Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-22T11:22:38.905Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability of continental shelf waves I. Side band instability and long wave resonance

Published online by Cambridge University Press:  17 February 2009

R. Grimshaw
R. Grimshaw
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Vic. 3052, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Continental shelf waves are examined for side band instability. It is shown that a modulated shelf wave is described by a nonlinear Schrödinger equation, from which the stability criterion is derived. Long shelf waves are stable to side band modulations, but as the wavenumber is increased there are regions of instability (in wavenumber space). A change of stability occurs at each long wave resonance, defined by the condition that the group velocity of the shelf wave equals a long wave speed. Equations describing the long wave resonance are derived.

Type
Research Article
Information
The ANZIAM Journal , Volume 20 , Issue 1 , June 1977 , pp. 13 - 30
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Benney, D. J. and Newell, A. C., “The propagation of nonlinear wave envelopes”, J. Math. and Physics 46 (1967), 133–139.CrossRefGoogle Scholar
[2]Buchwald, V. T. and Adams, J. K., “The propagation of continental shelf waves”, Proc. Roy. Soc. 305A (1968), 235–250.Google Scholar
[3]Cartwright, D. E., “Extraordinary tidal currents near St Kilda”, Nature 223 (1969), 928–932.CrossRefGoogle Scholar
[4]Gill, A. E. and Schumann, E. H., “The generation of long shelf waves by the wind”, J. Phys. Ocean 4 (1974), 83–90.2.0.CO;2>CrossRefGoogle Scholar
[5]Grimshaw, R., “The modulation and stability of an internal gravity wave”, Mém. Soc. Roy. Sc. Liège, 6e serie, 10 (1976) 299–314.Google Scholar
[6]Grimshaw, R., “Nonlinear aspects of long shelf waves”, Geophys. Astrophys. Fluid Dynamics 8 (1977), 3–16.CrossRefGoogle Scholar
[7]Grimshaw, R., “The stability of continental shelf waves I. Sideband instability and long wave resonance”, School of Mathematical Sciences Research Report No. 24, University of Melbourne (1976).Google Scholar
[8]Grimshaw, R., “The modulation of an internal gravity wave packet, and the resonance with mean motion”, Stud. Appl. Maths. (1977) (to appear).CrossRefGoogle Scholar
[9]Hasimoto, H. and Ono, H., “Nonlinear modulation of gravity waves”, J. Phys. Soc. Japan 33(1972), 805–811.CrossRefGoogle Scholar
[10]Huthnance, J. M., “On trapped waves over a continental shelf”, J. Fluid Mech. 69 (1975), 689–704.CrossRefGoogle Scholar
[11]Kundu, P. K., Allen, J. S. and Smith, R. L., “Modal decomposition of the velocity field near the Oregon coast”, J. Phys. Ocean. 5 (1975), 683–704.2.0.CO;2>CrossRefGoogle Scholar
[12]McIntyre, M. E., “Mean motions and impulse of a guided internal gravity wave packet”, J. Fluid Mech. 60 (1973), 801–811.CrossRefGoogle Scholar
[13]Plumb, R. A., “The stability of small amplitude Rossby waves in a channel” J. Fluid Mech. (1977) (to appear).CrossRefGoogle Scholar
[14]Smith, R., “Nonlinear Kelvin and continental-shelf waves”, J. Fluid Mech. 52 (1972), 379–391.CrossRefGoogle Scholar
[15]Zakharov, V. E. and Shabat, A. B. “Exact theory of two-dimensional self-forming and one-dimensional self-modulation of waves in nonlinear media”, Soviet Physics JETP 34 (1972), 62–69.Google Scholar