Nonlinear Ginzburg-Landau equation and its applications to fluid-mechanics

Abstract

The nonlinear Ginzburg-Landau equation is a convenient tool for studying the growth of unstable perturbations in shear flows. In particular, it allows [1] to understand the link between the subcritical character of the instabilities and the formation of localized patches of turbulence. This is consistent with the early observations[6] that those patches grow at constant speed in space with a rather homogeneous state inside. However when applied to real flows, this picture may be changed for two reasons. First there is a kind of very long range interaction through the pressure field in incompressible fluid mechanics and this, together with symmetry considerations explains the remarkable phenomenon of spiral turbulence [2] between counterrotating Taylor-Couette cylinders. Then other complications arise because of the basically non variational structure of the equations of fluid mechanics as well as of Ginzburg-Landau. This has two main consequences, both related to some experimental features: 1) The sign of the velocity of expansion alluded above may depend on the orientation in a basically anisotropic system as a shear flow [3]. 2) Then the Ginzburg -Landau equations may have stable solutions in the form of solitary waves [4,5]. This can be understood [5] by pertubation near the two opposite limits: unitary evolution and pure gradient flow and those solitary waves merge continuously with expanding domains by becoming infinitely wide.