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Weakly nonlinear Kelvin–Helmholtz waves

Published online by Cambridge University Press:  21 April 2006

John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093, USA

Abstract

The Lagrangian L for gravity waves of finite amplitude in an N-layer, stratified shear flow is constructed as a functional of the generalized coordinates qv(t)≡ {qnv(t)} of the N + 1 interfaces, where the qnv are the Fourier coefficients in the expansion of the interfacial displacement ηv(x, t) in a complete, orthogonal set {ψn(x)}. The explicit expansion of L is constructed through fourth-order in the qnν and $\dot{q}_{n}^{\nu}$. Progressive interfacial waves and Kelvin–Helmholtz instability in a two-layer fluid are examined, and the earlier results of Drazin (1970), Nayfeh & Saric (1972) and Weissman (1979) are extended to finite depth. It is found that the pitchfork bifurcation associated with the critical point for Kelvin–Helmholtz instability, which is supercritical for infinitely deep layers, may be subcritical (inverted) for finite depths. The evolution equations that govern Kelvin–Helmholtz waves in the parametric neighbourhood of this critical point are shown to be equivalent to those for a particle in a two-parameter, central force field. The effect of surface tension is examined in an Appendix. Finally, the wave motion forced by flow over a sinusoidal bottom (as in Thorpe's tilting tank) is examined and the corresponding resonance curves and Hopf bifurcations determined. Numerical integrations reveal that stable limit cycles exist in some parametric neighbourhoods of these bifurcations. Period doubling was observed but did not lead to chaotic motion.

Type
Research Article
Copyright
© 1991 Cambridge University Press

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References

Altman D. B.1985 Laboratory studies of internal gravity wave critical layers. Ph.D. dissertation, University of California, San Diego.
Drazin P. G.1970 Kelvin-Helmholtz instability of finite amplitude. J. Fluid Mech. 42, 321335.Google Scholar
Goldstein H.1980 Classical Mechanics. Addison-Wesley.
Keulegan, G. H. & Carpenter L. H.1961 An experimental study of internal progressive oscillatory waves. Natl Bur. Stand. Rep. 7319.Google Scholar
Maslowe, S. A. & Kelly R. E.1970 Finite amplitude oscillations in a Kelvin—Helmholtz flow. Intl. J. Non-Linear Mech. 5, 427435.Google Scholar
Miles J. W.1986 Weakly nonlinear waves in a stratified fluid: a variational formulation. J. Fluid Mech. 172, 499512.Google Scholar
Nayfeh, A. H. & Saric W. S.1972 Nonlinear waves in Kelvin—Helmholtz flow. J. Fluid Mech. 55, 311327.Google Scholar
Saffman, P. G. & Yuen H. C.1982 Finite-amplitude interfacial waves in the presence of a current. J. Fluid Mech. 123, 459476.Google Scholar
Stewartson K.1981 Marginally stable inviscid flows with critical layers. I M A J. Appl. Maths 27, 133175.Google Scholar
Stoker J. J.1950 Nonlinear Vibrations. Interscience.
Thorpe S. A.1968 A method of producing shear flow in a stratified fluid. J. Fluid Mech. 32, 693704.Google Scholar
Weissman M. A.1979 Nonlinear wave packets in the Kelvin—Helmholtz instability Phil. Trans. R. Soc. Lond. A 290, 639685.Google Scholar
Yuen H. C.1984 Nonlinear dynamics of interfacial waves. Physica 12D, 7182.Google Scholar