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

The effect of pressure anisotropy on ballooning modes in tokamak plasmas

Edge Localised Modes are thought to be caused by a spectrum of magnetohydrodynamic instabilities, including the ballooning mode. While ballooning modes have been studied extensively both theoretically and experimentally, the focus of the vast majority of this research has been on isotropic plasmas. The prevalence of pressure anisotropy in modern tokamaks thus motivates further study of these modes. This paper presents a numerical analysis of ballooning modes in anisotropic equilibria. The investigation was conducted using the newly-developed codes HELENA+ATF and MISHKA-A, which adds anisotropic physics to equilibria and stability analysis. We have examined the impact of anisotropy on the stability of an n = 30 ballooning mode, confirming results conform to previous calculations in the isotropic limit. Growth rates of ballooning modes in equilibria with different levels of anisotropy were then calculated using the stability code MISHKA-A. The key finding was that the level of anisotropy had a significant impact on ballooning mode growth rates. For ${T}_{\perp }\gt {T}_{| | }$, typical of ICRH heating, the growth rate increases, while for ${T}_{\perp }\lt {T}_{| | }$, typical of neutral beam heating, the growth rate decreases

Long range frequency chirping of Alfven eigenmodes

A theoretical framework has been developed for an NBI scenario to model the hard non-linear evolution of global Alfven eigenmodes (GAEs) where the adiabatic motion of phase-space structures (holes and clumps), associated with frequency chirping, occurs in generalised phase-space of slowing down energetic particles. The radial profile of the GAE is expanded using finite elements which allows update of the mode structure as the mode frequency chirps. Constants of motion are introduced to track the dynamics of energetic particles during frequency chirping by implementing proper action-angle variables and canonical transformations which reduce the dynamics essentially to 1D. Consequently, we specify whether the particles are drifting inward/outward as the frequency deviates from the initial MHD eigenfrequency. Using the principle of least action, we have derived the non-linear equation describing the evolution of the radial profile by varying the total Lagrangian of the system with respect to the weights of the finite elements. For the choice of parameters in this work, it is shown that the peak of the radial profile is shifted and also broadens due to frequency chirping. The time rate of frequency change is also calculated using the energy balance and we show that the adiabatic condition remains valid once it is satisfied. This model clearly illustrates the theoretical treatment to study the long range adiabatic frequency sweeping events observed for Alfven gap modes in real experiments.This work was funded by the Australian Research Council through Grant No. DP140100790

Bursting toroidal Alfven eigenmodes in KSTAR plasmas

We report on observations of bursty mode activity during early neutral beam heating in KSTAR plasmas, before current flat top while the q profile is still evolving. The magnitude of the activity increases with early beam heating, and reduces with the addition of resonant magnetic perturbation magnetic field coils. A mode analysis yields a toroidal mode number of n = 2. The mode is observed to be downward chirping in frequency, and exists for the duration of the slowing-down of the beam. Motional Stark effect constrained equilibrium reconstructions are available at adjacent time slices: we have rescaled the total current to the measured value to obtain the q profile during the mode activity. From this we have computed the mode spectrum and identified a number of candidate gap modes. Wave-particle simulations with plausible distribution functions are computed, which demonstrate that the lowest frequency mode satisfies the condition for wave drive , where is the fast ion diamagnetic drift frequency. An interesting finding is the change from exponential growth of the mode above , whereby the mode continues to nonlinearly grow at a reduced rate over a period of 100 wave periods up to final saturated amplitude. We believe that this may be because the two spatial resonances at s 0.4 and s 0.8 overlap for sufficiently high fast ion density, and so the phase-space volume and fast ion density available to drive the mode increases.This work was partly funded by the Australian Government through Australian Research Council grants FT0991899, DP140100790, as well as National Research Foundation of Korea grants NRF 2011-0018742 and NRF 2012-0000590 under the KSTAR project