The transition to turbulence in parallel flows: transition to turbulence or to regular structures

Abstract

We propose a scenario for the formation of localized turbulent spots 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 hand" a 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 and 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 while the fluctuations of the modulus are smaller and slaved to the phase (as already proved for the supercritical case). On the opposite, below the Benjamin-Feir instability threshold, the bifurcated state does loose 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.