On Formal Stability of Stratified Shear Flows

Abstract

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 $L\equiv E+C_E$ 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 $L$ 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 $L$ (${\delta }^{2}L=0$) 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 $O$ being an isolated equilibrium point in terms of orbital connections.