Robert Nyqvist

Energetic Particle Instabilities in Fusion Plasmas

Remarkable progress has been made in diagnosing energetic particle instabilities on present-day machines and in establishing a theoretical framework for describing them. This overview describes the much improved diagnostics of Alfven instabilities and modelling tools developed world-wide, and discusses progress in interpreting the observed phenomena. A multi-machine comparison is presented giving information on the performance of both diagnostics and modelling tools for different plasma conditions outlining expectations for ITER based on our present knowledge.

Adiabatic description of long range frequency sweeping

A theoretical framework is developed to describe long range frequency sweeping events in the 1D electrostatic bump-on-tail model with fast particle sources and collisions. The model includes three collision operators (Krook, drag (dynamical friction) and velocity space diffusion), and allows for a general shape of the fast particle distribution function. The behaviour of phase space holes and clumps is analysed in the absence of diffusion, and the effect of particle trapping due to separatrix expansion is discussed. With a fast particle distribution function whose slope decays above the resonant phase velocity, hooked frequency sweeping is found for holes in the presence of drag collisions alone.

Modeling of long range frequency sweeping for energetic particle modes

Long range frequency sweeping events are simulated numerically within a one-dimensional, electrostatic bump-on-tail model with fast particle sources and collisions. The numerical solution accounts for fast particle trapping and detrapping in an evolving wave field with a fixed wavelength, and it includes three distinct collisions operators: Drag (dynamical friction on the background electrons), Krook-type collisions, and velocity space diffusion. The effects of particle trapping and diffusion on the evolution of holes and clumps are investigated, and the occurrence of non-monotonic (hooked) frequency sweeping and asymptotically steady holes is discussed. The presented solution constitutes a step towards predictive modeling of frequency sweeping events in more realistic geometries.

Asymmetric radiative damping of low shear toroidal Alfven eigenmodes

An important issue for alpha-particle transport due to Alfven instabilities in burning ITER plasmas [1] is the number of unstable Alfven eigenmodes (AEs) and their properties at high plasma pressure. Multiple low shear toroidal Alfv´en eigenmodes (LSTAEs) [2, 3, 4] are of major concern for ITER scenarios, where extended regions of low magnetic shear may exist in e.g. sawtoothing plasmas or hybrid regimes [1, 5]. In contrast to the conventional TAE, which has a single eigenfrequency per TAE gap, the number of LSTAEs per gap is given by l ~= Є/S [4], and can be quite large in a low-shear region. Moreover, LSTAEs are less affected by high plasma pressure than TAEs: in fact, the very discovery of LSTAEs was galvanised by DT experiments on TFTR, where unstable AEs were detected at very high plasma pressure [2]. More recently, LSTAEs were recognized as important features of sawtoothing tokamak plasmas. On JET, core localized ”tornado” modes (LSTAEs inside the q = 1 radius) were found to precede monster sawtooth crashes [6], and on Alcator C-Mod, frequency chirping AEs associated with very low shear were observed during the sawtooth cycle [7]. In this contribution, we develop a theory of LSTAEs by incorporating non-ideal effects associated with finite ion Larmor gyroradius into the MHD model [4]. The radiative damping for LSTAEs at the top and bottom of the TAE gap is calculated in the limit of high mode number and first order gyroradius, using the characteristic LSTAE ordering d/dr > m/r. We also estimate the LSTAE drive due to ICRH-accelerated fast ions [8] and obtain the mode excitation threshold for ICRH-driven TAEs.

The least uncomfortable journey from A to B

A short introduction is given about direct variational methods and their relation to Galerkin and moment methods, all flexible and powerful approaches for finding approximate solutions to difficult physical equations. An application of these methods is given in the form of the variational problem of minimizing the discomfort experienced during different journeys, between two fixed horizontal points while keeping the travel time constant. The analysis is shown to provide simple, yet accurate, approximate solutions of the problem and illustrates the usefulness and the power of direct variational and moment methods. It also demonstrates the problem of a priori assessing the accuracy of the approximate solutions and illustrates that the variational solution does not necessarily provide a more accurate solution than that obtained by moment methods.

Kinetic theory of phase space plateaux in a non-thermal energetic particle distribution

The transformation of kinetically unstable plasma eigenmodes into hole-clump pairs with temporally evolving carrier frequencies was recently attributed to the emergence of an intermediate stage in the mode evolution cycle, that of an unmodulated plateau in the phase space distribution of fast particles. The role of the plateau as the hole-clump breeding ground is further substantiated in this article via consideration of its linear and nonlinear stability in the presence of fast particle collisions and sources, which are known to affect the production rates and subsequent frequency sweeping of holes and clumps. In particular, collisional relaxation, as mediated by e.g. velocity space diffusion or even simple Krook-type collisions, is found to inhibit hole-clump generation and detachment from the plateau, as it should. On the other hand, slowing down of the fast particles turns out to have an asymmetrically destabilizing/stabilizing effect, which explains the well-known result that collisional drag enhances holes and their sweeping rates but suppresses clumps. It is further demonstrated that relaxation of the plateau edge gradients has only a minor quantitative effect and does not change the plateau stability qualitatively, unless the edge region extends far into the plateau shelf and the corresponding Landau pole needs to be taken into account.

On wave-particle interaction in axisymmetric toroidal systems

A general formalism is developed to describe the interaction of charged particles with electromagnetic waves in terms of coupled finite difference mapping equations that incorporate tokamak topology. The approach is based on considering non-adiabatic changes in the constants of particle motion and it covers a range of wave-particle resonance frequencies, from the precessional to cyclotron frequencies of both passing and trapped ions. The concept of overlapping resonances is used to estimate the threshold for a single plane wave to cause stochastic particle motion. In the stochastic regime, the process is Markovian, and particle diffusion in three-dimensional phase space takes place. Estimations of diffusion coefficients are carried out in the two cases of waves interacting with passing and trapped ions by means of the cyclotron and bounce resonances, respectively, and previously known results are recovered in the proper limits.

Spreading of Collimated Particle Beams Within a Generalized Fokker-Planck Diffusion Description

Abstract Recently, an expansion of the Boltzmann scattering operator describing the angular spreading of particle beams was given that included the effects of large angle scattering processes, thus generalizing the classical Fokker-Planck equation, valid in the limit of small angle scattering. The present work aims at making an analytical comparison between predictions based on the classical Fokker-Planck equation and those based on a generalized one, which includes a first-order correction term in the expansion of the Boltzmann scattering operator. The analysis is carried out for thin slabs where backscattering effects can be neglected and makes use of a moment approach, which leads to an infinite system of recursively coupled ordinary differential equations. The system is truncated in a consistent manner, and the effects of large angle scattering on the evolution of the moments are determined in explicit analytical form. An approximate similarity solution of the generalized Fokker-Planck equation is also found, and the results of both approaches provide a clear picture of the increased diffusive beam spreading due to large angle scattering. A comparison with previously published Monte Carlo simulation results shows good agreement.