Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T16:05:29.793Z Has data issue: false hasContentIssue false
  • English
  • Français

Forced resonant second-order interaction between damped internal waves

Published online by Cambridge University Press:  29 March 2006

A. D. Mcewan ,
D. W. Mander  and
R. K. Smith
A. D. Mcewan
Affiliation:
C.S.I.R.O. Division of Atmospheric Physics, Aspendale, Victoria, Australia
D. W. Mander
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria, Australia
R. K. Smith
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria, Australia Present address: Department of Applied Mathematics, University of Edinburgh.
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical and experimental study is made of the second-order resonant interaction between triads of linearly damped waves, one common member of which is continuously forced. In the case of a single triad, if the forced wave exceeds a critical amplitude defined by properties of the triad members, energy proceeds irreversibly to the other two waves. A stable limit state is reached where all power in excess of that required to sustain a critical amplitude in the forced wave is transferred to the other waves, which also reach steady terminal amplitudes.

It is shown that when two or more triads are simultaneously at resonance the only stable limit state is one wherein the forced wave has fallen to the lowest critical amplitude, and the only other two waves remaining are those of the triad possessing this critical amplitude. Regardless of their initial amplitudes, all other waves not externally forced ultimately disappear.

The theory is applied to the interaction of standing internal gravity waves in a linearly stratified liquid. The experiments described here quantitatively confirm the major predictions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 55 , Issue 4 , 24 October 1972 , pp. 589 - 608
Copyright
© 1972 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

Ball, F. K. 1964 Energy transfer between external and internal waves. J. Fluid Mech. 19, 465–480.Google Scholar
Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Blaisdell.
Craik, A. D. D. 1971 Nonlinear resonant instability in boundary layers. J. Fluid Mech. 50, 393–413.Google Scholar
Davis, R. E. & Acrivos, A. 1967 The stability of oscillatory internal waves. J. Fluid Mech. 30, 723–36.Google Scholar
Hasselmann, K. 1966 Feynman diagrams and interaction rules of wave–wave scattering processes. Rev. Geophys. 4, 1–32.Google Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30, 737–39.Google Scholar
McEwan, A. D. 1971 Degeneration of resonantly-excited internal gravity waves. J. Fluid Mech. 50, 431–48.Google Scholar
McGoldrick, L. F. 1965 Resonant interactions among capillary–gravity waves. J. Fluid Mech. 21, 305–31.Google Scholar
McGoldrick, L. F. 1970 An experiment on second-order capillary–gravity resonant wave interactions. J. Fluid Mech. 40, 251–71.Google Scholar
Martin, S., Simmons, W. & Wunsch, C. 1972 The excitation of resonant triads by single internal waves. J. Fluid Mech. 53, 17–44.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.