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Multiple linear instability of layered stratified shear flow

Published online by Cambridge University Press:  26 April 2006

Colm-Cille P. Caulfield
Colm-Cille P. Caulfield
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, CB3 9EW, UK Present address: Department of Physics, 60 St George St, University of Toronto, Toronto, Ontario, Canada, M5S 1A7.
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Abstract

We develop a simple model for the behaviour of an inviscid stratified shear flow with a thin mixed layer of intermediate fluid. We find that the flow is simultaneously unstable to oscillatory disturbances that are a generalization of those discussed by Holmboe (1962), purely unstable modes analogous to those considered by Taylor (1931), and a new type of oscillatory disturbance at large wavelength. The relative significance of these different types of instability depends on the ratio R of the depth of the intermediate layer to the depth of the shear layer. For small values of R, the new type of oscillatory wave has both the largest growthrate for given bulk Richardson number Ri0, and is also primarily unstable to disturbances propagating at an angle to the mean flow, i.e. such modes violate the conditions of Squire's theorem (1933), and thus the assumption of initial two dimensionality of such flows is invalid. For intermediate values of R, the Holmboe-type modes and the Taylor-types modes may have wavelengths and phase speeds conducive to the formation of a resonant triad over a wide range of Ri0. Thus the presence of an intermediate layer in a stratified shear flow markedly changes its stability properties.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 258 , 10 January 1994 , pp. 255 - 285
Copyright
© 1994 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Browand, F. K. & Winant, C. D. 1973 Laboratory observations of shear layer instability in a stratified fluid. Boundary-Layer Met. 5, 67–87.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. 1968 Resonant gravity-wave interactions in a shear flow. J. Fluid Mech. 34, 531–549.Google Scholar
Craik, A. D. D. 1985 Wave interactions and fluid flows. Cambridge University Press.
Gill, A. E. 1982 Atmosphere Ocean Dynamics. Academic.
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
Hoiland, E. 1948 Stability and instability waves in sliding layers with internal stability. Arch. Math. og Nat. L, no. 3Google Scholar
Holmboe, J. 1962 On the behaviour of symmetric waves in stratified shear layers. Geofys. Publ. 24, 67–113.Google Scholar
Howard, L. N. & Maslowe, S. A. 1973 Stability of stratified shear flow. Boundary-Layer Met. 4, 511–523.Google Scholar
Lawrence, G. A., Browand, F. K. & Redekopp, L. G. 1991 The stability of a sheared density interface. Phys. Fluids A 3, 2360–2370.Google Scholar
Linden P. F. 1979 Mixing in stratified fluids. Geophys. Astrophys. Fluid Dyn. 13, 1–23.Google Scholar
Narimousa, S. & Fernando, H. J. S. 1987 On the sheared density interface of an entraining stratified fluid. J. Fluid Mech. 174, 1–22.Google Scholar
Rayleigh, Lord 1994 The Theory of Sound, 2nd edn. Macmillan.
Redondo, J. M. R. 1989 Internal and external mixing in stratified-shear flow. In Advances in Turbulence 2 (ed. H.-H. Fernholz & H. E. Fiedler).Springer.
Smyth, W. D. & Peltier, W. R. 1990 Three-dimensional primary instability of a stratified dissipative parallel flow. Geophys. Astrophys. Fluid Dyn. 52, 249–261.Google Scholar
Smyth, W. D., Klaasen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43, 181–222.Google Scholar
Squire, H. B. 1933 On the stability of three-dimensional disturbances of viscous flow between parallel walls. Proc. R. Soc. Lond. A 142, 621–628.Google Scholar
Stephenson, P. W. & Fernando, H. J. S. 1991 Turbulence and mixing in a stratified shear flow. Geophys. Astrophys. Fluid Dyn. 59, 147–164.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
Thorpe, S. A. 1973 Turbulence in stably stratified fluids: a review of laboratory experiments. Boundary-Layer Met. 5, 95–119.Google Scholar
Thorpe, S. A. 1987 Transitional phenomena and the development of turbulence in stratified fluids: a review. J. Geophys. Res. 92, J. Geophys. Res. C, 5231–5248.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Yih, C.-S. 1955 Stability of two-dimensional parallel flows for three-dimensional disturbances. Q. Appl. Maths 12, 434–435.Google Scholar