Bounds on the Growth of Perturbations to Non-Parallel Steady Flow on the Barotropic Beta Plane

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  • 1 Department of Meteorology and Physical Oceanography, Massachusetts Institute of Technology
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Abstract

Based on consideration of the perturbation enstrophy and energy equations, we have derived a general family of bounds on the growth rates of perturbations to non-parallel (vortex-like or wave-like) flow on the barotropic beta-plane, allowing for the effects of forcing, Ekman friction, and topography. The family of bounds generalizes Arnol’d's stability criterion. A number of specific applications of the family of bounds are explored. In particular, the formulas are used to demonstrate that the growth rate of the perturbations must vanish if the perturbation length-scale approaches zero or infinity. The distinction between transient and sustained growth of perturbation energy is discussed in light of our results. It is suggested that the bounds are most useful for estimating transient growth rates.

Abstract

Based on consideration of the perturbation enstrophy and energy equations, we have derived a general family of bounds on the growth rates of perturbations to non-parallel (vortex-like or wave-like) flow on the barotropic beta-plane, allowing for the effects of forcing, Ekman friction, and topography. The family of bounds generalizes Arnol’d's stability criterion. A number of specific applications of the family of bounds are explored. In particular, the formulas are used to demonstrate that the growth rate of the perturbations must vanish if the perturbation length-scale approaches zero or infinity. The distinction between transient and sustained growth of perturbation energy is discussed in light of our results. It is suggested that the bounds are most useful for estimating transient growth rates.

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