Global and absolute instabilities of spatially developing open flows and media with algebraically decaying tails

Abstract

In this paper we extend our recently developed theory of absolute instability of spatially developing localized open flows and media to treating the case when the physical properties of the base state tend algebraically to constant properties at ±∞. The Laplace–transformed problem Z x ( x , ω ) = [ A ( ω ) + ( x ) ] Z ( x , ω ) + g ( x , ω ) , governing the perturbation dynamics of the flow is treated as a dynamical system. Here, Z ( x, ω ) is the Laplace–transformed perturbation, x e R is the spatial coordinate, ω e C is a frequency (and a Laplace transform parameter) and g( x, ω ) is the source function. The analysis assumes that the entries of the tail matrix R ( x ) decay as | x |– α , when x → ± ∞, where α > 0 is sufficiently large. We impose no restriction on the rate of variability of R ( x ) in the finite domain. The Levinson theorem is used for obtaining decompositions of the fundamental matrix of the system, Φ ( x , ω ) = B ± ( x , ω ) e A ( ω ) x [ B ± ( 0 , ω ) ] - 1 with the asymptotics B±( x, ω ) = I + 0 |x|− ϵ), ϵ > 0, when x → ± ∞ , respectively, where I is the identity matrix, which parallel the Floquet decomposition in the spatially periodic case. By using these decompositions, the boundary conditions of decay of Z ( x, ω ), when x → ±∞, are formulated in terms of B ±( x,ω ) and of two projectors on the subspaces spanned by the eigenvectors and generalized eigenvectors of A ( ω ) having the eigenvalues with positive and, correspondingly, negative real parts. The boundary–value problem for Z ( x, ω ) is solved formally, and the dispersion relation functions, Dn ( ω ), for the global normal modes, for the corresponding regions, R n ⊂ C, n ⩾ 1, are expressed in terms of the projectors and the matrices B ±(0, ω ). When the associated uniform state, i.e. the one with R ( x ) being zero, is stable, the flow is shown to be globally unstable if and only if the function D 1( ω ) has a root in the upper ω –half–plane. A formal solution of the initial–value linear stability problem is obtained, and it is shown that the flow is absolutely unstable if and only if either the analytic continuation, D∼ 1( ω ), of D 1( ω ) has a root or one of the matrix functions B ±(0, ω ) has a singularity, for ω with Im ω > 0, or the associated uniform flow is absolutely unstable or a combination of the above holds. It is argued that the concept of local stability cannot be consistently defined for open inhomogeneous flows treated. We present a procedure for analysing a spatially developing open flow on global and absolute instabilities, and suggest a frequency–selection criterion for open flows with self–sustained oscillations.