ON ARNOLDS 2ND NONLINEAR STABILITY THEOREM FOR 2-DIMENSIONAL QUASI-GEOSTROPHIC FLOW

Arnol'd's second hydrodynamical stability theorem, proven originally f or the two-dimensional Euler equations, can establish nonlinear stabil ity of steady flows that are maxima of a suitably chosen energy-Casimi r invariant. The usual derivations of this theorem require an assumpti on 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 t he disturbances are allowed to have non-zero circulation. New nonlinea r stability criteria are derived, and explicit bounds are obtained on both the disturbance energy and potential enstrophy which are expresse d in terms of the initial disturbance fields. While Arnol'd's stabilit y method relies on the second variation of the energy-Casimir invarian t being sign-definite, the new criteria can be applied to cases where the second variation is sign-indefinite because of the disturbance cir culations. A version of Andrew's theorem is also established for this problem. The case of uniform potential vorticity flow is not treatable using Arnol'd's theorems. Nevertheless, using the inequalities develo ped in this paper, uniform potential vorticity flow is shown to be non linearly stable.