This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

The modification of particle distributions by low amplitude magnetohydrodynamic modes is an important topic for magnetically confined plasmas. Low amplitude modes are known to be capable of producing significant modification of injected neutral beam profiles, and the same can be expected in burning plasmas for the alpha particle distributions. Flattening of a distribution due to phase mixing in an island or due to portions of phase space becoming stochastic is a process extremely rapid on the time scale of an experiment but still very long compared to the time scale of guiding center simulations. Thus it is very valuable to be able to locate significant resonances and to predict the final particle distribution produced by a given spectrum of magnetohydrodynamic modes. In this paper we introduce a new method of determining domains of phase space in which good surfaces do not exist and use this method for quickly determining the final state of the particle distribution without carrying out the full time evolution leading to it.

/pdf/modification-of-particle-distributions-by-mhd-instabilities-3kkuhazy2p.pdf

Modification of particle distributions by MHD instabilities

The results of particle-in-cell simulations of a negative neutral beam heated Alfven-mode experiment in the Japan Atomic Energy Research Institute Tokamak-60 Upgrade (JT-60U) [H. Ninomiya et al., Fusion Sci. Technol. 42, 7 (2002); A. Kitsunezaki et al., Fusion Sci. Technol. 42, 179 (2002)] are presented. They seem to match quite well the dynamics of the abrupt large-amplitude event (ALE) experimentally observed in the reference JT-60U discharge. The time scale and frequency spread of the ALE are well reproduced too. The issue of the weaker Alfven fluctuation phase following the ALEs, characterized by fast frequency sweeping modes, is also investigated and an interpretation of the full JT-60U bursting-mode phenomenology is presented. Finally, the simulation tool is exploited by ad hoc synthetic diagnostics on the fast ion distribution function to get a deeper insight into the ALE nonlinear dynamics. The underlying fast-growing energetic particle mode saturates as resonant energetic ions are scattered out o...

http://absimage.aps.org/image/DPP06/MWS_DPP06-2006-000272.pdf

Particle simulation of bursting Alfv\'{e}n modes in JT-60U

Following the analysis on linear spectra of shear Alfven fluctuations excited by energetic particles (EPs) in the Divertor Tokamak Test (DTT) facility plasmas [T. Wang et al., Phys. Plasmas 25, 062509 (2018)], in this work, nonlinear dynamics of the corresponding mode saturation and the fluctuation induced EP transport is studied by hybrid magnetohydrodynamic-gyrokinetic simulations. For the reversed shear Alfven eigenmode driven by magnetically trapped EP precession resonance in the central core region of DTT plasmas, the saturation is mainly due to radial decoupling of resonant trapped EPs. Consistent with the wave-EP resonance structure, EP transport occurs in a similar scale to the mode width. On the other hand, passing EP transport is analyzed in detail for toroidal Alfven eigenmode in the outer core region, with mode drive from both passing and trapped EPs. It is shown that passing EPs experience only weak redistributions in the weakly unstable case; and the transport extends to meso-scale diffusion in the strongly unstable case, due to orbit stochasticity induced by resonance overlap. Here, weakly/strongly unstable regime is determined by Chirikov condition for resonance overlap. This work then further illuminates rich and diverse nonlinear EP dynamics related to burning plasma studies, and the capability of DTT to address these key physics.

Nonlinear dynamics of Shear Alfv\'en fluctuations in Divertor Tokamak Test facility plasmas

Spontaneous nonlinear excitation of geodesic acoustic mode (GAM) by toroidal Alfven eigenmode (TAE) is investigated using nonlinear gyrokinetic theory. It is found that, the nonlinear decay process depends on thermal ion \beta value. Here, \beta is the plasma thermal to magnetic pressure ratio. In the low-\beta limit, TAE decays into a GAM and a lower TAE sideband in the toroidicity induced shear Alfven wave continuous spectrum gap; while in the high-\beta limit, TAE decays into a GAM and a propagating kinetic TAE in the continuum. Both cases are investigated for the spontaneous decay conditions. The nonlinear saturation levels of both GAM and daughter wave are derived. The corresponding power balance and wave particle power transfer to thermal plasma are computed. Implications on thermal plasma heating are also discussed.

Nonlinear excitation of geodesic acoustic mode by toroidal Alfv\'en eigenmodes and impact on plasma performance

The Divertor Tokamak Test (DTT) facility is proposed for studying power exhaust solutions as well as integrated physics and technology aspects for the demonstration power plant. To illuminate the richness of new novel plasma physics that can be explored in this device, linear stability properties and shear Alfven fluctuation spectra of a typical DTT reference scenario are investigated by self-consistent hybrid magnetohydrodynamic-gyrokinetic simulations. The DTT core plasmas can be divided into two regions, characterized by reverse shear Alfven eigenmode in the central core and by toroidal Alfven eigenmode in the outer core region. The non-perturbative effect of energetic particles (EPs) and the wave-EP resonance condition as well as power transfer is analyzed in great detail, demonstrating the peculiar role played by EPs in multi-scale dynamics. The most unstable mode numbers of dominant Alfvenic fluctuations are shown to be of the order of 10, consistent with the typical orbit widths of the EPs normalized to the plasma minor radius and the DTT target design.

/pdf/shear-alfven-fluctuation-spectrum-in-divertor-tokamak-test-46q0m8sbtc.pdf

Shear Alfvén fluctuation spectrum in divertor tokamak test facility plasmas

Following the analysis on linear spectra of shear Alfven fluctuations excited by energetic particles (EPs) in the Divertor Tokamak Test (DTT) facility plasmas [Wang et al., Phys. Plasmas 25, 062509 (2018)], in this work, nonlinear dynamics of the corresponding mode saturation and the fluctuation induced EP transport is studied by hybrid magnetohydrodynamic-gyrokinetic simulations. For the reversed shear Alfven eigenmode driven by magnetically trapped EP precession resonance in the central core region of DTT plasmas, the saturation is mainly due to radial decoupling of resonant trapped EPs. Consistent with the wave-EP resonance structure, EP transport occurs in a similar scale to the mode width. On the other hand, passing EP transport is analyzed in detail for the toroidal Alfven eigenmode in the outer core region, with mode drive from both passing and trapped EPs. It is shown that passing EPs experience only weak redistributions in the weakly unstable case, and the transport extends to meso-scale diffusio...

/pdf/nonlinear-dynamics-of-shear-alfven-fluctuations-in-divertor-iqaixywclb.pdf

Nonlinear dynamics of shear Alfvén fluctuations in divertor tokamak test facility plasmas

Hybrid MHD-gyrokinetic code simulations are used to investigate the dynamics of frequency sweeping reversed shear Alfven eigenmode (RSAE) strongly driven by energetic particles (EPs) during plasma current ramp-up in a conventional tokamak configuration. A series of weakly reversed shear equilibria representing time slices of long timescale MHD equilibrium evolution is considered, where the self-consistent RSAE-EP resonant interactions on the short timescale are analyzed in detail. Both linear and nonlinear RSAE dynamics are shown to be subject to the non-perturbative effect of EPs by maximizing wave-EP power transfer. In linear stage, EPs induce evident mode structure and frequency shifts; meanwhile, RSAE saturates by radial decoupling with resonant EPs due to weak magnetic shear, and gives rise to global EP convective transport and non-adiabatic frequency chirping. The spatiotemporal scales of phase space wave-EP interactions are characterized by the perpendicular wavelength and wave-particle trapping time. The simulations provide insights into general as well as specific features of RSAE spectra and EP transport from experimental observations, and illustrate the fundamental physics of wave-EP resonant interaction with the interplay of magnetic geometry, plasma non-uniformity and non-perturbative EPs.

Dynamics of reversed shear Alfvén eigenmode and energetic particles during current ramp-up

Hybrid MHD-gyrokinetic code simulations are used to investigate the dynamics of frequency sweeping reversed shear Alfven eigenmode (RSAE) strongly driven by energetic particles (EPs) during plasma current ramp-up in a conventional tokamak configuration A series of weakly reversed shear equilibria representing time slices of long timescale MHD equilibrium evolution is considered, where the self-consistent RSAE-EP resonant interactions on the short timescale are analyzed in detail Both linear and nonlinear RSAE dynamics are shown to be subject to the non-perturbative effect of EPs by maximizing wave-EP power transfer In linear stage, EPs induce evident mode structure and frequency shifts; meanwhile, RSAE saturates by radial decoupling with resonant EPs due to weak magnetic shear, and gives rise to global EP convective transport and non-adiabatic frequency chirping The spatiotemporal scales of phase space wave-EP interactions are characterized by the perpendicular wavelength and wave-particle trapping time The simulations provide insights into general as well as specific features of RSAE spectra and EP transport from experimental observations, and illustrate the fundamental physics of wave-EP resonant interaction with the interplay of magnetic geometry, plasma non-uniformity and non-perturbative EPs

Tao Wang

Papers

Shear Alfvén fluctuation spectrum in divertor tokamak test facility plasmas

Nonlinear dynamics of Shear Alfv\'en fluctuations in Divertor Tokamak Test facility plasmas

Nonlinear dynamics of shear Alfvén fluctuations in divertor tokamak test facility plasmas

Dynamics of reversed shear Alfvén eigenmode and energetic particles during current ramp-up

Dynamics of reversed shear Alfv\'en eigenmode and energetic particles during current ramp-up