Critical Nonlinear Phenomena for Kinetic Instabilities Near Threshold

A universal integral equation has been derived and solved for the nonlinear evolution of collective modes driven by kinetic wave particle resonances just above the threshold for instability. The dominant nonlinearity stems from the dynamics of resonant particles which can be treated perturbatively near the marginal state of the system. With a resonant particle source and classical relaxation processes included, the new equation allows the determination of conditions for a soft nonlinear regime, where the saturation level is proportional to the increment above threshold, or a hard nonlinear regime, where the saturation level is independent of the closeness to threshold. It has been found, both analytically and numerically, that in the hard regime the system exhibits explosive behavior and rapid oscillations of the mode amplitude. When the kinetic response is a requirement for the existence of the mode, this explosive behavior is accompanied by frequency chirping. The universality of the approach suggests that the theory applies to many types of resonant particle driven instabilities, and several specific cases, viz. energetic particle driven Alfvén wave excitation, the fishbone oscillation, and a collective mode in particle accelerators, are discussed. PACS numbers: 52.35.-g, 52.35.Mw, 52.40.Mj