The Most Unstable Profiles of a Plane-Parallel Flow in a Channel

It is well known (after Rayleigh) that a plan-parallel flow in a channel can be unstable only if the basic velocity profile U(z) possesses inflection points. The profile determines (via the Rayleigh equation) the maximal increment ∣αci∣ of small perturbations and 'eigenvalues' c (see Equation (1)). The increment and the imaginary parts ci of the eigenvalues c provide a quantitative characterization of the basic stability properties of the flow. Here we find some best possible bounds for these values. The bounds are determined by the following parameters: wave number α; enstrophy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1/2\int_0^L {[U'(z)]^2 } {\text{d}}z$$ \end{document} of the basic flow; width of the channel L. A similar approach can be applied to models of atmosphere, ocean, plasma etc.