On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow
Mu, M. and Shepherd, T. G. (1994) On Arnol'd's second nonlinear stability theorem for two-dimensional quasi-geostrophic flow. Geophysical & Astrophysical Fluid Dynamics, 75 (1). pp. 21-37. ISSN 0309-1929
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To link to this article DOI: 10.1080/03091929408203645
Arnol'd's second hydrodynamical stability theorem, proven originally for the two-dimensional Euler equations, can establish nonlinear stability of steady flows that are maxima of a suitably chosen energy-Casimir invariant. The usual derivations of this theorem require an assumption of zero disturbance circulation. In the present work an analogue of Arnol'd's second theorem is developed in the more general case of two-dimensional quasi-geostrophic flow, with the important feature that the disturbances are allowed to have non-zero circulation. New nonlinear stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expressed in terms of the initial disturbance fields. While Arnol'd's stability method relies on the second variation of the energy-Casimir invariant being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance circulations. A version of Andrews' theorem is also established for this problem.