Primary and secondary nonlinear global instability

Abstract

We study oscillating solutions of the complex Ginzburg–Landau equation in a semi-infinite domain asymptotic to a saturated traveling wave at +∞ and subject to a homogeneous upstream boundary condition at x=0. This inlet condition breaks the Galilean invariance, giving the advection velocity the role of a control parameter. We give a criterion for the existence of a nontrivial solution called a nonlinear global mode or a self sustained resonance, and we obtain the selected frequency. The threshold and the frequency are shown to be determined by the linear absolute instability transition. We undertake a singular perturbation analysis which first proves the validity of the criterion and secondly yields scaling laws for the frequency, the growth length and the slope at the origin of the nonlinear global modes as functions of the criticality parameter. Comparisons with direct numerical simulation of the Ginzburg–Landau model validate these predictions. Furthermore, the results are in excellent agreement with numerical simulation of the Taylor–Couette problem with throughflow by Büchel et al. [P. Büchel, M. Lücke, D. Roth, R. Schmitz, Pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem: Taylor vortices in axial flow, Phys. Rev. E 53 (1996) 4764–4777] and of Rayleigh–Bénard flow with an added Poiseuille flow by Müller et al. [H.W. Müller, M. Lücke, M. Kamps, Convective patterns in horizontal flow, Europhys. Lett. 10 (1989) 451–456; H.W. Müller, M. Lücke, M. Kamps, Transversal convection patterns in horizontal shear flow, Phys. Rev. A 45 (1992) 3714–3726]. The numerical simulations indicate that the nonlinear global modes are stable if the saturated traveling wave to which the nonlinear global mode is asymptotic (i.e. the asymptotic downstream part of the global modes) is not absolutely unstable to perturbations. A complete analysis of the dispersion relation of the secondary instability of saturated traveling waves is given.