Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T10:29:01.931Z Has data issue: false hasContentIssue false
  • English
  • Français

Fourth-order nonlinear evolution equations for 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 Applied Mathematics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Calcutta 700009, INDIA
K. P. Das
Affiliation:
Department of Applied Mathematics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Calcutta 700009, INDIA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Two coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

Type
Research Article
Information
The ANZIAM Journal , Volume 39 , Issue 2 , October 1997 , pp. 214 - 229
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Ball, F. K., “Energy transfer between external and internal waves”, J. Fluid. Mech. 19 (1964) 465–478.CrossRefGoogle Scholar
[2]Brinch-Nielsen, U. and Jonsson, I. G., “Fourth order evolution equations and stability analysis for Stokes waves on arbitrary water depth”, Wave Motion 8 (1986) 455–472.CrossRefGoogle Scholar
[3]Das, K. P., “On evolution equations for a three dimensional surface gravity wave packet in a two layer fluid”, Wave Motion 8 (1986) 191–204.CrossRefGoogle Scholar
[4]Davey, A. and Stewartson, K., “On three-dimensional packets of surface waves”, Proc. R. Soc. Lond. A 338 (1974) 101–110.Google Scholar
[5]Dhar, A. K. and Das, K. P., “A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water”, Phys. Fluids A 2 (1990) 778–783.CrossRefGoogle Scholar
[6]Dhar, A. K. and Das, K. P., “Fourth order nonlinear evolution equation for two Stokes wave trains in deep water”, Phys. Fluids A 3 (1991) 3021–3026.Google Scholar
[7]Dhar, A. K. and Das, K. P., “Stability analysis from fourth order evolution equation for small but finite amplitude interfacial waves in the presence of a basic current shear”, J. Austral. Math. Soc. B 35 (1994) 348–365.Google Scholar
[8]Dysthe, K. B., “Note on a modification to the nonlinear Schrödinger equation for application to deep water waves”, Proc. R. Soc. Lond. A 369 (1979) 105–114.Google Scholar
[9]Dysthe, K. B. and Das, K. P., “Coupling between a surface wave spectrum and an internal wave: modulational interaction”, J. Fluid Mech. 104 (1981) 483–503.Google Scholar
[10]Funakoshi, M. and Oikawa, M., “The resonant interaction between a long internal gravity wave and a surface gravity wave packet”, J. Phys. Soc. Japan 52 (1983) 1982.Google Scholar
[11]Hara, T. and Mei, C. C., “Frequency downshift in narrowbanded surface waves under the influence of wind”, J. Fluid Mech. 230 (1991) 429–177.CrossRefGoogle Scholar
[12]Hasselman, K., “Nonlinear interactions treated by method of theoretical physics (with application to generation of waves by wind)”, Proc. R. Soc. Lond. A 229 (1967) 77.Google Scholar
[13]Hogan, S. J., “Fourth order evolution equation for deep water gravity-capillary waves”, Proc. R.Soc. Lond. A 402 (1985) 359–372.Google Scholar
[14]Janssen, P. A. E. M., “On a fourth order envelope equation for deep water waves”, J. Fluid Mech 126 (1983) 1–11.CrossRefGoogle Scholar
[15]Longuet-Higgins, M. S., “The instabilities of gravity waves of finite amplitude in deep water. I Super harmonics”, Proc. R. Soc. Lond. A 360 (1978) 471–488.Google Scholar
[16]Longuet-Higgins, M. S., “The instabilities of gravity waves of finite amplitude in deep water. II Subharmonics”, Proc. R. Soc. Lond. A 360 (1978) 489–505.Google Scholar
[17]Ma, Y. C., “A study of resonant interaction between internal and surface waves based on a two layer fluid model”, Wave Motion 5 (1983) 145.Google Scholar
[18]Olbers, D. J. and Herterich, K., “The spectral energy transfer from surface waves to internal waves”, J. Fluid Mech. 92 (1979) 349–379.CrossRefGoogle Scholar
[19]Phillips, O. M., The Dynamics of the Upper Ocean (Cambridge University Press, 1977).Google Scholar
[20]Risk, M. H. and Ko, D. R. S., “Interaction between small scale surface waves and large scale internal waves”, Phys. Fluids 21 (1978) 1900–1907.Google Scholar
[21]Stiassnie, M., “Note on modified nonlinear Schrodinger equation for deep water waves”, Wave Motion 6 (1984) 431–433.CrossRefGoogle Scholar
[22]Thorpe, S. A., “On wave interaction in a stratified fluid”, J. Fluid Mech. 24 (1966) 737.Google Scholar
[23]Watson, K. M., West, B. J. and Cohen, B. I., “Coupling of surface and internal gravity waves: a mode coupling model”, J. Fluid Mech. 77 (1976) 185.CrossRefGoogle Scholar