Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-19T00:07:50.824Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability of small amplitude Rossby waves in a channel

Published online by Cambridge University Press:  11 April 2006

R. A. Plumb
R. A. Plumb
Affiliation:
Geophysical Fluid Dynamics Laboratory, Meteorological Office, Bracknell, Berkshire Present address: CSIRO, Division of Atmospheric Physics, Aspendale, Victoria 3195, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The breakdown of Rossby waves in a bounded system is studied for the case in which the wave amplitude is small. In a very long, laterally bounded, channel all waves are unstable via second-order resonant interactions except those of wavenumber π/L in the cross-channel direction (where L is the channel width), which are stable if their longitudinal wavenumber is greater than 0·681π/L. These waves are, however, unstable to weaker side-band interactions, so that all waves with non-zero longitudinal wavenumber are unstable. The transition from sideband to triad instability occurs where the group velocity of the basic wave is equal to the velocity of long waves.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 80 , Issue 4 , 23 May 1977 , pp. 705 - 720
Copyright
© 1977 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

Baines, P. G. 1976 The stability of planetary waves on a sphere. J. Fluid Mech. 73, 193–213.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Bretherton, F. P. 1964 Resonant interactions between waves: the case of discrete oscillations. J. Fluid Mech. 20, 457–479.Google Scholar
Fjørtoft, R. 1953 On changes in the spectral distribution of kinetic energy in twodimensional, non-divergent flow. Tellus, 5, 225–230.Google Scholar
Gill, A. E. 1974 The stability of planetary waves on an infinite beta-plane. Geophys. Fluid Dyn. 6, 29–47.Google Scholar
Grimshaw, R. H. J. 1975 The modulation and stability of an internal gravity wave. Mem. Soc. Roy. Sci. Liège, 10(6), 299–314.Google Scholar
Hasselmann, K. 1967 A criterion for non-linear wave stability. J. Fluid Mech. 30, 737–739.Google Scholar
Hide, R. 1958 An experimental study of thermal convection in a rotating liquid. Phil. Trans. 250, 441–478.Google Scholar
Hide, R. & Mason, P. J. 1975 Sloping convection in a rotating fluid. Adv. in Phys. 24, 47–100.Google Scholar
Hoskins, B. J. 1973 Stability of the Rossby — Haurwitz wave. Quart. J. Roy. Met. Soc. 99, 732–745.Google Scholar
Loesch, A. Z. 1974 Resonant interactions between unstable and neutral baroclinic waves. Part I. J. Atmos. Sci. 31, 1177–1201.Google Scholar
Longuet-Higgins, M. S. & Gill, A. E. 1967 Resonant interactions between planetary waves. Proc. Roy. Soc. A 299, 120–140.Google Scholar
Mcewan, A. D., Mander, D. W. & Smith, R. K. 1972 Forced resonant interaction between damped internal waves. J. Fluid Mech. 55, 589–608.Google Scholar
Mcewan, A. D. & Robinson, R. M. 1975 Parametric instability of internal gravity waves. J. Fluid Mech. 67, 667–687.Google Scholar
Mcintyre, M. E. 1973 Mean motions and impulse of a guided internal gravity wave packet. J. Fluid Mech. 60, 801–811.Google Scholar
Martin, S., Simmons, W. & Wunsch, C. 1972 The excitation of resonant triads by single internal waves. J. Fluid Mech. 53, 17–44.Google Scholar
Newell, A. C. 1969 Rossby wave packet interactions. J. Fluid Mech. 35, 255–271.Google Scholar
Plumb, R. A. 1976 Wave interactions in a bounded fluid. Submitted to J. Fluid Mech.Google Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.Google Scholar