Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T17:40:18.298Z Has data issue: false hasContentIssue false
  • English
  • Français

Interaction of internal waves in a continuous thermocline model

Published online by Cambridge University Press:  17 February 2009

N. T. Hung  and
S. A. Maslowe
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.

Weak nonlinear interactions are studied for systems of internal waves when the Brunt-Väisälä frequency is proportional to sech z, where z = 0 is the centre of the thermocline. Explicit results expressed in terms of gamma functions have been obtained for the interaction coefficients appearing in the amplitude evolution equations. The cases considered include resonant triads as well as second and third harmonic resonance. In the non-resonant situation, the Stokes frequency correction due to finite-amplitude effects has been computed and the extension to wave packets is outlined. Finally, the effect of a mean shear on resonant interactions is discussed.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 1 , July 1983 , pp. 94 - 109
Copyright
Copyright © Australian Mathematical Society 1983

References

[1]Bretherton, F. P., “Resonant interactions between waves. The case of discrete oscillations”, Fluid Mech. 20 (1964), 457–479.CrossRefGoogle Scholar
[2]Case, K. M. and Chiu, S. C., “Bäcklund transformation for the resonant three-wave process”, Phys. Fluids 20 (1977), 746–749.CrossRefGoogle Scholar
[3]Davis, R. E. and Acrivos, A., “The stability of oscillatory internal waves”, J. Fluid Mech. 30 (1967), 723–736.CrossRefGoogle Scholar
[4]Grimshaw, R., “The modulation of an internal gravity wave packet and the resonance with mean motion”, Stud. Appl. Math. 56 (1977), 241–266.CrossRefGoogle Scholar
[5]Koop, C. G. and Redekopp, L. G., “The interaction of long and short internal gravity theory and experiment”, J. Fluid Mech. 111 (1981), 367–410.CrossRefGoogle Scholar
[6]Liu, A. K. and Benney, D. J., “The evolution of nonlinear wave trains in stratified shear flows”, Stud. Appl. Math. 64 (1981), 247–269.CrossRefGoogle Scholar
[7]Lucas, R. D., Metcalfe, R. W. and Maslowe, S. A., “Effects of shear on resonantly interacting internal waves”, Flow Research Report (Unpublished) (1978), 98 pp.Google Scholar
[8]McEwan, A. D., “Degeneration of resonantly excited standing internal gravity waves”, J. Fluid Mech. 50 (1971), 431–448.CrossRefGoogle Scholar
[9]McEwan, A. D. and Plumb, R. A., “Off-resonant amplification of finite internal wave packets”, Dyn. Atmos. and Oceans 2(1977), 83–105.CrossRefGoogle Scholar
[10]McGoldrick, L. F., “An experiment on second-order capillary gravity resonant wave interactions”, J. Fluid Mech. 40 (1970), 251–271.CrossRefGoogle Scholar
[11]McGoldrick, L. F., “On Wilton's ripple: a special case of resonant interactions’, J. Fluid Mech. 42 (1970), 193–200.CrossRefGoogle Scholar
[12]Mied, R. P. and Dugan, J. P., “Internal wave reflection by a velocity shear and density anomaly’, J. Phys. Ocean. 5 (1975), 279–287.2.0.CO;2>CrossRefGoogle Scholar
[13]Phillips, O. M., “The interaction trapping of internal gravity waves’, J. Fluid Mech., 34 (1968), 407–416.CrossRefGoogle Scholar
[14]Phillips, O. M., “Wave interaction’, Chapter 7 in Nonlinear waves (eds. Leibovich, S. and Seebass, R.) (Cornell University Press, 1974).Google Scholar
[15]Simmons, W. F., “A variational method for weak resonant wave interactions”, Proc. Roy. Soc. London Ser. A 309 (1969), 551–579.Google Scholar
[16]Thorpe, S. A., “On the shape of progressive internal waves”, Phil. Trans. Roy. Soc. London Ser. A 263 (1968), 563–613.Google Scholar
[17]Yuen, H. C. and Lake, B., “Nonlinear deep water: theory and experiment”, Phys. Fluids 18 (1975), 956–960.CrossRefGoogle Scholar