Against the wind

The linear and the nonlinear dynamics of open unstable flow in a finite domain of size L is studied on a modified supercritical Ginzburg-Landau equation. When the advection term is nonzero, the bifurcation to a finite-amplitude state occurs when the instability is absolute, even for large L. The standard weakly nonlinear theory is limited to a control parameter domain of size varying as L^-5 due to the nonnormality of the linear evolution operator. The fully nonlinear solution is given and two generic cases are discussed: a supercritical case in which the instability is absolute and a subcritical case in which the instability is solely convective. The subcritical case gives a mathematical example of a bypass transition due to transient growth. The supercritical case allows a fully quantitative comparison, including the effect of the domain size, with results obtained by Büchel et al. for the size of the bifurcated solutions in the Taylor-Couette problem with throughflow.