On formal stability of stratified shear flows

Hirofumi Sakuma, Yasuhide Fukumoto

Research output: Contribution to journalArticlepeer-review

Abstract

A novel linear stability criterion is established for the equilibria of general three-dimen- sional (3D) rotating ows 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 in- variants. As the conventional ECC method cannot be applied directly to stratified shear ows, in our new approach, rather than checking the local convexity of a Lyapunov func- tional L ≡ E + CE defined as a sum of the total energy and a certain Casimir, we seek the condition for nonexistence of unstable manifolds: orbits (physically realisable ow 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 (δ2L = 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.

Original languageEnglish
Pages (from-to)605-633
Number of pages29
JournalPublications of the Research Institute for Mathematical Sciences
Volume51
Issue number4
DOIs
Publication statusPublished - Jan 1 2015

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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