Pattern Selection in the Presence of a Cross Flow

We study the pattern selection and the dynamics of a bifurcating system such as Taylor-Couette flow or Rayleigh-Bénard convection, subject to an externally imposed cross flow using the complex Ginzburg-Landau equation as a qualitative model. We show that the bifurcation scenario is radically modified by the introduction of a cross flow, and that a nonlinear global mode, i.e., a nonlinear oscillating solution in a semi-infinite domain $[0,+\infty )$, with a homogeneous condition at $x=0\mathrm{}$, exists only when the basic state is linearly absolutely unstable. We derive the scaling law for the characteristic growth size, which varies as ${\epsilon }^{-1\mathrm{}/2}$ ( $\epsilon$ being the criticality parameter), and compares satisfactorily with numerical and experimental results from the literature.