Pushed global modes in weakly inhomogeneous subcritical flows

Abstract

A new type of nonlinear global mode (or fully nonlinear synchronized solution), arising in the dynamics of open shear flows which behave as oscillators [Annu. Rev. Fluid Mech. 22 (1990) 473] and in optical parametric oscillators [J. Opt. Soc. Am. B 17 (2000) 997], is identified in the context of the real subcritical Ginzburg–Landau equation with slowly varying coefficients. The nonlinear global modes satisfy a boundary condition accounting for the inlet of the flow. We show that the spatial structure of these new nonlinear global modes consists of a localized state limited by a pushed upstream front that withstand the mean advection and a fast return to zero downstream achieved either by a second stationary front facing backward, or by a saddle-node bifurcation driven by the non-parallelism of the flow. We derive scaling laws for the slope of the nonlinear global modes at the inlet and for the position of the maximum amplitude which are in agreement with similar scaling obtained in experiments with a shear layer in a Hele–Shaw cell [Phys. Rev. Lett. 82 (1999) 1442]. Extension to the complex Ginzburg–Landau equation is discussed. In a large region of parameters, the nonlinear global modes have the same spatial structure as in the real case and oscillate at a global frequency selected at threshold by the pushed front. In parameters region where these pushed global modes are not selected, new states with a non-periodic in time behavior are exhibited.