Absolute and convective instabilities of spatially periodic flows

Stability of monochromatic waves and wave packet evolution are of fundamental importance in the transition to turbulence in open-flow systems. Although the analysis of monochromatic wave stability and the evolution of linear wave packets in spatially homogeneous or parallel flows is generally well understood, such analysis for spatially inhomogeneous flows is not so well understood. In this paper we consider the problem of linear wave packet evolution in a spatially periodic medium. The mathematical formalism of absolute and convective instabilities in spatially homogeneous flows is generalized to the spatially periodic case. The Laplace transform is used to reduce the initial-value problem to a system of ordinary differential equations with periodic coefficients which is then completely analysed using the Floquet theory and the Fourier transform. We define a generalized dispersion relation by $\Delta $($\mu $, $\omega $) = 0, where $\mu \in \text{C}\backslash ${0} is a spatial Floquet multiplier and $\omega \in \text{C}$ is a frequency (and a Laplace transform parameter). We find that a spatially periodic flow is absolutely unstable if and only if $\Delta $($\mu $, $\omega $) has a double root (or more generally a multiple root) in $\mu $ at ($\mu _{0}$, $\omega _{0}$) with Im $\omega _{0}$ > 0 that satisfies the collision criterion: i.e. the double root splits under perturbation in such a way that in the limit, as Im($\omega $ - $\omega _{0}$) $\rightarrow $ +$\infty $, with Re($\omega $ - $\omega _{0}$) = 0, one of the roots goes interior and the other exterior to the unit circle {$|\mu|$ = 1} in the complex $\mu $-plane. Further results are obtained on the long-time asymptotics of absolutely unstable and absolutely stable flows and the signalling problem for periodic media is introduced. In general the asymptotic states of unstable periodic waves are quasi-periodic in space and/or time. The theory is applied to the finite-amplitude periodic travelling wave states of the real and complex Ginzburg-Landau equation. We find that the Eckhaus instability is always absolutely unstable whereas in the complex case there is an interesting decomposition of the region of unstable finite-amplitude travelling waves into an absolutely unstable region and a convectively unstable but absolutely stable region. The problem of unstable wave packets for the Navier-Stokes equations, linearized about a spatially periodic state, is also formulated.