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Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows

Published online by Cambridge University Press:  21 April 2006

Theodore G. Shepherd
Theodore G. Shepherd
Affiliation:
Department of Applied Mathematies and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: Department of Physics, University of Toronto, Toronto M5S 1A7, Canada.
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Abstract

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 196 , November 1988 , pp. 291 - 322
Copyright
© 1988 Cambridge University Press

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Footnotes

With an appendix by P. H. Haynes.

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