Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-18T18:42:30.304Z Has data issue: false hasContentIssue false
  • English
  • Français

The instabilities of finite-amplitude barotropic Rossby waves

Published online by Cambridge University Press:  12 April 2006

Richard P. Mied
Richard P. Mied
Affiliation:
Ocean Sciences Division, Naval Research Laboratory, Washington, D.C. 20375
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of a plane Rossby wave in a homogeneous fluid is considered. When the two-dimensional equation which governs fluid flow on a beta-plane is linearized in the disturbance stream function, a partial differential equation with a periodic coefficient results. Substitution of a solution dictated by the Floquet theory leads to a determinant equation, and it may be shown from its symmetry properties that disturbances to a Rossby wave may be of only two types: (i) neutrally stable modes not necessarily contiguous to a stability boundary and (ii) a pair of temporally unstable waves, one growing and the other decaying.

The determinant is solved numerically for the neutral-stability boundaries and curves of constant disturbance growth rate; two distinct types of instability emerge. The first is the parametric instability, which renders all waves unstable, and is shown to be asymptotic to the classical nonlinear resonant interaction in the limit of vanishing basic-state amplitude. The details of the disturbance frequency bifurcation for zero-amplitude basic-state waves are presented, and calculations for waves with eastward and westward group velocities are made and discussed in the context of Rhines’ (1975) results for waves and turbulence on a beta-plane. In addition, a second type of instability is computed which is separate and distinct from the parametric instability. The very limited evidence presented suggests that this second kind of instability may possess characteristics which are identifiable in part with the shearing of the fluid by the large-amplitude basic state and in part with the overturning of the ambient vorticity gradient.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 86 , Issue 2 , 29 May 1978 , pp. 225 - 246
Copyright
© 1978 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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Baines, P. G. 1976 The stability of planetary waves on a sphere. J. Fluid Mech. 73, 193.Google Scholar
Carnahan, B., Luther, H. A. & Wilkes, J. O. 1969 Applied Numerical Methods. Wiley.
Coaker, S. A. 1977 The stability of a Rossby wave. Geophys. Astrophys. Fluid Dyn. 9, 1.Google Scholar
Davis, R. E. & Acrivos, A. 1967 The stability of oscillatory internal waves. J. Fluid Mech. 30, 723.Google Scholar
Firing, E. & Beardsley, R. C. 1976 The behavior of a barotropic eddy on a β-plane. J. Phys. Ocean. 6, 57.Google Scholar
Gavrilin, B. L. & Zhmur, V. V. 1977 Stability of Rossby waves in a baroclinic ocean. Oceanology 16, 330.Google Scholar
Gill, A. E. 1974 The stability of planetary waves on an infinite beta-plane. Geophys. Fluid Dyn. 6, 29.Google Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30, 737.Google Scholar
Hoskins, B. J. & Hollingsworth, A. 1973 On the simplest example of the barotropic instability of Rossby wave motion. J. Atmos. Sci. 30, 150.Google Scholar
Lorenz, E. N. 1972 Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29, 258.Google Scholar
Longuet-Higgins, M. S. & Gill, A. E. 1967 Resonant interactions between planetary waves. Proc. Roy. Soc. A 299, 120.Google Scholar
Mcewan, A. D. & Robinson, R. M. 1975 Parametric instability of internal gravity waves. J. Fluid Mech. 67, 667.Google Scholar
Mcgoldrick, L. F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21, 305.Google Scholar
Mied, R. P. 1976 The occurrence of parametric instabilities in finite-amplitude internal gravity waves. J. Fluid Mech. 78, 763.Google Scholar
Noble, B. 1969 Applied Linear Algebra. Prentice-Hall.
Plumb, R. A. 1977 The stability of small amplitude Rossby waves in a channel. J. Fluid Mech. 80, 705.Google Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417.Google Scholar