Frequency chirping of energetic particle driven modes in tokamaks; Self-consistent modelling and simulation
Long range frequency chirping waves have been observed in various plasma experiments. Theoretical toy models and simulations show that the nonlinear saturation of an initially unstable plasma wave driven by energetic particles may lead to the emergence of chirping waves. In toroidal configurations, these waves can lead to ejection of the particles from the hot core of the plasma and degrade the machine performance. Therefore, it is essential to develop theory and simulation models to better understand and control chirping waves in future fusion plasmas, such as ITER. Fast particles of tokamaks can destabilize plasma waves. Previous numerical simulations have revealed that an isolated Alfven eigenmode can evolve into chirping signals if the mode is subject to intrinsic damping into the cold plasma. In 1997, Berk-Breizman and co-workers studied the nonlinear stage of an excited electrostatic plasma wave where chirping signals were observed and interpreted as Bernstein-Greene-Kruskal waves with sweeping frequencies. These chirping waves are associated with coherent structures, the so-called "holes" and "clumps", in phase space of energetic particles. Such structures found to move adiabatically which implies the particles can be carried with the waves; so called convective transport. During long ranges of frequency chirping, the spatial profile of the wave can change considerably and hence new models are required to study long range frequency chirping of Alfvenic perturbations. In 2010, a 1D electrostatic nonperturbative model was developed by Breizman for an adiabatic study of chirping waves as a 1D paradigm of their electromagnetic counterparts. In order to build more realistic models, nonperturbative theory needs to be extended to electromagnetic waves and also fast particle orbital dynamics should be captured. In this work, a model is developed to investigate the effect of fast particle dynamics in tokamaks using a 1D description. This new trapped/passing model demonstrates how energetic particle orbits can impact the non-linear behaviour of the waves. The evolution of the wave radial profile as well as the rate of frequency chirping as a function of the fast particles orbits are studied. As the next step, the theory of adiabatic frequency chirping is developed for Alfvenic-type perturbations in realistic configurations. The impact of long range frequency deviations on the radial profile of a global Alfven eigenmode is studied and the chirping rate analysed. In this model, the radial component of the wave is described using the method of finite elements and the total Lagrangian of the system is varied with respect to the weight of each finite element. Also, exact constants of motion during the long range frequency chirping are introduced. In this work, the theory of chirping waves is extended further by investigating the impact of particle trapping in phase-space for a growing wave potential in the orbital model introduced above. In this respect, a phase space waterbag model is developed by using Lagrangian contours to discretize the phase-space island in adiabatic invariants. In addition, a study of the influence of higher particle resonances on the behaviour of chirping waves is also performed. Finally, self-consistent simulations are performed using the hybrid model of the MEGA code to illustrate convective transport of energetic particles in tokamaks. A novel phase-space analysis tool is built which enables a reduction of particle dynamics to a 1D picture. This is based on a new conservation law for particle dynamics which remains valid even when frequency changes. Therefore, it is possible to observe the evolution of holes and clumps on appropriate sub-slices of fast particles phase-space. For a toroidicity-induced Alfven eigenmode, the mechanism of frequency chirping phenomenon has been clarified. The observations of the wave behaviour are consistent with the adiabatic theory.
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