Instability of flows in spatially developing media

This paper studies specific forms of partial differential equations which represent the amplitude evolution of disturbances to flows in media whose properties vary slowly in space (over a length scale $\epsilon ^{-1}$). A method is described for finding series solutions in the small parameter $\epsilon $, and general results for the Green's function and the time-periodic response are obtained. With the aid of a computer-based symbolic algebra manipulation package, the method permits results to be obtained to any desired accuracy for any specified spatial variation. The method is then applied to find the Green's function for the linearized Ginzburg-Landau equation (which is a good simple model of instability growth in many marginally unstable fluid-dynamical systems) in the case where the instability parameter $\mu $ (normally a constant in the equation) varies with the spatial coordinate. Even though the method produces only power series solutions in $\epsilon $, it is shown that with its help exact analytic solutions, valid for all $\epsilon $ (however large), can be found in some cases. The exact solutions for the cases of linear and quadratic spatial variation of $\mu $ are obtained and discussed, and criteria are found for the onset of global instability. Comparisons are made between these criteria and the local absolute and convective instability criteria, and the results are found to be in qualitative agreement with other authors' numerical and analytic results; in particular, it is shown that a small region of local absolute instability (up to a maximal size which is known analytically) may exist and yet that the flow may be globally stable. It is also shown that in the case of quadratic spatial variation of $\mu $ with an infinite region of local absolute instability, the Green's function may (for some particular parameter ranges) become infinite at all spatial locations simultaneously at some finite time.