On Formal Stability of Stratified Shear Flows

A novel linear stability criterion is established for the equilibria of general three-dimensional (3D) rotating flows of an ideal gas satisfying Boyle Charles' law by a newly refined energy-Casimir convexity (ECC) method that can exploit a larger class of Casimir invariants. As the conventional ECC method cannot be applied directly to stratified shear flows, in our new approach, rather than checking the local convexity of a Lyapunov functional L equivalent to E + C-E defined as a sum of the total energy and a certain Casimir, we seek the condition for nonexistence of unstable manifolds: orbits (physically realisable flow in phase space) on the leaves of invariants including L as well as other Casimirs connecting a given equilibrium point O and other points in the neighbourhood of it. We argue that the separatrices of the second variation of L (delta L-2 = 0) generally consist of such unstable manifolds as well as pseudo unstable ones for which either the total energy or Casimirs actually serve as a barrier for escaping orbits. The significance of the new method lies in the fact that it eliminates the latter so as to derive a condition for O being an isolated equilibrium point in terms of orbital connections.