Mean Evolution of Wrinkle Wavelengths in a Model of Weakly-Turbulent Premixed Flame

New informations on weakly-turbulent premixed flames in the cusped-flame regime are obtained upon studying a Michelson-Sivashinsky equation whose solution (the instantaneous flame shape) is excited by the influence of a weak, pseudo-random additive forcing, that is meant to mimic incoming velocity fluctuations. Attention is focused on the time-dependent mean spacing l c {t) between crests, in cases where the front is flat before forcing is switched on. Despite the complexity of each individual run, a surprisingly simple history A c (t) always emerges upon averaging l c (t) over many realizations of the forcing function, especially in the limit of weak forcings: a noise-triggered linear stage is followed by a stage of crest-coalescence during which dAJdt is a constant and the direct role of forcing is minor; then, rather abruptly, an equilibrium stage is reached where A c (t) acquires a time-independent value because the crest implants induced by the noise statistically equilibrate the coalescences. Improved or new scaling-laws, including dynamical ones, are suggested numerically and theoretically for each of the aforementioned stages.