The Berk-Breizman Model as a Paradigm for Energetic Particle-driven Alfven Eigenmodes

The achievement of sustained nuclear fusion in magnetically confined plasma relies on efficient confinement of high-energy ions produced by the fusion reaction. Such particles can excite Alfven Eigenmodes (AEs), which significantly degrade their confinement and threatens the vacuum vessel of future reactors. To develop diagnostics and control schemes, a better understanding of linear and nonlinear features of resonant interactions between plasma waves and high-energy particles, is required. In the case of an isolated single resonance, the problem is homothetic to the so-called Berk-Breizman (BB) problem, which is an extension of the classic bump-on-tail electrostatic problem, including external damping to a thermal plasma, and collisions. A semi-Lagrangian simulation code, COBBLES, is developed to solve the initial-value BB problem. The nonlinear behavior of instabilities in experimentally-relevant conditions is categorized into steady-state, periodic, chaotic, and frequency-sweeping (chirping) regimes. The chaotic regime is shown to extend into a linearly stable region, and a mechanism for such subcritical instabilities is proposed. Analytic and semi-empirical laws for nonlinear chirping characteristics, such as sweeping-rate, lifetime, and asymmetry, are developed and validated. Long-time simulations demonstrate the existence of a quasi-periodic chirping regime. Collisional drag and diffusion are shown to be essential to reproduce the alternation between major chirping events and quiescent phases, which is observed in experiments. Based on these findings, a fitting procedure between COBBLES simulations and chirping AE experiments is developped. This procedure, which yields local linear drive and external damping rate, is applied to Toroidicity-induced AEs (TAEs) on JT-60U and MAST tokamaks. This suggests the existence of TAEs relatively far from marginal stability

中性流体およびプラズマにおける亜臨界不安定性について

International audience亜臨界不安定性は，非線形不安定性の一種である．亜臨界不安定な系は，線形安定であっても非線形的に不安定となる．特徴として，不安定性が生じるための初期摂動の大きさに閾値が存在し，閾値以下の摂動は減衰し安定化する．亜臨界不安定性は，流体やプラズマにおいて広くみられる現象である．亜臨界不安定性は，乱流や構造形成，異常抵抗性や乱流輸送に本質的なインパクトを与えるため重要な問題である．この解説では，亜臨界不安定性の概念について解説し，様々な物理的局面における研究について紹介する

Subcritical Instabilities in Neutral Fluids and Plasmas

International audienceIn neutral fluids and plasmas, the analysis of perturbations often starts with an inventory of linearly unstable modes. Then, the nonlinear steady-state is analyzed or predicted based on these linear modes. A crude analogy would be to base the study of a chair on how it responds to infinitesimaly small perturbations. One would conclude that the chair is stable at all frequencies, and cannot fall down. Of course, a chair falls down if subjected to finite-amplitude perturbations. Similarly, waves and wave-like structures in neutral fluids and plasmas can be triggered even though they are linearly stable. These subcritical instabilities are dormant until an interaction, a drive, a forcing, or random noise pushes their amplitude above some threshold. Investigating their onset conditions requires nonlinear calculations. Subcritical instabilities are ubiquitous in neutral fluids and plasmas. In plasmas, subcritical instabilities have been investigated based on analytical models and numerical simulations since the 1960s. More recently, they have been measured in laboratory and space plasmas, albeit not always directly. The topic could benefit from the much longer and richer history of subcritical instability and transition to subcritical turbulence in neutral fluids. In this tutorial introduction, we describe the fundamental aspects of subcritical instabilities in plasmas, based on systems of increasing complexity, from simple examples of a point-mass in a potential well or a box on a table, to turbulence and instabilities in neutral fluids, and finally, to modern applications in magnetized toroidal fusion plasmas

Self-consistent gyrokinetic modelling of turbulent and neoclassical tungsten transport in toroidally rotating plasmas

The effect of toroidal rotation on both turbulent and neoclassical transport of tungsten (W) in tokamaks is investigated using the flux-driven, global, nonlinear 5D gyrokinetic code GYSELA. Nonlinear simulations are carried out with different levels of momentum injection that drive W to the supersonic regime, while the toroidal velocity of the main ions remains in the subsonic regime. The numerical simulations demonstrate that toroidal rotation induces centrifugal forces that cause W to accumulate in the outboard region, generating an in-out poloidal asymmetry. This asymmetry enhances neoclassical inward convection, which can lead to central accumulation of W in cases of strong plasma rotation. The core accumulation of W is mainly driven by inward neoclassical convection. However, as momentum injection continues, roto-diffusion, proportional to the radial gradient of the toroidal velocity, becomes significant and generate outward turbulent flux in the case of ion temperature gradient (ITG) turbulence. Overall, the numerical results from nonlinear GYSELA simulations are in qualitative agreement with the theoretical predictions for impurity transport, as well as experimental observations.Comment: 26 pages, 10 figures, to be publishe

Nonlinear excitation of subcritical fast ion-driven modes

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On the relationship between residual zonal flows and bump-on tail saturated instabilities

A connection is established between two classical problems: the non linear saturation of a bump-on tail instability in collisionless regime, and the decay of a zonal flow towards a finite amplitude residual. Reasons for this connection are given and commented

Effects of collisions on energetic particle-driven chirping bursts

International audienceIn the presence of an energetic particle population in a dissipative plasma, self-trapped structures in phase-space (holes and clumps) emerge from nonlinear wave-particle interactions. Their dynamics can lead to a nonlinear continuous shifting of the wave frequency (chirping). The effects of collisions on chirping characteristics are investigated, with a one-dimensional kinetic model. Existing analytic theory is extended to account for Krook-like collisions, which quantitatively explains a significant departure from widely accepted square-root time dependency. Relaxation oscillations, associated with chirping bursts, are investigated in the presence of dynamical friction and velocity-diffusion. The period increases with decreasing drag and weakly increases with decreasing diffusion. The mechanism is clarified with a simple semi-analytic model of hole/clump pair, which satisfies a Fokker-Planck equation. The model shows that the linear growth rate cannot be obtained simply by fitting an exponential to the amplitude time-series. V C 2013 AIP Publishing LLC

Gyrokinetics

DoctoralGyrokinetics is a self-consistent kinetic model of magnetised plasmas that applies to dynamical systems characterised by typical frequencies lower than the cyclotron frequency. Any gyrokinetic theory proceeds in two steps. The first one is the derivation of a gyrokinetic Vlasov equation for each charged species. This is done by building a new adiabatic invariant of motion, the magnetic moment, associated with a virtual particle, the gyrocentre, slightly shifted from the particle guiding-centre. It relies on a near-identity change of variables in a 8D extended phase space. This change of variables is not unique. Several options are discussed in this lecture note. The second part is the derivation of particle charge and current densities that enter the Maxwell equations, knowing the gyrocentre distribution functions. It appears that a magnetised plasma behaves as a medium that is both electrically polarised and magnetised. The resulting model encompasses one kinetic equation per species and the Maxwell equations. It can be used to address any self-consistent electromagnetic problem in magnetised plasmas, in particular instabilities and turbulent transport. Sections labelled with a star "*" can be skipped in a first reading. Notations can be found in Appendix A