Relaxation oscillations in weak turbulence

A model for nonlinear Landau stabilization of a linear instability is treated analytically. The model may apply to any dissipative system described by: (i) a linear dissipation function γ( k ), of one degree of freedom k , which presents a limited unstable domain δ K and a damped region for lower values of k , and (ii) a transition probability for the nonlinear Landau damping (nonlinear wave–particle interaction) proportional to k – K '. For a sharply pointed γ( k ), the model leads to a kinetic regime presenting periodic relaxation oscillations of large and constant amplitude. Starting from weak turbulence, the system organizes itself into a periodic regime, which may be of interest in connexion with research in morphogenesis . For a smoothed γ( k ), the relaxation oscillations damp, slowing down towards a turbulent energy level.