A novel linear stability criterion is established for the equilibria of general three-dimensional (3D) rotating flows of an ideal gas satisfying Boyle-Charles\rq\ law by a newly refined energy-Casimir convexity (ECC) method that can exploit a larger class of Casimir invariants. As a conventional ECC method cannot be applied directly to stratified shear flows, in our new approach, rathern than checking the local convexity of a Lyapunov functional defined as a sum of the total energy and a ceratin Casimir, we seek the condition for non-existence of unstable manifolds: orbits (physically realisable flow in phase space) on the leaves of invariants including as well as other Casimirs connecting a given equilibrium point and other points in the neighbourhood of it. We argue that the separatrices of the second variation of () generally consist of such unstable manifolds as well as pseudo unstable ones for which either the total energy or Casimirs actually serves as a barrier for escaping orbits. The significance of new method lies in the fact that it eliminates the latter so as to derive a condition that being an isolated equilibrium point in terms of orbital connections.
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Hirofumi Sakuma, Yasuhide Fukumoto, On Formal Stability of Stratified Shear Flows. Publ. Res. Inst. Math. Sci. 51 (2015), no. 4, pp. 605–633DOI 10.4171/PRIMS/166