Arnol'd's second stability theorem for the equivalent barotropic model

Arnol'd's second stability theorem is proved for arbitrary perturbations of the potential vorticity field δ q and the circulation (s) δγ. The formal stability condition is essentially the same as that for δγ ≡ 0, which is much easier to obtain. Similarly, the condition obtained assuming q 0 (the overbar denoting a horizontal average) is found to be also valid for q 0. It is argued that a Lyapunov functional that is extreme only on the sheet of constant Casimirs (and other integrals of motion) also proves stability for perturbations off the sheet, even though its second variation may not be sign definite for general perturbations. This conjecture is illustrated by means of a very simple mechanical problem: a point particle subject to the action of a central force. For the case of Phillips’ problem in a periodic channel, formal stability conditions on the isovortical sheet coincide with the criteria obtained from normal modes analysis.