Nonlinear drift-wave and energetic particle long-time behaviour in stellarators: solution of the kinetic problem

We propose a theoretical scheme for the study of the nonlinear interaction of drift-wave-like turbulence and energetic particles in stellarators. The approach is based on gyrokinetics, and features a separation of time and scales, for electromagnetic fluctuations, inspired by linear ballooning theory. Two specific moments of the gyrokinetic equation constitute the main equations of the system, which requires a full kinetic nonlinear solution. This is found iteratively, expanding in the smallness of the bounce-average radial drift frequency, and nonlinear E x B drift frequency, compared with the inverse time scales of the resonantly interacting energetic particles. Our analysis is therefore valid for neoclassically optimised stellators. The resummation of all iterative and perturbative nonlinear kinetic solutions is discussed in terms of Feynman diagrams. Particular emphasis is put on the role of collisionlessly undamped large-scale structures in phase space, the kinetic equivalent of zonal flows, i.e. phase-space zonal structures, and on wave-like fluctuations generated by energetic particles.

Automated summary

We propose a theoretical scheme for studying the nonlinear interaction of drift-wave-like turbulence and energetic particles in stellarators. The scheme is based on gyrokinetics and separates time and scales for electromagnetic fluctuations using linear ballooning theory. The main equations of the system are derived from two specific moments of the gyrokinetic equation, which are then solved iteratively for a full kinetic nonlinear solution. Our analysis is valid for neoclassically optimized stellarators, and we discuss the resummation of all iterative and perturbative nonlinear kinetic solutions using Feynman diagrams, with emphasis on collisionlessly undamped large-scale structures and wave-like fluctuations generated by energetic particles.