Resonances In Phase Space Research Articles

Resonant transport of fusion alpha particles in quasisymmetric stellarators

In modern, highly optimized stellarator configurations where prompt fusion alpha particle losses from the plasma core are absent, alpha particles can still be lost due to stochastic orbits which result in delayed losses. One mechanism leading to stochastic orbits are changes in the particle trapping class during drift motion along the contours of the parallel adiabatic invariant J ∥ leading to jumps in J ∥ when crossing class boundaries. Another mechanism, which is of main interest here, is the resonance between particle drift and bounce motion (drift-orbit resonance). The first mechanism affects mainly trapped particles near the trapped-passing boundary in the phase space of quasi-symmetric and quasi-isodynamic devices, and can be minimized there by aligning local magnetic field maxima on a given flux surface. The second mechanism may affect a broader range in the trapped particle domain where contours of J ∥ still remain closed. Drift-orbit resonances modify the topology of orbits leading to island-like structures on Poincaré plots where these islands may overlap thus leading to the stochastic transport. In this report, we study this stochastization mechanism in quasi-symmetric stellarator configurations with help of the Hamiltonian drift-kinetic code NEO-RT as well as orbit classification and direct computation of fusion alpha losses within the symplectic orbit following code SIMPLE. The width and overlap of resonances in phase-space is studied using Hamiltonian perturbation theory. Based on optimized reactor configurations we assess if this approach can be used as a fast metric for fusion alpha losses in stellarator optimization.

Effects of rotating magnetic island on the transport of trapped fast ions

The enhanced transport of trapped energetic ions (TEIs) in the presence of resonant interactions between trapped fast ions and a rotating magnetic island is investigated within a drift-kinetic framework. Gyro-orbit banana center model equations of resonances between the island rotation, the bounce motion of trapped fast ions, and their precession frequency (poloidal precession and precession in the helical direction) are constructed. There are two solutions for resonances in phase space for different mode numbers, with only one solution having low-energy resonant lines (<100 keV); the other has not only low-energy resonant lines but also high-energy lines (≥100 keV). Island rotation plays an important role in the low-energy region, especially near the trapped-passing boundary. The precession frequency is more important when resonances occur in the high-energy area. Thus, the effect of islands on TEI transport in a low-energy region is the focus of this paper. Transport fluxes caused by collisions, resonances, and symmetry breaking induced by an island are obtained. We divide transport fluxes into two types: Γc arising from magnetic drift and Γb arising from the island rotation. There is a discontinuity in Γc with different island widths near the island separatrix. On the right-hand side of the (m = 2, n = 1) rational surface, Γc is more important than Γb, and at the plasma boundary, the flux due to drift can suppress Γb, which makes fast ions move toward inner plasma. On the left-hand side of the rational surface, Γb is dominant. When the island width is larger than a certain threshold, the fluxes oscillate, and Γb is far larger than Γc.

Quantitative modeling of neoclassical tearing mode driven fast ion transport in integrated TRANSP simulations

The TRANSP-‘Kick’ energetic particle transport model has been extended to study neoclassical tearing mode (NTM) driven fast ion (FI) transport with zero free parameters. FI transport induced by the NTM is obtained by calculating perturbed FI orbits via the guiding center particle following code orbit in the magnetic field of an experimentally characterized island chain. Transport probabilities are then used in TRANSP’s Monte Carlo module NUBEAM to modify the FI distribution in every time step of the full TRANSP analysis. This procedure retains all TRANSP functionality and self-consistently predicts the NTM impact on beam torque, current drive and heating. Comparisons to DIII-D steady state, hybrid and ITER baseline plasmas are encouraging with the model quantitatively recovering the measured neutron rates. FI effects depend on the location and width of FI phase space resonances which are not accounted for by the former ad hoc beam diffusivity model of TRANSP. Resonance overlaps result in a transport threshold, for the cases investigated herein, at . When the FI transport is dominated by energy and momentum redistribution. In particular, m/n = 3/2 NTMs lead to hollow profile at q = m/n and broadened profile in the core. When FI losses increase, the neutral beam torque () and driven current () decrease across the entire plasma. The effects on NB profiles can strongly depend on the NTM frequency and mode numbers with the 3/2 (2/1) broadening (peaking) near the magnetic axis.

Fractional Resonances between Waves and Energetic Particles in Tokamak Plasmas

From numerical simulation and analytical modeling it is shown that fast ions can resonate with plasma waves at fractional values of the particle drift-orbit transit frequency when the plasma wave amplitude is sufficiently large. The fractional resonances, which are caused by a nonlinear interaction between the particle orbit and the wave, give rise to an increased density of resonances in phase space which reduces the threshold for stochastic transport. The effects of the fractional resonances on spatial and energy transport are illustrated for an energetic particle geodesic acoustic mode but they apply equally well to other types of MHD activity.

Super-diffusion in optical realizations of Anderson localization

We discuss the dynamics of particles in one dimension in potentials that are random in both space and time. The results are applied to recent optics experiments on Anderson localization, in which the transverse spreading of a beam is suppressed by random fluctuations in the refractive index. If the refractive index fluctuates along the direction of the paraxial propagation of the beam, the localization is destroyed. We analyze this broken localization in terms of the spectral decomposition of the potential. When the potential has a discrete spectrum, the spread is controlled by the overlap of Chirikov resonances in phase space. As the number of Fourier components is increased, the resonances merge into a continuum, which is described by a Fokker–Planck equation. We express the diffusion coefficient in terms of the spectral intensity of the potential. For a general class of potentials that are commonly used in optics, the solutions to the Fokker–Planck equation exhibit anomalous diffusion in phase space, implying that when Anderson localization is broken by temporal fluctuations of the potential, the result is transport at a rate similar to a ballistic one or even faster. For a class of potentials which arise in some existing realizations of Anderson localization, atypical behavior is found.

Rotational dynamics of planetary satellites: A survey of regular and chaotic behavior

The problem of observability of chaotic regimes in the rotation of planetary satellites is studied. The analysis is based on the inertial and orbital data available for all satellites discovered up to now. The Lyapunov spectra of the spatial chaotic rotation and the full range of variation of the spin rate are computed numerically by integrating the equations of the rotational motion; the initial data are taken inside the main chaotic layer near the separatrices of synchronous resonance in phase space. The model of a triaxial satellite in a fixed elliptic orbit is adopted. A short Lyapunov time along with a large range of variation of the spin rate are used as criteria for observability of the chaotic motion. Independently, analysis of stability of the synchronous state with respect to tilting the axis of rotation provides a test for the physical opportunity for a satellite to rotate chaotically. Finally, a calculation of the times of despinning due to tidal evolution shows whether a satellite's spin could evolve close to the synchronous state. Apart from Hyperion, already known to rotate chaotically, only Prometheus and Pandora, the 16th and 17th satellites of Saturn, pass all these four tests.

Experimental tests of quantum nonlinear dynamics in atom optics

Cold atoms in optical potentials provide an ideal test bed to explore quantum nonlinear dynamics. Atoms are prepared in a magneto-optic trap or as a dilute Bose-Einstein condensate and subjected to a far detuned optical standing wave that is modulated. They exhibit a wide range of dynamics, some of which can be explained by classical theory while other aspects show the underlying quantum nature of the system. The atoms have a mixed phase space containing regions of regular motion which appear as distinct peaks in the atomic momentum distribution embedded in a sea of chaos. The action of the atoms is of the order of Planck's constant, making quantum effects significant. This tutorial presents a detailed description of experiments measuring the evolution of atoms in time-dependent optical potentials. Experimental methods are developed providing means for the observation and selective loading of regions of regular motion. The dependence of the atomic dynamics on the system parameters is explored and distinct changes in the atomic momentum distribution are observed which are explained by the applicable quantum and classical theory. The observation of a bifurcation sequence is reported and explained using classical perturbation theory. Experimental methods for the accurate control of the momentum of an ensemble of atoms are developed. They use phase space resonances and chaotic transients providing novel ensemble atomic beamsplitters. The divergence between quantum and classical nonlinear dynamics is manifest in the experimental observation of dynamical tunnelling. It involves no potential barrier. However a constant of motion other than energy still forbids classically this quantum allowed motion. Atoms coherently tunnel back and forth between their initial state of oscillatory motion and the state 180° out of phase with the initial state.

Experimental study of the quantum driven pendulum and its classical analog in atom optics

We present experimental results for the dynamics of cold atoms in a far detuned amplitude-modulated optical standing wave. Phase-space resonances constitute distinct peaks in the atomic momentum distribution containing up to 65% of all atoms resulting from a mixed quantum chaotic phase space. We characterize the atomic behavior in classical and quantum regimes and we present the applicable quantum and classical theory, which we have developed and refined. We show experimental proof that the size and the position of the resonances in phase space can be controlled by varying several parameters, such as the modulation frequency, the scaled well depth, the modulation amplitude, and the scaled Planck's constant of the system. We have found a surprising stability against amplitude noise. We present methods to accurately control the momentum of an ensemble of atoms using these phase-space resonances which could be used for efficient phase-space state preparation.

Atoms in an amplitude-modulated standing wave - dynamics and pathways to quantum chaos

Cold rubidium atoms are subjected to an amplitude-modulated far-detuned standing wave of light to form a quantum-driven pendulum. Here we discuss the dynamics of these atoms. Phase space resonances and chaotic transients of the system exhibit dynamics which can be useful in many atom optics applications as they can be utilized as means for phase space state preparation. We explain the occurrence of distinct peaks in the atomic momentum distribution, analyse them in detail and give evidence for the importance of the system for quantum chaos and decoherence studies.

Excitation of Alfvén cyclotron instability by charged fusion products in tokamaks

The spectrum of ion cyclotron emission (ICE) observed in tokamak experiments shows narrow peaks at multiples of the edge cyclotron frequency of background ions. A possible mechanism of ICE based on the fast Alfvén Cyclotron Instability (ACI) resonantly excited by high energy charged products (α-particles or protons) is presented here. Two-dimensional eigenmode analysis of ACI mode structure and eigenfrequency are obtained in the large tokamak aspect ratio limit. The ACI is excited via wave-particle resonances in phase space by tapping the fast ion velocity space free energy. The instability growth rates are computed perturbatively from the perturbed fast particle distribution function, which is obtained by integrating the high frequency gyrokinetic equation along the particle orbit. Numerical examples of ACI growth rates are presented for Tokamak Fusion Test Reactor (TFTR) [World Survey of Activities in Controlled Fusion Research [Nuclear Fusion special supplement 1991] (International Atomic Energy Agency, Vienna, 1991)] supershot plasmas. The fast ion distribution function is assumed to be singular in pitch angle near the plasma edge. The results are employed to understand the ICE in Deuterium-Deuterium (DD) and Deuterium-Tritium (DT) tokamak experiments.

How Robust are Bottlenecks to Unimolecular Fragmentation?

AbstractSome properties of classical mechanical phase space resonances that serve as bottlenecks to the redistribution of internal molecular energy are investigated. Resonances of a two degree of freedom model van der Waals (vdW) molecule are subjected to a linear harmonic perturbation from an additional degree of freedom. The resulting effects on the phase portrait of the subsystem appear to be independent of the frequency of the third oscillator but relatively small coupling critically affects most quasi‐periodic resonances. The approximate separatrix in the vdW degree of freedom is used as a dividing surface in an analysis of the vdW molecule fragmentation rate. The influence on the separatrix of coupling to a third oscillator supports the validity of the two degree of freedom model for systems with small to moderate coupling. The golden mean resonance remains the most robust resonance of the two degree of freedom phase portrait even when the third oscillator coupling is nonzero, an observation which supports the recognition of golden mean resonances as the main intramolecular bottlenecks.