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Relaxation oscillations in weak turbulence. The case of nonlinear wave-particle interaction
Published online by Cambridge University Press: 13 March 2009
Abstract
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 δK 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.
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