Variational Necessary and Sufficient Stability Conditions for Inviscid Shear Flow

Abstract

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-selfadjoint eigenvalue problem) can be associated with positive eigenvalues of a certain selfadjoint 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 understandings of Krein signature for continuous spectrum and is applicable to other stability problems of infinite-dimensional Hamiltonian systems.