Linear global modes in spatially developing media

Selection criteria for self-excited global modes in doubly infinite one-dimensional domains are examined in the context of the linearized Ginzburg-Landau equation with slowly varying coefficients. Following Lynn & Keller (1970), uniformly valid approximations are sought in the complex plane in a region containing all relevant turning points. A mapping transformation is introduced to reduce the original Ginzburg-Landau equation to an exactly solvable comparison equation which qualitatively preserves the geometry of the Stokes line network. The specific case of two turning points with counted multiplicity is analysed in detail, particular attention being paid to the allowable configurations of the Stokes line network. It is shown that all global modes are either of type-1, with two simple turning points connected by a common Stokes line, or of type-2, with a single double-turning point. Explicit approximations are derived in both instances, for the global frequencies and associated eigenfunctions. It is argued, on geometrical grounds, that type-1 global modes may, in principle, be more unstable than type-2 global modes. This paper is a continuation and extension of the earlier study of Chomaz, Huerre & Redekopp (1991), where only type-2 global modes were investigated via a local WKBJ approximation scheme.