Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-19T02:56:21.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Parametric instability of internal gravity waves

Published online by Cambridge University Press:  29 March 2006

A. D. McEwan  and
R. M. Robinson
A. D. McEwan
Affiliation:
CSIRO Division of Atmospheric Physics, Aspendale, Victoria 3195, Australia
R. M. Robinson
Affiliation:
CSIRO Division of Atmospheric Physics, Aspendale, Victoria 3195, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

A continuously stratified fluid, when subjected to a weak periodic horizontal acceleration, is shown to be susceptible to a form of parametric instability whose time dependence is described, in its simplest form, by the Mathieu equation. Such an acceleration could be imposed by a large-scale internal wave field. The growth rates of small-scale unstable modes may readily be determined as functions of the forcing-acceleration amplitude and frequency. If any such mode has a natural frequency near to half the forcing frequency, the forcing amplitude required for instability may be limited in smallness only by internal viscous dissipation. Greater amplitudes are required when boundaries constrain the form of the modes, but for a given bounding geometry the most unstable mode and its critical forcing amplitude can be defined.

An experiment designed to isolate the instability precisely confirms theoretical predictions, and evidence is given from previous experiments which suggest that its appearance can be the penultimate stage before the traumatic distortion of continuous stratifications under internal wave action.

A preliminary calculation, using the Garrett & Munk (197%) oceanic internal wave spectrum, indicates that parametric instability could occur in the ocean at scales down to that of the finest observed microstructure, and may therefore have a significant role to play in its formation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 67 , Issue 4 , 25 February 1975 , pp. 667 - 687
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions, 20, pp. 722–730.
Barcilon, V. 1968 Axi-symmetric inertial oscillations of a rotating ring of fluid. Mathematika, 15, 93–102.Google Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of a plane free surface of a liquid in vertical periodic motion. Proc. Roy. Soc. A 225, 505–515.Google Scholar
Bowen, A. J. & Inman, D. L. 1969 Rip currents 2. Laboratory and field observations.
J. Geophys. Res. 74, 5479–5490.
Brillouin, L. 1953 Wave Propagation in Periodic Structures. Dover.
Garrett, C. J. R. 1970 On cross-waves. J. Fluid Mech., 41, 837–849.Google Scholar
Garrett, C. & Munk, W. 1972a Space-time scales of internal waves. Geophys. Fluid Dyn., 3, 225–264.Google Scholar
Garrett, C. & Munk, W. 1972b Oceanic mixing by breaking internal waves. Deep-sea Res., 19, 823–832.Google Scholar
Mcewan, A. D. 1971 Degeneration of resonantly-excited standing internal gravity waves. J. Fluid Mech., 50, 431–448.Google Scholar
Mcewan, A. D. 1973 Interactions between internal gravity waves and their traumatic effect on a continuous stratification. Boundary Layer Met., 5, 159–175.Google Scholar
Mcewan, A. D., Mander, D. W. & Smith, R. K. 1972 Forced resonant second-order interaction between damped internal waves. J. Fluid Mech., 55, 589–608.Google Scholar
Mahony, J. J. 1972 Cross-waves. Part 1. Theory. J. Fluid Mech., 55, 229–244.Google Scholar
Martin, S., Simmons, W. & Wunsch, C. 1972 The excitation of resonant triads by single
Mowbray, D. E. 1967 The use of schlieren and shadowgraph techniques in the study of flow patterns in density stratified liquids. J. Fluid Mech., 27, 595–608.Google Scholar
Orlanski, I. 1972 On the breaking of standing internal gravity waves. J. Fluid Mech., 27 595–608.Google Scholar
Orlanski, I. 1973 Trapeze instability as a source of internal gravity waves. Part I. J. Atmos. Sci., 30, 1007–1016.Google Scholar
Orlanski, I. & Bryan, K. 1969 Formulation of the thermoclinic step structure by large a,[;otide omternal waves. J. Geophys. Res., 74, 6975–6983.Google Scholar
Osborn, T. R. & Cox, C. S. 1972 Oceanic fine structure. Geophys. Fluid Dyn., 3, 321–345.Google Scholar
Rhines, P. B. 1970 Wave propagation in a periodic medium with application to the ocean. Rev. Geophys. Space Phys., 8, 303–319.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.