Laminar-Turbulent Patterns in Shear Flows: Evasion of Tipping, Saddle-Loop Bifurcation, and Log Scaling of the Turbulent Fraction

Spatial pattern formation can be a signal for tipping points and abrupt transitions in complex systems. In wall shear flows, the homogeneous turbulent state is disconnected from the laminar one and disappears in a tipping catastrophe scenario. It, however, linearly destabilizes before tipping, giving rise to laminar-turbulent banded patterns. The subcritical transition to turbulence is thus a promising candidate for investigating the evasion of tipping and its consequences in a well-controlled setting. To do so, we analyze a one-dimensional two-scalar fields advection diffusion reaction model of the transition. We characterize the multistability of the nonlinear solutions emerging from the instability and show that the pattern wavelength is selected by turbulent fluctuations. At lower Reynolds numbers, the pattern follows a cascade of destabilizations toward larger and larger, eventually infinite wavelengths. In that limit, the periodic limit cycle associated with the spatial pattern hits the laminar fixed point, resulting in a saddle loop, also called homoclinic, global bifurcation and the emergence of solitary pulse solutions. This saddle-loop scenario predicts a logarithmic divergence of the wavelength, which captures available experimental and numerical data.