On formal stability of stratified shear flows (Q899101)

Summary: 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 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.