Linear Global Modes in Spatially-Developing Media
Stéphane Le Dizès, Patrick Huerre, Jean-Marc Chomaz
Laboratoire d'Hydrodynamique (LadHyX),
Ecole Polytechnique, F-91128 Palaiseau cedex, France.
Peter A. Monkewitz
Département de Mécanique, IMHEF,
Ecole Polytechnique Fédérale de Lausanne, ME-Ecublens,
CH-1015 Lausanne, Switzerland.
Phil. Trans. R. Soc. Lond. 354, 169-212 (1996).
Abstract:
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.
Article (Gzipped Tar Postcript version with figures, 177K)