On the interpretation of Andrews’ theorem
Carnevale, G. F. and Shepherd, T. G. (1990) On the interpretation of Andrews’ theorem. Geophysical & Astrophysical Fluid Dynamics, 51 (1-4). pp. 1-17. ISSN 1029-0419
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To link to this article DOI: 10.1080/03091929008219847
Andrews (1984) has shown that any flow satisfying Arnol'd's (1965, 1966) sufficient conditions for stability must be zonally-symmetric if the boundary conditions on the flow are zonally-symmetric. This result appears to place very strong restrictions on the kinds of flows that can be proved to be stable by Arnol'd's theorems. In this paper, Andrews’ theorem is re-examined, paying special attention to the case of an unbounded domain. It is shown that, in that case, Andrews’ theorem generally fails to apply, and Arnol'd-stable flows do exist that are not zonally-symmetric. The example of a circular vortex with a monotonic vorticity profile is a case in point. A proof of the finite-amplitude version of the Rayleigh stability theorem for circular vortices is also established; despite its similarity to the Arnol'd theorems it seems not to have been put on record before.