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The effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline

Published online by Cambridge University Press:  17 February 2009

Sudebi Bhattacharyya  and
K. P. Das
Sudebi Bhattacharyya
Affiliation:
Department of Mathematics, Scottish Church College, 1&3, Urquhart Square, Calcutta-700006, India.
K. P. Das
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Calcutta-700009, India
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Abstract

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The effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline is studied. A previously derived fourth order nonlinear evolution equation is used to find a spectral transport equation for a narrow band of surface gravity wave trains. This equation is used to study the stability of an initially homogeneous Lorentz shape of spectrum to small long wave-length perturbations for a range of spectral widths. The growth rate of the instability is found to decrease with the increase of spectral widths. It is found that the fourth order term in the evolution equation produces a decrease in the growth rate of the instability. There is stability if the spectral width exceeds a certain critical value. For a vanishing bandwidth the deterministic growth rate of the instability is recovered. Graphs have been plotted showing the variations of the growth rate of the instability against the wavenumber of the perturbation for some different values of spectral width, thermocline depth, angle of perturbation and wave steepness.

Type
Research Article
Information
The ANZIAM Journal , Volume 40 , Issue 2 , October 1998 , pp. 190 - 206
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Alber, I. E., “The effects of randomness on the stability of two-dimensional wavetrains”, Proc. R. Soc. Lond. A363 (1978) 525–546.Google Scholar
[2]Bhattacharyya, S. and Das, K. P., “Fourth order non-linear evolution equations for surface gravity waves in the presence of a thin thermocline”, J. Austral. Math. Soc. Ser. B 39 (1997), to be published.CrossRefGoogle Scholar
[3]Crawford, D. R., Saffman, P. G. and Yuen, H. C., “Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves”, Wave Motion 2 (1980) 1–16.CrossRefGoogle Scholar
[4]Dysthe, K. B., “Note on a modification to the nonlinear Schrödinger equation for application to deep water waves”, Proc. R. Soc. Lond. A369 (1979) 105–115.Google Scholar
[5]Longuet-Higgins, M. S., “The instabilities of gravity waves of finite amplitude in deep water I. Super harmonics”, Proc. R. Soc. Lond. A360 (1978) 471–488.Google Scholar
[6]Longuet-Higgins, M. S., “The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics”, Proc. R. Soc. Lond. A360 (1978) 489–505.Google Scholar
[7]Zakharov, V. E., “Stability of periodic waves of finite amplitude on the surface of deep fluid”, J. Appl. Meh. Tech. Phys. 2 (1968) 190–194.Google Scholar