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

Formation of Phase Space Holes and Clumps

It is shown that the formation of phase space holes and clumps in kinetically driven, dissipative systems is not restricted to the near threshold regime, as previously reported and widely believed. Specifically, we observe hole-clump generation from the edges of an unmodulated phase space plateau, created via excitation, phase mixing and subsequent dissipative decay of a linearly unstable bulk plasma mode in the electrostatic bump-on-tail model. This has now allowed us to elucidate the underlying physics of the hole-clump formation process for the first time. Holes and clumps develop from negative energy waves that arise due to the sharp gradients at the interface between the plateau and the nearly unperturbed, ambient distribution and destabilize in the presence of dissipation in the bulk plasma. We confirm this picture by demonstrating that the formation of such nonlinear structures in general does not rely on a "seed" wave, only on the ability of the system to generate a plateau. In addition, we observe repetitive cycles of plateau generation and erosion, the latter due to hole-clump formation and detachment, which appear to be insensitive to initial conditions and can persist for a long time. We present an intuitive discussion of why this continual regeneration occurs

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

Fast Particle Driven Instabilities in Tokamak Plasmas

The burning plasmas of the next generation tokamak fusion experiment ITER will contain signicant populations of highly energetic ions. Both fusion gener- ated alpha particles and fast ions accelerated by auxiliary heating schemes are capable of exciting instabilities in the Alfven frequency range, which may in turn cause redistribution of energetic particles and lead to deleterious events such as e.g. monster sawtooth crashes. On present day machines, many aspects of fast ion collective eects are observed and well understood, including e.g. ex- citation of toroidal Alfven eigenmodes (TAEs) and frequency sweeping Alfven cascades. However, a currently hot topic is the role of kinetic and nonlinear Alfvenic instabilities, including explanations for rapid frequency sweeping and unambiguous identication of energetic particle modes. This thesis presents theoretical research carried out on the linear and non- linear dynamics of fast particle driven instabilities, with a major focus on such events in tokamaks. In the linear regime, we investigate particle behavior in the presence of a given electromagnetic perturbation. In particular, we formu- late a condition on the prescribed wave amplitude to cause stochastic particle motion and we calculate the rate coecients for the associated diusive ran- dom walks. We also derive analytic expressions for the damping rate of even and odd TAEs, which arises as a result of non-ideal coupling to radially prop- agating waves. The damping is found to be weak for the odd mode, which raises concern for the connement of energetic particles on ITER, where an- tiballooning instabilities are likely to be resonantly driven by passing, fusion born alpha particles. Nonlinearly, we develop a simple one-dimensional model to investigate the impact of signicant frequency shifts on the dynamics of frequency sweeping, weakly driven modes. We include a number of previously neglected long range eects that are all capable of signicantly altering the temporal frequency sweeping patterns. In the case when the fast particle collision operator con- tains both a drag-like slowing-down term, due to binary, small-angle Coulomb collisions, and velocity space diusion, the model predicts transiently hooked and steady state frequency sweeping patterns. The latter scenario turns out to be analytically tractable

Behavioural syndrome in a solitary predator is independent of body size and growth rate.

Models explaining behavioural syndromes often focus on state-dependency, linking behavioural variation to individual differences in other phenotypic features. Empirical studies are, however, rare. Here, we tested for a size and growth-dependent stable behavioural syndrome in the juvenile-stages of a solitary apex predator (pike, Esox lucius), shown as repeatable foraging behaviour across risk. Pike swimming activity, latency to prey attack, number of successful and unsuccessful prey attacks was measured during the presence/absence of visual contact with a competitor or predator. Foraging behaviour across risks was considered an appropriate indicator of boldness in this solitary predator where a trade-off between foraging behaviour and threat avoidance has been reported. Support was found for a behavioural syndrome, where the rank order differences in the foraging behaviour between individuals were maintained across time and risk situation. However, individual behaviour was independent of body size and growth in conditions of high food availability, showing no evidence to support the state-dependent personality hypothesis. The importance of a combination of spatial and temporal environmental variation for generating growth differences is highlighted

Automatiserad rättning av beräknings och programmeringsuppgifter

Molntjänsten CoCalc erbjuder en smidig administration av inlämnings- och laborationsuppgifter direkt i den miljö där studenterna ska använda vid problemlösning. CoCalc kombinerar ett stort antal fria matematikprogram och programmeringsspråk tillsammans med en Linuxterminal

Algebraic Dynamical Systems, Analytical Results and Numerical Simulations

In this thesis we study discrete dynamical system, given by a polynomial, over both finite extension of the fields of p-adic numbers and over finite fields. Especially in the p-adic case, we study fixed points of dynamical systems, and which elements that are attracted to them. We show with different examples how complex these dynamics are. For certain polynomial dynamical systems over finite fields we prove that the normalized average of the numbers of linear factors modulo prime numbers exists. We also show how to calculate the average, by using Chebotarev's Density Theorem. The non-normalized version of the average of the number of linear factors of linearized polynomials modulo prime numbers, tends to infinity, so in that case we find an asymptotic formula instead. We have also used a computer to study different behaviors, such as iterations of polynomials over the p-adic fields and the asymptotic relation mention above. In the last chapter we present the computer programs used in different part of the thesis