Hostname: page-component-5c6d5d7d68-wp2c8 Total loading time: 0 Render date: 2024-08-14T03:49:35.826Z Has data issue: false hasContentIssue false
  • English
  • Français

On the mechanism of shear flow instabilities

Published online by Cambridge University Press:  26 April 2006

Peter G. Baines  and
Humio Mitsudera
Peter G. Baines
Affiliation:
CSIRO Division of Atmospheric Research, Aspendale, Australia
Humio Mitsudera
Affiliation:
CSIRO Division of Atmospheric Research, Aspendale, Australia Present address: JAMSTEC, Yokosuka, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

In homogeneous and density-stratified inviscid shear flows, the mechanism for instability that is most commonly invoked and discussed is Kelvin–Helmholtz instability, as it occurs for a simple velocity discontinuity. There is a second mechanism, the wave-interaction mechanism, which is much more general, and is the subject of this paper. This mechanism depends on two free waves that propagate in opposite directions in a stratified shear flow, and which may become stationary relative to each other because of the shear. If this occurs, and their relative phase is suitably chosen, the velocity field of each wave increases the displacement of the other, and so the disturbance grows.

We show that this mechanism is responsible for instability in a general class of symmetric but otherwise arbitrary velocity and density profiles, provided that the Richardson number Ri < ¼ in a central region of arbitrarily small thickness. A critical layer exists in this central region for the growing disturbance, but its role in the instability process is incidental. When Ri > ¼ everywhere, the flow is stable because the free waves described above are absorbed by the critical layer, and hence are heavily damped. The necessary criteria of Rayleigh and Fjortoft for instability in homogeneous fluid are seen to provide a suitable geometry for two interacting waves. Some specific examples are given, including a succinct explanation of Holmboe waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 276 , 10 October 1994 , pp. 327 - 342
Copyright
© 1994 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

Baines, P. G. 1994 Topographic Effects in Stratified Flows. Cambridge University Press (to appear).
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press, 615 pp.
Bell, T. H. 1974 Effects of shear on the properties of internal gravity wave modes. Deutsche Hydro. Zeits. 27, 57–62.Google Scholar
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
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. 1985 Wave Interactions and Fluid Flows. Cambridge University Press, 322 pp.
Drazin, P. G. 1989 Internal gravity waves and shear instability. In Waves and Stability in Continuous Media (ed. A. Donato & S. Giambò), commenda di Rende, Italy: EditEl, pp. 61–73.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press, 525 pp.
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic Press, 662 pp.
Goldstein, S. 1931 On the stability of superposed streams of fluids of different densities. Proc. R. Soc. Lond. A 132, 524–548.Google Scholar
Hayashi, Y.-Y. & Young, W. R. 1987 Stable and unstable shear modes of rotating parallel flows in shallow water. J. Fluid Mech. 184, 477–504.Google Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39–61.Google Scholar
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geophys. Publik. 24, 67–113.Google Scholar
Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps. Q. J. R. Met. Soc. 111, 877–946.Google Scholar
Howard, L. N. & Maslowe, S. A. 1973 Stability of stratified shear flows. Boundary-Layer Met. 4, 511–523.Google Scholar
Huppert, H. E. 1973 On Howard's technique for perturbing neutral solutions of the Taylor–Goldstein equation. J. Fluid Mech. 57, 361–368.Google Scholar
Ince, E. L. 1926 Ordinary Differential Equations. Reproduced by Dover (1956), 558 pp.
Lindzen, R. S. 1988 Instability of plane parallel shear flow (towards a mechanistic picture of how it works). Pageoph. 126, 103–121.Google Scholar
Ostrovskii, L. A., Rybak, S. A. & Tsimring, L. Sh. 1986 Negative energy waves in hydrodynamics. Sov. Phys. Usp. 29, 1040–1052.Google Scholar
Rayleigh, Lord, 1896 The Theory of Sound, vol. 2. Reproduced by Dover (1945), 504 pp.
Sakai, S. 1989 Rossby–Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves. J. Fluid Mech. 202, 149–176.CrossRefGoogle Scholar
Smyth, W. D., Klaassen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43, 181–222.Google Scholar
Smyth, W. D. & Peltier, W. R. 1989 The transition between Kelvin–Helmholtz and Holmboe instability: an investigation of the overreflection hypothesis. J. Atmos. Sci. 46, 3698–3720.Google Scholar
Takehiro, S.-I. & Hayashi, Y. Y. 1992 Over-reflection and shear instability in a shallow-water model. J. Fluid Mech. 236, 250–279.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499–523.Google Scholar
Yih, C.-S. 1974 Instability of stratified flows as a result of resonance. Phys. Fluids 17, 1483–1488.Google Scholar