Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T08:45:42.363Z Has data issue: false hasContentIssue false
  • English
  • Français

A stability analysis for interfacial waves using a Zakharov equation

Published online by Cambridge University Press:  26 April 2006

A. Dixon
A. Dixon
Affiliation:
University of New South Wales, Kensington, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

An amplitude equation for weak interactions of waves is derived to describe the time evolution of disturbances on an interface between fluids of differing densities, with rigid upper and lower boundaries. This equation is analogous to the Zakharov equation for water waves and is used to investigate the stability of a periodic wave. It is found that, for small wave steepnesses, the instabilities are due to resonant quartets with perturbation wavenumbers of the order of, or less than, that of the main wave. A second instability is found for large perturbation wavenumbers and moderately high wave steepnesses. This is restricted to the case when the Boussinesq parameter is small. It is shown that this is a Kelvin–Helmholtz instability caused by a wave-induced jump in the fluid velocity across the interface.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 214 , May 1990 , pp. 185 - 210
Copyright
© 1990 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

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.
Crawford, D. R., Lake, B. M., Saffman, P. G. & Yuen, H. C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177–191.Google Scholar
Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves.. Proc. R. Soc. Lond. A 338, 101–110.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Grimshaw, R. H. J. & Pullin, D. I. 1985 Stability of finite-amplitude interfacial waves. Part 1. Modulational instability for small-amplitude waves. J. Fluid Mech. 160, 297–315.Google Scholar
Holyer, J. H. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433–448.Google Scholar
Longuet-Higgins, M. S. 1978a The instability of gravity waves of finite amplitude in deep water. I. Superharmonics.. Proc. R. Soc. Lond. A 360, 471–488.Google Scholar
Longuet-Higgins, M. S. 1978b The instability of gravity waves of finite amplitude in deep water. II. Superharmonics.. Proc. R. Soc. Lond. A 360, 489–505.Google Scholar
McLean, J. W. 1982a Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
McLean, J. W. 1982b Instabilities of finite-amplitude gravity water waves on water of finite depth. J. Fluid Mech. 114, 331–341.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9, 193–217.Google Scholar
Phillips, O. M. 1981 Wave interactions—the evolution of an idea. J. Fluid Mech. 106, 215–227.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1983a Nonlinear interfacial progressive waves near a boundary in a Boussinesq fluid. Phys. Fluids 26, 897–905.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1983b Interfacial progressive waves in a two layer shear flow. Phys. Fluids 26, 1731–1739.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1985 Stability of finite-amplitude interfacial waves. Part 2. Numerical results. J. Fluid Mech. 160, 317–386.Google Scholar
Pullin, D. I. & Grimshaw, R. H. J. 1986 Stability of finite-amplitude interfacial waves. Part 3. The effect of basic current shear for one-dimensional instabilities. J. Fluid Mech. 172, 277–306.Google Scholar
Saffman, P. G. & Yuen, H. C. 1982 Finite amplitude interfacial waves in the presence of a current. J. Fluid Mech. 123, 459–476.Google Scholar
Stiassnie, M. & Shemer, L. 1984 On modifications of the Zakharov equation for surface gravity waves. J. Fluid Mech. 143, 47–67.Google Scholar
Whitham, G. B. 1967 Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399–412.Google Scholar
Yuen, H. C. 1984 Nonlinear dynamics of interfacial waves. Physica 12D, 71–82.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Fiz. 9, 86. (Engl. transl. J. Appl. Mech. Tech. Phys. 2, 190–194.)Google Scholar
Zakharov, V. E. & Kharitonov, V. G. 1970 Instability of monochromatic waves on the surface of a liquid of arbitrary depth. Zh. Prikl. Mekh. Fiz. 5, 45–49 (Engl. transl. J. Appl. Mech. Tech. Phys. 11, 747–751).Google Scholar