In toroidal magnetically confined plasmas, eigenmodes of Alfvén waves can be destablized by energetic ions with velocities comparable to the Alfvén velocity. With the advent of tokamak experiments in which populations of energetic ions can be introduced by neutral beam injection, radio frequency wave heating or by fusion reactions, major advances have been made in Alfvén eigenmode research in the past 10 years. After introducing the basic concepts on the Alfvén eigenmode instability, data on this subject from various toroidal devices are described, emphasizing the interplay between experiment and theory. Experimental results on mode identification, instability drive, mode damping and saturation, and energetic ion redistribution are compared with theory.

No abstract

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 that 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, characterized by explosive behavior, where the saturation level is independent of the closeness to threshold. In the hard regime, rapid oscillations typically arise that lead to large frequency shifts in a fully developed nonlinear stage. 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.