Hostname: page-component-7dd5485656-bt4hw Total loading time: 0 Render date: 2025-10-30T23:24:35.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonant wave interactions in a stratified shear flow

Published online by Cambridge University Press:  21 April 2006

R. Grimshaw
R. Grimshaw
Affiliation:
School of Mathematics, University of New South Wales, PO Box 1, Kensington, NSW 2033, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 190 , May 1988 , pp. 357 - 374
Copyright
© 1988 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.)

Article purchase

Temporarily unavailable

References

Ball, F. K. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19, 465–478.Google Scholar
Bers, A., Kaup, D. J. & Reiman, A. 1976 Nonlinear interactions of three wave packets in a homogeneous medium. Phys. Rev. Lett. 37, 182–185.Google Scholar
Bloembergen, N. 1965 Nonlinear Optics. New York: Benjamin.
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513–539.Google Scholar
Brown, S. N. & Stewartson, K. 1980 On the nonlinear reflection of a gravity wave at a critical level. Part 1. J. Fluid Mech. 100, 577–595.Google Scholar
Brown, S. N. & Stewartson, K. 1982a On the nonlinear reflection of a gravity wave at a critical level. Part 2. J. Fluid Mech. 115, 217–230.Google Scholar
Brown, S. N. & Stewartson, K. 1982b On the nonlinear reflection of a gravity wave at a critical level. Part 3. J. Fluid Mech. 115, 231–250.Google Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.Google Scholar
Craik, A. D. D. 1978 Evolution in space and time of resonant wave triads. II. A class of exact solutions. Proc. R. Soc. Lond. A 363, 257–269.Google Scholar
Craik, A. D. D. 1986 Wave Interactions and Fluid Flows. Cambridge University Press.
Craik, A. D. D. & Adam, J. A. 1979 ‘Explosive’ resonant wave interactions in a three-layer fluid flow. J. Fluid Mech. 92, 15–33.Google Scholar
Davidson, R. C. 1972 Methods in Nonlinear Plasma Theory. Academic.
Grimshaw, R. 1974 Internal gravity waves in a slowly varying dissipative medium. Geophys. Fluid Dyn. 6, 131–148.Google Scholar
Grimshaw, R. 1981 Evolution equations for long, nonlinear internal waves in stratified shear flows. Stud. Appl. Maths 65, 159–188.Google Scholar
Grimshaw, R. 1984 Wave action and wave-mean flow interaction, with application to stratified shear flows. Ann. Rev. Fluid Mech. 16, 11–44.Google Scholar
Grimshaw, R. 1987 Triad resonance for weakly coupled, slowly varying oscillators. Stud. Appl. Maths 77, 1–35.Google Scholar
Hasselman, K. 1966 Feynman diagrams and interaction rules of wave–wave scattering processes. Rev. Geophys. 4, 1–32.Google Scholar
Hasselman, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30, 737–739.Google Scholar
Kaup, D. J., Reiman, A. & Bers, A. 1979 Space–time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium. Rev. Mod. Phys. 51, 275–309.Google Scholar
Reiman, A. 1979 Space–time evolution of nonlinear three-wave interactions. II. Interaction in an inhomogeneous medium. Rev. Mod. Phys. 51, 311–330.Google Scholar
Reiman, A., Bers, A. & Kaup, D. J. 1977 Nonlinear interaction of three wave packets in an inhomogeneous medium. Phys. Rev. Lett. 39, 245–248.Google Scholar
Thorpe, S. A. 1966 On wave interactions in a stratified fluid. J. Fluid Mech. 24, 737–751.Google Scholar
Tsutahara, M. 1984 Resonant interactions of internal waves in a stratified shear flow. Phys. Fluids 27, 1942–1947.Google Scholar
Tsutahara, M. 1986 Three-wave resonant interactions and multiple resonances in two-layer and three-layer flows. Phys. Fluids 29, 2812–2818.Google Scholar
Weiland, J. & Wilhelmsson, H. 1977 Coherent non-linear interaction of waves in plasma. Pergamon.
Zakharov, V. E. & Manakov, S. V. 1975 The theory of resonance interaction of wave packets in nonlinear media. Zh. Eksp. Teor. Fiz. 69, 1654–1673. [ Sov. Phys., J. Exp. Theor. Phys. 42, 842–850 (1976)].Google Scholar