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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
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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

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