On the normal modes of parallel flow of inviscid stratified fluid. Part 2. Unbounded flow with propagation at infinity

Abstract

The linear perturbations of the flow of a non-diffusive fluid are considered. The classification of the normal modes of parallel flow of an inviscid stratified fluid presented by Banks, Drazin & Zaturska (1976) is here extended to encompass modes which propagate at infinity. When the basic flow is unbounded and the buoyancy frequency is non-zero at infinity the five classes presented earlier are augmented by three further classes: for a given flow and wavenumber they are (a) a continuous class of non-singular stable modes which are modifications of internal gravity waves by shear; (b) a continuous class of stable modes which are singular at each critical layer but otherwise similar to those of class (a); and (c) a finite number of marginally stable singular modes with over-reflexion. This classification is illustrated by many new results. Some asymptotic properties of the stable and unstable modes are found for large values of the Richardson number and for long waves. Two prototype problems, in which the basic flows are a piecewise-linear shear layer and a triangular jet, are solved analytically. The modified internal gravity waves for a Bickley jet with uniform buoyancy frequency are treated to illustrate the complementary nature of the propagating and evanescent modes. This treatment is both analytical and numerical. The general ideas are further illustrated by a numerical study of the stability characteristics of a hyperbolic-tangent shear layer. Finally the modes for a basic flow of boundary-layer type are found in exact terms of a hypergeometric function.