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

Shepherd, T. G. ORCID: https://orcid.org/0000-0002-6631-9968 (1988) Rigorous bounds on the nonlinear saturation of instabilities to parallel shear flows. Journal Of Fluid Mechanics, 196. pp. 291-322. ISSN 0022-1120 doi: 10.1017/S002211208800271X

Abstract/Summary

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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/32907
Item Type	Article
Refereed	Yes
Divisions	No Reading authors. Back catalogue items Science > School of Mathematical, Physical and Computational Sciences > Department of Meteorology
Publisher	Cambridge University Press
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