Transition to turbulence or to periodic patterns in parallel flows

We propose a scenario for the formation of localized turbulentspots in transition flows, which is known as resulting from the subcritical character of the transition. We show that it is not necessary to add by handa term of random noise in the equations, in order to describe the existence of long wavelength fluctuations as soon as the bifurcated state is beyond the Benjamin-Feir instability threshold. We derive the instability threshold for generalized complex Ginzburg-Landau equation which displays subcriticality. Beyond but close to the Benjamin-Feir threshold we show that the dynamics is mainly driven by the phase of the complex amplitude which obeys Kuramoto-Sivashinsky equation. As already proved for the supercritical case, the fluctuations of the intensity are smaller than those of the phase and slaved to the phase. On the opposite, below the Benjamin-Feir instability threshold, the bifurcated state does lose the randomness associated to turbulence so that the transition becomes of the mean-field type as in noiseless reaction-diffusion systems and leads to pulse-like patterns.

Automated summary

We propose a scenario for the formation of localized turbulent spots in transition flows, which does not require the addition of random noise term in the equations. We derive the instability threshold for generalized complex Ginzburg-Landau equation and show that the dynamics is driven by the phase of the complex amplitude. Above the Benjamin-Feir threshold, the fluctuations of the intensity are smaller than those of the phase and slaved to the phase; below the threshold, the transition becomes of the mean-field type and leads to pulse-like patterns.