Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-29T21:16:39.872Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical-level behaviour and wave amplification of a gravity wave incident upon a shear layer

Published online by Cambridge University Press:  29 March 2006

I. A. Eltayeb  and
J. F. McKenzie
I. A. Eltayeb
Affiliation:
Department of Mathematics, University of Khartoum, Sudan
J. F. McKenzie
Affiliation:
Department of Mathematics, University of Khartoum, Sudan
Get access Rights & Permissions [Opens in a new window]

Abstract

The properties of reflexion, refraction and absorption of a gravity wave incident upon a shear layer are investigated. It is shown that one must expect these properties to be very different depending upon the parameters (such as the Richardson number Ri, the wavelength normalized by the length scale of the shear and the ratio of the flow speed to the phase speed of the wave) characterizing the interaction of a gravity wave with a shear layer. In particular, it is shown that for all Richardson numbers there is a discontinuity in the net wave-action flux across the critical level, i.e. at a height where the flow speed matches the horizontal phase speed of the wave. When Ri > ¼, this is accompanied by absorption of part of the energy of the incident wave into the mean flow. In addition it is shown that the phenomenon of wave amplification (over-reflexion) can arise provided that the ultimate shear flow speed exceeds the horizontal phase speed of the wave and Ri is less than a certain critical value Ric ≃ 0·1129, in which case the reflected wave extracts energy from the streaming motion. It is also pointed out that wave amplification can lead to instability if the boundary conditions are altered in such a way that the system can behave like an ‘amplifier’.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 72 , Issue 4 , 23 December 1975 , pp. 661 - 671
Copyright
© 1975 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

Baldwin, P. & Roberts, P. H. 1970 Mathematika, 17, 102.
Booker, M. R. & Bretherton, F. P. 1967 J. Fluid Mech. 27, 513.
Bretherton, F. P. & Garrett, C. J. R. 1968 Proc. Roy. Soc. A 302, 529.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, p. 492. Oxford: Clarendon Press.
Drazin, P. G. & Howard, L. N. 1966 Adv. Appl. Mech. 9, 1.
Eliassen, A. & Palm, E. 1960 Geof. Publ. 22, (3), 1.
Fejer, J. A. 1964 Phys. Fluids, 7, 449.
Fejer, J. A. & Miles, J. W. 1963 J. Fluid Mech. 15, 335.
Hayes, W. D. 1970 Proc. Roy. Soc. A 320, 187.
Jones, W. L. 1968 J. Fluid Mech. 34, 609.
Mcintyre, M. E. 1975 To be submitted to. Nature. (See also J. Fluid Mech. 71, (1975) 246.)Google Scholar
Mckenzie, J. F. 1970 Planet Space Sci. 18, 1.
Mckenzie, J. F. 1972 J. Geophys. Res. 77, 2915.
Miles, J. W. 1957 J. Acoust. Soc. Am. 29, 226.
Miles, J. W. 1961 J. Fluid Mech. 10, 496.
Sturrock, P. A. 1962 In th Lockheed Symp. in Magnetohydrodyn. Plasma Hydro-magnetics (ed. D. Bershader). Stanford University Press.