grimshaw, R. 1977b Nonlinear aspects of long shelf waves. Geophys. Astrophys. Fluid Dyn. 8, 3–16.Google Scholar
Hamon, B. V. 1962 The spectrums of mean sea level at Sydney, Coff's Harbour, and Lord Howe Island. J. Geophys. Res. 67, 5147–5155 (Corrigendum J. Geophys. Res. 68, (1963), 4635).
Hsieh, W. W. 1982 On the detection of continental shelf waves J. Phys. Oceanogr. 12, 414–427.Google Scholar
Hsieh, W. W. & Mysak, L. A. 1980 Resonant interactions between shelf waves, with applications to the Oregon shelf J. Phys. Oceanogr. 10, 1729–1741.Google Scholar
Hsueh, Y. 1980 Scattering of continental shelf waves by longshore variations in bottom topography J. Geophys. Res. 85, 1147–1150.Google Scholar
Leblond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier Oceanography Series no. 20.
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. McGraw-Hill.
Mysak, L. A. 1968 Edge waves on a gently sloping continental shelf of finite width J. Mar. Res. 36, 24–33.Google Scholar
Mysak, L. A. 1980 Recent advances in shelf wave dynamics Rev. Geophys. Space Phys. 18, 211–241.Google Scholar
Robinson, A. R. 1964 Continental shelf waves and the response of sea level to weather systems J. Geophys. Res. 69, 367–368.Google Scholar
Smith, R. 1972 Nonlinear Kelvin and continental shelf waves J. Fluid Mech. 52, 379–391.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.