Variational necessary and sufficient stability conditions for inviscid shear flow

M. Hirota, P. J. Morrison, Y. Hattori

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle, where the velocity profile is assumed to be monotonic and analytic. It is shown that unstable eigenvalues of Rayleigh's equation (which is a non-self-adjoint eigenvalue problem) can be associated with positive eigenvalues of a certain self-adjoint operator. The stability is therefore determined by maximizing a quadratic form, which is theoretically and numerically more tractable than directly solving Rayleigh's equation. This variational stability criterion is based on the understanding of KreѤn signature for continuous spectra and is applicable to other stability problems of infinite-dimensional Hamiltonian systems.

Original languageEnglish
Article number20140322
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2172
Publication statusPublished - 2014 Dec 8


  • Continuum Hamiltonian Hopf bifurcation
  • Flow stability
  • KreѤn signature
  • Variational method


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