Phase Space Transport Research Articles

Application of non-axisymmetric magnetic field for control of Alfvén eigenmodes in KSTAR

We report an experimental examination of non-axisymmetric (3D) magnetic fields for the control of Alfvén eigenmodes (AEs) in KSTAR. Application of the phase-sweeping n = 1 3D magnetic field identifies the effective 3D field phase and threshold amplitude for suppression of toroidal AEs. Such observations indicate that at least two conditions on the 3D field phase and amplitude should be satisfied for the AE suppression. The phase window of AE suppression is largely resonant and thereby overlapped with that of mode locking, while the threshold of mode locking is slightly higher than that of AE suppression, which implies a narrow 3D configuration window for AE suppression. Numerical analyses on the AE stability and fast ion phase-space transport suggest that the key mechanism of the AE suppression is the reduction of the AE drive through redistribution of fast ion phase-space distribution by strong resonant interactions of the fast ions with the 3D magnetic field.

Effects of multi-dimensionality and energy exchange on electrostatic current-driven plasma instabilities and turbulence

Large-amplitude current-driven plasma instabilities, which can transition to the Buneman instability, were observed in one-dimensional simulations to generate high-energy back-streaming ions. We investigate the saturation of multi-dimensional plasma instabilities and its effects on energetic ion formation. Such ions directly impact spacecraft thruster lifetimes and are associated with magnetic reconnection and cosmic ray inception. An Eulerian Vlasov–Poisson solver employing the grid-based direct kinetic method is used to study the growth and saturation of 2D2V collisionless, electrostatic current-driven instabilities spanning two dimensions each in the configuration (D) and velocity (V) spaces supporting ion and electron phase-space transport. Four stages characterise the electric potential evolution in such instabilities: linear modal growth, harmonic growth, accelerated growth via quasi-linear mechanisms alongside nonlinear fill-in and saturated turbulence. Its transition and isotropisation process bears considerable similarities to the development of hydrodynamic turbulence. While a tendency to isotropy is observed in the plasma waves, followed by electron and then ion phase spaces after several ion-acoustic periods, the formation of energetic back-streaming ions is more limited in the 2D2V than in the 1D1V simulations. Plasma waves formed by two-dimensional electrostatic kinetic instabilities can propagate in the direction perpendicular to the net electron drift. Thus, large-amplitude multi-dimensional waves generate high-energy transverse-streaming ions and eventually limit energetic backward-streaming ions along the longitudinal direction. The multi-dimensional study sheds light on interactions between longitudinal and transverse electrostatic plasma instabilities, as well as fundamental characteristics of the inception and sustenance of unmagnetised plasma turbulence.

Nonlinear equilibria and transport processes in burning plasmas

In this work, we put forward a general phase space transport theory in axisymmetric tokamak plasmas based upon the concept of zonal state (ZS). Within this theoretical framework, the ZS corresponds to a renormalized plasma nonlinear equilibrium consisting of phase space zonal structures (PSZS) and zonal electromagnetic fields (ZFs) which evolve self-consistently with symmetry breaking fluctuations and sources/collisions. More specifically, our approach involves deriving governing equations for the evolution of particle distribution functions (i.e, PSZS), which can be used to compute the corresponding macro-/meso-scale evolving magnetized plasma equilibrium adopting the Chew Goldberger Low description, separating the spatiotemporal microscale structures. The nonlinear physics of ZFs and of geodesic acoustic modes (GAMs)/energetic particle driven GAMs is then analyzed to illustrate the applications of our theory.

Navigating phase space transport with the origin-fate map.

We introduce and demonstrate the usage of the origin-fate map (OFM) as a tool for the detailed investigation of phase space transport in reactant-product-type systems. For these systems, which exhibit clearly defined start and end states, it is possible to build a comprehensive picture of the lobe dynamics by considering backward and forward integration of sets of initial conditions to index their origin and fate. We illustrate the method and its utility in the study of a two degrees of freedom caldera potential with four exits, demonstrating that the OFM not only recapitulates results from classical manifold theory but even provides more detailed information about complex lobe structures. The OFM allows the detection of dynamically significant transitions caused by the creation of new lobes and is also able to guide the prediction of the position of unstable periodic orbits (UPOs). Further, we compute the OFM on the periodic orbit dividing surface (PODS) associated with the transition state of a caldera entrance, which allows for a powerful analysis of reactive trajectories. The intersection of the manifolds corresponding to this UPO with other manifolds in the phase space results in the appearance of lobes on the PODS, which are directly classified by the OFM. This allows computations of branching ratios and the exploration of a fractal cascade of lobes as the caldera is stretched, which results in fluctuations in the branching ratio and chaotic selectivity. The OFM is found to be a simple and very useful tool with a vast range of descriptive and quantitative applications.

Visualization of fast ion phase-space flow in plasmas well-below, near and well-above Alfvén eigenmode stability threshold in tokamak

Transport of fast ions along certain local phase space paths, referred to as fast ion phase-space flow, has been systematically measured by an imaging neutral particle analyzer (INPA) in three plasma regimes, which are well-below, near, and well-above the Alfvén eigenmode stability threshold (Du et al 2021 Phys. Rev. Lett. 127 235002). (1) In plasmas well-below the Alfvén eignenmodes (AE) stability threshold, fast ions are well-confined on passing particle orbits without noticeable transport over the phase space. The observed INPA images agree well with the synthetic INPA images, using the fast ion distribution predicted by neoclassical theory. (2) In plasmas near the AE stability threshold, INPA images in the presence of AE activity moderately deviate from those without AE activity. The image difference can be well interpreted by AE-driven, phase-space fast ion flow. Paths of this flow over the velocity space (or streamlines) are reconstructed by the intersection lines of curved Eʹ and µ surfaces, referred to as Eʹ and µ line (where ; E, P ζ and µ are the energy, canonical toroidal momentum and magnetic moment of ions; ω and n are the angular frequency and toroidal mode number of AEs, respectively). Resonant fast ions move radially inward by gaining energy and move radially outward by losing energy and the trajectory well aligns with the Eʹ and µ lines that pass through the mode resonances near the injection energy of neutral beams. (3) In plasmas well-above the AE stability threshold, fast ion phase-space dynamics shows additional features. Fast ions are transported out of the birth positions so promptly along the streamlines that the slowing-down process from the injection energy is not observable, exhibiting strong critical gradient behavior at local phase space. As a result, the increase of electron temperature is very small, in spite of an increase of beam power by . It should be emphasized that the directions of phase-space transport, induced by AEs with different frequencies, structures and mode numbers, do not largely differ.

Hybrid-kinetic simulation of synergy between fishbone/sawtooth and tearing mode-induced energetic-ion transport in a tokamak plasma

The resonant interaction between tearing mode (TM) and energetic ions redistributed by fishbone/sawtooth collapse has been observed recently in HL-2A neutral beam injection-heated high-density plasmas, where m is the poloidal mode number and n is the toroidal mode number. This event essentially results from a synergistic effect with a multiple-mode nature. Understanding such experimental phenomena, especially when using high-power auxiliary heating, is crucial for achieving high-performance plasmas with high confinement and high β N in the prospective thermonuclear fusion reactors. In this work, based on the typical experimental phenomenon of synergy between fishbone/sawtooth and TM on HL-2A, a self-consistent nonlinear hybrid-kinetic simulation is conducted utilizing the experimental parameters and profiles before the occurrence of this synergy as the code M3D-K input. In particular, we focus chiefly on the phase-space transport, loss and redistribution of energetic ions induced by two cases, one with only TM and one with synergy between fishbone/sawtooth and TM. It is found that the synergy of the fishbone/sawtooth and TM on energetic-ion transport, loss and redistribution is important and can dramatically enhance the loss rate of energetic ions compared with the case with only TM. It is due to the fact that such synergy can open up an efficient energetic-ion loss channel which connects the plasma center and edge. The phase space transport and redistribution of energetic ions induced by the two different cases are also respectively analyzed and compared in detail. Because of the overestimation of amplitudes of fishbone/sawtooth and TM in the simulations, the obtained energetic-ion transport level is only limited in the context of the numerical framework. These findings indicate that the synergy triggering the sawtooth collapse may lead to an enhancement of the energetic-ion transport and can help us understand the underlying mechanisms of synergy-induced energetic-ion transport.

Saddle transport and chaos in the double pendulum

Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. In this work, we explore the existence and implications of codimension-1 invariant manifolds in the double pendulum, which originate from unstable periodic orbits around saddle equilibria and act as separatrices that mediate the global phase space transport. Motivated in part by similar studies on the three-body problem, we are able to draw a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic, saddle-mediated transport, with direct relevance to energy-efficient space mission design. The analytical results of this work provide an existence result, concerning arbitrarily long itineraries in phase space, that is applicable to a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum. This manuscript details a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting.

Global dynamics visualisation from Lagrangian Descriptors. Applications to discrete and continuous systems

This paper introduces a new global dynamics and chaos indicator based on the method of Lagrangian Descriptor apt for discriminating ordered and deterministic chaotic motions in multidimensional systems. The selected implementation of this method requires only the knowledge of orbits on finite time windows and is free of the computation of the tangent vector dynamics (i.e.,variational equations are not needed). To demonstrate its ability in visualising different dynamical behaviours, in particular for highlighting chaotic regions, several stability maps of classical systems, obtained with different phase space methods, are reproduced. The benchmark examples are rooted in discrete and continuous nearly-integrable dynamical systems, with prominent features played by resonances. These include the Chirikov standard map, higher dimensional symplectic and volume preserving maps, fundamental models of resonances, and a 3 degrees-of-freedom nearly-integrable Hamiltonian system with a dense web of resonances. The indicator thus appears to be relevant for understanding phase space transport mediated by resonances in nearly-integrable system, as ubiquitous in celestial mechanics or astrodynamics.

Phase Space Transport in a Symmetric Caldera Potential with Three Index-1 Saddles and No Minima

We apply the method of Lagrangian Descriptors (LDs) to a symmetric Caldera-type potential energy surface which has three index-1 saddles surrounding a relatively flat region that contains no minimum. Using this method we show the phase space transport mechanism that is responsible for the existence and nonexistence of the phenomenon of dynamical matching for this form of Caldera potential energy surface.

Electron Dynamics in a Chorus Wave Field Generated From Particle‐In‐Cell Simulations

AbstractHow to properly describe resonant interactions between electrons and quasi‐coherent chorus waves is still an open question. Previous studies have progressed from modeling chorus as a single wave to considering effects such as amplitude modulation or phase decoherence in wave particle resonance. However, incorporating realistic features of chorus waves in test particle calculations has always been a challenging but critically important step to evaluate their nonlinear effects. In this work, we use a rising‐tone chorus element generated by a particle‐in‐cell simulation in test‐particle simulations. The used chorus wave naturally has features including amplitude modulation, phase decoherence and dynamic evolution during propagation. Our results suggest that, while being latitudinal dependent, realistic features of chorus generally lead to significant suppression of nonlinear effects. This result should be important to understand phase space transport of electrons due to interactions with chorus waves.

Visualization of Fast Ion Phase-Space Flow Driven by Alfvén Instabilities.

Fast ion phase-space flow, driven by Alfvén eigenmodes (AEs), is measured by an imaging neutral particle analyzer in the DIII-D tokamak. The flow firstly appears near the minimum safety factor at the injection energy of neutral beams, and then moves radially inward and outward by gaining and losing energy, respectively. The flow trajectories in phase space align well with the intersection lines of the constant magnetic moment surfaces and constant E-(ω/n)P_{ζ} surfaces, where E, P_{ζ} are the energy and canonical toroidal momentum of ions; ω and n are angular frequencies and toroidal mode numbers of AEs. It is found that the flow is so destructive that the thermalization of fast ions is no longer observed in regions of strong interaction. The measured phase-space flow is consistent with nonlinear hybrid kinetic-magnetohydrodynamics simulation. Calculations of the relatively narrow phase-space islands reveal that fast ions must transition between different flow trajectories to experience large-scale phase-space transport.

Nonlinear dynamics and phase space transport by chorus emission

Chorus emission in planetary magnetospheres is taken as working paradigm to motivate a short tutorial trip through theoretical plasma physics methods and their applications. Starting from basic linear theory, readers are first made comfortable with whistler wave packets and their propagation in slowly varying weakly non-uniform media, such as the Earth’s magnetosphere, where they can be amplified by a population of supra-thermal electrons. The nonlinear dynamic description of energetic electrons in the phase space in the presence of self-consistently evolving whistler fluctuation spectrum is progressively introduced by addressing renormalization of the electron response and spectrum evolution equations. Analytical and numerical results on chorus frequency chirping are obtained and compared with existing observations and particle-in-cell simulations. Finally, the general theoretical framework constructed during this short trip through chorus physics is used to draw analogies with condensed matter and laser physics as well as magnetic confinement fusion research. Discussing these analogies ultimately presents plasma physics as an exciting cross-disciplinary field to study.

Support vector machines for learning reactive islands.

We develop a machine learning framework that can be applied to data sets derived from the trajectories of Hamilton's equations. The goal is to learn the phase space structures that play the governing role for phase space transport relevant to particular applications. Our focus is on learning reactive islands in two degrees-of-freedom Hamiltonian systems. Reactive islands are constructed from the stable and unstable manifolds of unstable periodic orbits and play the role of quantifying transition dynamics. We show that the support vector machines are an appropriate machine learning framework for this purpose as it provides an approach for finding the boundaries between qualitatively distinct dynamical behaviors, which is in the spirit of the phase space transport framework. We show how our method allows us to find reactive islands directly in the sense that we do not have to first compute unstable periodic orbits and their stable and unstable manifolds. We apply our approach to the Hénon-Heiles Hamiltonian system, which is a benchmark system in the dynamical systems community. We discuss different sampling and learning approaches and their advantages and disadvantages.

Phase space transport in the interaction between shocks and plasma turbulence

The interaction of collisionless shocks with fully developed plasma turbulence is numerically investigated. Hybrid kinetic simulations, where a turbulent jet is slammed against an oblique shock, are employed to address the role of upstream turbulence on plasma transport. A technique, using coarse graining of the Vlasov equation, is proposed, showing that the particle transport strongly depends on upstream turbulence properties, such as strength and coherency. These results might be relevant for the understanding of acceleration and heating processes in space plasmas.

Sticky islands in stochastic webs and anomalous chaotic cross-field particle transport by E×B electron drift instability

The ExB electron drift instability, present in many plasma devices, is an important agent in cross-field particle transport. In presence of a resulting low frequency electrostatic wave, the motion of a charged particle becomes chaotic and generates a stochastic web in phase space. We define a scaling exponent to characterise transport in phase space and we show that the transport is anomalous, of super-diffusive type. Given the values of the model parameters, the trajectories stick to different kinds of islands in phase space, and their different sticking time power-law statistics generate successive regimes of the super-diffusive transport.

Nonlinear radial envelope evolution equations and energetic particle transport in tokamak plasmas

This work provides a general description of the self-consistent energetic particle phase space transport in burning plasmas, based on nonlinear gyrokinetic theory. The self consistency is ensured by the evolution equations of the Alfvénic fluctuations by means of nonlinear radial envelope evolution equations, while energetic particle fluxes in the phase space are explicitly constructed from long-lived phase space zonal structures, which are undamped by collisionless processes. As a result, this work provides a viable route to computing fluctuation induced energetic particle transport on long time scales in realistic tokamak plasmas.

Study of the Electron Distribution Function Downstream of the Solar Wind Termination Shock: A Touch with the Interstellar Medium

We theoretically describe the evolution of the solar wind electron distribution function downstream of the termination shock under the influence of electron impact ionizations of interstellar H- and He- atoms that enter the heliosheath from the upwind hemisphere and continue to move along the heliotail region. We start from a kinetic phasespace transport equation in the bulk frame of the heliosheath plasma flow that takes into account convective changes, cooling processes, whistler wave-induced energy diffusion, and electron injection into and removal from velocity-space cells due to electron impact ionization processes of interstellar neutral atoms. From this kinetic equation we can ascend to an associated pressure moment of the electron distribution function and there with arrive at a so-called pressure transport equation describing the evolution of the electron pressure in the bulk-velocity frame of the plasma flow. Assuming that the local electron distribution can be represented by a kappa function with a K- parameter that varies with the streamline coordinate s, we obtain an ordinary differential equation for K as function of s. With this result we first gain the heliosheath electron distribution function downstream of the termination shock, and at second, obtain a newly based estimate of the ionization probability of interstellar neutral atoms like H and He at the passage over the heliosheath. The latter information will enable us to quantitatively predict the interstellar inflow of neutral He- and H- atoms into the heliosheath. As we shall show the effect of H- and He impact ionizations especially gives its signature to the electron distribution along the extended down-tail region of the heliosheath. This is why we especially study this 100AU- extended down-tail region here in this article.

Constitutive relations for polar continua based on statistical mechanics and spatial averaging

The purpose of the current work is the formulation of macroscopic constitutive relations, and in particular continuum flux densities, for polar continua from the underlying mass point dynamics. To this end, generic microscopic continuum field and balance relations are derived from phase space transport relations for expectation values of point fields related to additive mass point quantities. Given these, microscopic energy, linear momentum and angular momentum, balance relations are obtained in the context of the split of system forces into non-conservative and conservative parts. In addition, divergence–flux relations are formulated for the conservative part of microscopic supply-rate densities. For the case of angular momentum, two such relations are obtained. One of these is force-based, and the other is torque-based. With the help of physical and material theoretic restrictions (e.g. material frame-indifference), reduced forms of the conservative flux densities are obtained. In the last part of the work, formulation of macroscopic constitutive relations from their microscopic counterparts is investigated in the context of different spatial averaging approaches. In particular, these include (weighted) volume-averaging based on a localization function, surface averaging of normal flux densities based on Cauchy flux theory and volume averaging with respect to centre of mass.

Time series analysis for multi-dimensional dynamical systems combining wavelet transformation and local principal component analysis

Abstract To analyze trajectories for systems of many degrees of freedom, we propose a new method called wavelet local principal component analysis (WlPCA) combining the wavelet transformation and local PCA in time. Our method enables us to reduce the dimensionality of time series both in degrees of freedom and frequency so that characteristic features of oscillatory behavior can be captured. To test the new method, we apply WlPCA to a non-autonomous model of multiple degrees of freedom, the Froeschlé maps of $N=2$ and $N=4$, which correspond to autonomous systems of three and five degrees of freedom, respectively. The eigenvalues and eigenvectors obtained by WlPCA reveal those times when frequency variation exhibits switching between relatively stationary features. Moreover, further analyses indicate which degrees of freedom and frequencies are involved in the switching. We confirm that the switching corresponds to the onset of transport in phase space. These findings imply that, even for systems of larger degrees of freedom, barriers can exist in phase space that block transport for a finite time, thereby dividing the phase space into multiple quasi-stationary regions. Thus, our method is promising for understanding dynamics in systems of many degrees of freedom, such as vibrational-energy redistribution in molecules.

Relativistic Wigner Function for Quantum Walks

A relativistic Wigner function for free Discrete Time Quantum Walks (DTQWs) on the square 2D space-time lattice is defined. The transport equation obeyed by the relativistic Wigner function is derived in terms of discrete derivatives and degenerates in the continuum limit into the transport equation obeyed by the Wigner function of 2D Dirac fermions. The first corrections to the continuous equation induced by the discreteness of the lattice are also determined and the similarities with non-quantum relativistic transport in phase space are discussed further.