Guiding Center Orbits Research Articles

Guiding-centre orbit-following simulations of charge exchange loss of NBI ions with the finite Larmor radius effect

Guiding-centre orbit-following simulations of the charge exchange (CX) loss of neutral beam injection (NBI) ions are presented. The finite Larmor radius (FLR) effect in the fast ion–neutral collision can be included in guiding-centre orbit-following simulations by using the gyroaverage method. It is proved that the neutralization probability of fast ions computed by using the gyroaverage method in the guiding-centre orbit simulation is roughly the same as that computed in the full-orbit simulation when the time step in the guiding-centre simulation is the order of the gyroperiod. The CX losses of NBI fast ions for two NBIs in the EAST tokamak have been simulated by the guiding-centre orbit-following code GYCAVA, and the FLR effect in the fast ion–neutral collision on CX losses has been numerically studied. The CX effect of the fast ion–neutral collision can significantly enhance NBI ion losses on EAST. The FLR effect in the fast ion–neutral collision can enhance the CX loss. Vertical asymmetry of localized heat loads induced by CX losses is found, which is related to the FLR effect of fast ions and the strong radial gradient of the neutral density near the plasma edge. Heat loads induced by CX losses are localized in the regions near the poloidal angle $\theta =-60^\circ$ , because the likelihood of exchanging charge is the largest at gyrophase $\xi ={\rm \pi}$ , and this leads to fast downwards moving neutrals. Fast ion loss fractions induced by CX increase with the neutral density increasing.

Diffusion–convection model of runaway electrons due to large magnetohydrodynamic perturbations in post-thermal quench plasmas

Systematic test particle tracing simulations for runaway electrons (REs) are performed for six post-thermal quench equilibria from DIII-D and ITER, where large scale, kink-like n = 1 (n is the toroidal mode number) magnetohydrodynamic (MHD) instabilities are found. The modeled particle guiding center orbits allow extraction of the effective diffusion–convection coefficients of REs in the presence of large three-dimensional (3D) perturbations up to 10% of the equilibrium toroidal field. With a fixed spatial distribution of the field perturbation, the RE transport coefficients along the plasma radial coordinate track reasonably well with the surface-averaged perturbation level. A substantial variation in the value of the transport coefficients—by three orders of magnitude in most cases, however, occurs with varying launching location of REs along the plasma radius. Large 3D perturbations almost always lead to comparable diffusion and convection processes, meaning that diffusion alone is insufficient to describe the particle motion. At lower (but still high) level of perturbation, the RE convection is found to be dominant over diffusion. A similar observation is made when the perturbation is too strong. In the presence of large perturbation, the dependence of the RE transport on the particle energy is sensitive to the spatial distribution of the perturbation. Based on numerically obtained RE transport coefficients, an analytic fitting model is proposed to quantify the particle diffusion and convection processes due to large MHD events in post-thermal quench plasmas. The model is shown to reasonably well reproduce the direct test particle tracing results for the RE loss fraction and can, thus, be useful for incorporating into other kinetic RE codes in order to simulate the RE beam evolution in the presence of large 3D perturbations.

GORILLA: Guiding-center ORbit Integration with Local Linearization Approach

GORILLA: Guiding-center ORbit Integration with Local Linearization Approach

Alpha particle confinement metrics based on orbit classification in stellarators

We present orbit classification schemes for use as fast metrics for fusion alpha particle losses implemented in the symplectic guiding-centre code SIMPLE. Two variants respectively based on conservation of the parallel adiabatic invariant, and topology of footprints in Poincaré sections are introduced. Like an existing approach based on the Minkowski fractal dimension, those methods estimate whether a guiding-centre orbit is regular and therefore expected to be confined for infinite time in the collisionless case, or chaotic, which might lead to its loss. Compared with the existing approach, the required orbit tracing time for the novel classifiers is shorter by at least an order of magnitude. This enables massive sampling of orbits across the whole phase space to identify regular and chaotic regions for the purpose stellarator optimization. Based on conservation of the perpendicular invariant, we demonstrate how extended regular regions may act as radial barriers for orbits from the chaotic regions on the radially inboard side. We propose to use a quantified version of this property as a new metric for collisionless fusion alpha losses. As pitch-angle scattering becomes only relevant after alphas have already deposited a significant fraction of their energy, such a metric remains useful also for the case with collisions. This is illustrated by comparison with collisional loss computations. Results are presented for applications to two quasi-isodynamic configurations, a quasi-helical configuration and two quasi-axisymmetric configurations. In addition, the Hamiltonian action-angle formalism is used in quasi-axisymmetric configurations to investigate the overlap of drift-orbit resonances leading to chaos. The respective analysis is performed with the NEO-RT code originally developed for investigation of neoclassical toroidal viscous torque in tokamak plasmas.

Particle and guiding-center orbits in crossed electric and magnetic fields

The problem of the charged-particle motion in crossed electric and magnetic fields is investigated, and the validity of the guiding-center representation is assessed in comparison with the exact particle dynamics. While the magnetic field is considered to be straight and uniform, the (perpendicular) radial electric field is nonuniform. The Hamiltonian guiding-center theory of charged-particle motion is presented for arbitrary radial electric fields, and explicit examples are provided for the case of a linear radial electric field.

Faithful guiding-center orbits in an axisymmetric magnetic field

The problem of the charged-particle motion in an axisymmetric magnetic geometry is used to assess the validity of higher-order Hamiltonian guiding-center theory, which includes higher-order corrections associated with gyrogauge invariance as well as guiding-center polarization induced by magnetic-field non-uniformity. Two axisymmetric magnetic geometries are considered: a magnetic mirror geometry and a simple tokamak geometry. When a magnetically confined charged-particle orbit is regular (i.e., its guiding-center magnetic moment is adiabatically invariant), the guiding-center approximation, which conserves both energy and azimuthal canonical angular momentum, is shown to be faithful to the particle orbit when higher-order corrections are taken into account.

Minimizing separatrix crossings through isoprominence

A simple property of magnetic fields that minimizes bouncing to passing type transitions of guiding center orbits is defined and discussed. This property, called isoprominence, is explored through the framework of a near-axis expansion. It is shown that isoprominent magnetic fields for a toroidal configuration exist to all orders in a formal expansion about a magnetic axis. Some key geometric features of these fields are described.

Effect of ideal internal MHD instabilities on NBI fast ion redistribution in ITER 15 MA scenario

Transport of fast ions is a crucial issue during the operation of ITER. Redistribution of neutral beam injection (NBI) fast ions by the ideal internal magnetohydrodynamic (MHD) instabilities in ITER is studied utilizing the guiding-center code ORBIT (White R B and Chance M S 1984 Phys. Fluids 27 2455). Effects of the perturbation amplitude of the internal kink, the perturbation frequency of the fishbone instability, and the toroidal mode number of the internal kink are investigated, respectively, in this work. The internal kink mode can cause NBI fast ions transporting in real space from regions of to where labels the normalized plasma radial coordinate. The transport of fast ions is greater as the perturbation amplitude increases. The maximum relative change of the number of fast ions approaches 5% when the perturbation amplitude rises to 500 G. A strong transport is generated between the regions of and in the presence of the fishbone instability. Higher frequency results in greater transport, and the number of fast ions in is reduced by 30% at the fishbone frequency of 100 kHz. Perturbations with higher will lead to the excursion of fast ion transport regions outward along the radial direction. The loss of fast ions, however, is not affected by the internal MHD perturbation. Strong transport from to does not influence the plasma heating power of ITER, since the NBI fast ions are still located in the plasma core. On the other hand, the influence of fast ion transport from to needs further study.

Testing the conservative character of particle simulations: I. Canonical and noncanonical guiding center model in Boozer coordinates

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating terms containing the covariant component BΨP of the magnetic field vector with respect to the poloidal flux function ΨP. In an unperturbed plasma, BΨP can be eliminated via exact coordinate transformations, but, in general, one relies on approximations, assuming that the effect of BΨP is small. Here, we are interested in the question whether Hamiltonian conservation laws are still satisfied when BΨP is retained in the presence of fluctuations. Considering fast ions in the presence of a shear Alfvén wave field with fixed amplitude, fixed frequency, and a single toroidal mode number n, we show that simulations using the code ORBIT with and without BΨP yield practically the same resonant and nonresonant GC orbits. The numerical results are consistent with theoretical analyses (presented in the appendix), which show that the unabridged GC Lagrangian with BΨP retained yields equations of motion that possess two key properties of Hamiltonian flows: (i) phase space conservation and (ii) energy conservation. As counter-examples, we also show cases where energy conservation (ii) or both conservation laws (i) and (ii) are broken by omitting certain small terms. When testing the conservative character of the simulation code, it is found to be beneficial to apply perturbations that do not resemble normal (eigen) modes of the plasma. The deviations are enhanced and, thus, more easily spotted when one inspects wave-particle interactions using nonnormal modes.

Guiding center orbit simulation of energetic particles in tokamak plasma

Guiding center orbit simulation of energetic particles in tokamak plasma

Solenoid-free current drive via ECRH in EXL-50 spherical torus plasmas

As a new spherical tokamak designed to simplify the engineering requirements of a possible future fusion power source, the EXL-50 experiment features a low aspect ratio vacuum vessel (VV), encircling a central post assembly containing the toroidal field coil conductors without a central solenoid. Multiple electron cyclotron resonance heating (ECRH) resonances are located within the VV to improve current drive effectiveness. Copious energetic electrons are produced and measured with hard x-ray detectors, carry the bulk of the plasma current ranging from 50–150 kA, which is maintained for more than 1 s duration. It is observed that over one ampere current can be sustained per watt of ECRH power issued from the 28 GHz gyrotrons. The plasma current reaches I p > 80 kA for high density (>5 × 1018 m−2) discharge with 150 kW ECRH. An analysis was carried out combining reconstructed multi-fluid equilibrium, guiding-center orbits of energetic electrons, and resonant heating mechanisms. It is verified that in EXL-50 a broadly distributed current of energetic electrons creates a smaller closed magnetic-flux surface of low aspect ratio that in turn confines the thermal plasma electrons and ions and participate in maintaining the equilibrium force balance.

Representation and modeling of charged particle distributions in tokamaks

Experimental diagnostics, analysis tools and simulations represent particle distributions in various forms and coordinates. Algorithms to manage these data are needed on platforms like the ITER Integrated Modeling & Analysis Suite (IMAS), performing tasks such as archiving, modeling, conversion and visualization. A method that accomplishes some of the required tasks for distributions of charged particles with arbitrarily large magnetic drifts in axisymmetric tokamak geometry is described here. Given a magnetic configuration, we first construct a database of guiding center orbits, which serves as a basis for representing particle distributions. The orbit database contains the geometric information needed to perform conversions between arbitrary coordinates, modeling tasks, and resonance analyses. Using that database, an imported or newly modeled distribution is mapped to an exact equilibrium, where the dimensionality is reduced to three constants of motion (CoM). The orbit weight is uniquely given when the input is a true distribution: one that measures the number of physical particles per unit of phase space volume. Less ideal inputs, such as distributions estimated without drifts, or models of particle sources, can also be processed. As an application example, we reconstruct the drift-induced features of a distribution of fusion-born alpha particles in a large tokamak, given only a birth profile, which is not a function of the alpha's CoM. Repeated back-and-forth transformations between CoM space and energy-pitch-cylinder coordinates are performed for verification and as a proof of principle for IMAS.

Numerical study on the loss of fast ions produced by minority ion cyclotron resonance heating in EAST

The classical prompt loss of fast ions produced by minority ion cyclotron resonance heating (ICRH) is studied by a guiding center orbit following code in the Experimental Advanced Superconducting Tokamak (EAST). It is found that the loss of fast ions produced by ICRH mainly appears in both ends of the resonance layer, while the loss of fast ions in the middle resonance layer is very small. The dominant fast loss comes from trapped ions, rather than from passing ions. Controlling the location of resonance layer at the plasma core may be more beneficial to the EAST tokamak ICRH. In addition, the loss distribution of fast ions is studied. The results show that the fast ions are mainly lost near the midplane in the poloidal direction, but almost uniformly in the toroidal direction. Moreover, we investigate the dependence of fast ion loss on the ICRH power. The simulation results show that the loss fraction of fast ions in both ends of the resonance region increases with the ion cyclotron range of frequencies (ICRF) power, but barely affects the loss of fast ions in the middle region.

Feasibility study of fast ion loss diagnostics for CFQS by beam ion loss calculation on vacuum vessel

A feasibility study for installing fast-ion loss diagnostics (FILDs) is initiated to understand energetic ion loss process in a quasi-axisymmetric stellarator CFQS. The calculations of beam ion orbit, the guiding center orbit code including the slowing down by bulk plasma in the Boozer coordinates and the collisionless Lorentz orbit code in the Cartesian coordinates, are performed to estimate the toroidal/poloidal distribution of beam ion losses on a vacuum vessel. We observed that loss points on the vacuum vessel are mainly located at the upper side of torus according to gradient magnetic field drift direction. Based on the loss distribution on the vacuum vessel, a possible location of FILD for CFQS is investigated. Although the energy of lost beam ions is ∼10 keV, relatively high-energy (> 20 keV) components can be measured on the upper side of the torus in the low-density case. Furthermore, the radial or vertical dependence of beam ion loss flux at candidate movable scintillator-type FILD positions is obtained. A considerable number of particles reach the scintillator-type FILD position. The energy and pitch angle distribution of beam ions reaching the FILD position shows that scintillator-type FILDs can measure co-going transit beam ions with energy of ∼25 keV and pitch angle of ∼140 degrees and transition beam ions with energy of 10–15 keV and pitch angle of ∼80–100 degrees.

Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields

A numerical integration method for guiding-center orbits of charged particles in toroidal fusion devices with three-dimensional field geometry is described. Here, high order interpolation of electromagnetic fields in space is replaced by a special linear interpolation, leading to locally linear Hamiltonian equations of motion with piecewise constant coefficients. This approach reduces computational effort and noise sensitivity, while the conservation of total energy, magnetic moment and phase space volume is retained. The underlying formulation treats motion in piecewise linear fields exactly and, thus, preserves the non-canonical symplectic form. The algorithm itself is only quasi-geometric due to a series expansion in the orbit parameter. For practical purposes, an expansion to the fourth order retains geometric properties down to computer accuracy in typical examples. When applied to collisionless guiding-center orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator configuration, the method demonstrates stable long-term orbit dynamics conserving invariants. In Monte Carlo evaluation of transport coefficients, the computational efficiency of quasi-geometric integration is an order of magnitude higher than with a standard fourth order Runge–Kutta integrator.

Observation of neutron emission anisotropy by neutron activation measurement in beam-injected LHD deuterium plasmas

Neutron emission anisotropy caused by neutral beam injection was observed by the neutron activation measurement in deuterium plasmas confined in the Large Helical Device. The shot-integrated fast neutron flux was measured by the irradiation of activation foils at two irradiation ends located at different ports. The ratio of the flux at the outside horizontal port to that at the lower port was used as an index of neutron emission anisotropy. The dependence of the ratio on the direction of the neutral beam injection was confirmed by comparing cases of tangential, perpendicular and both tangential and perpendicular neutral beam injections. Neutron emission anisotropy was numerically evaluated assuming these three cases of neutral beam injection. The analysis of neutron emission anisotropy comprises the evaluation of the velocity distribution function of energetic deuterons following guiding-centre orbits, calculation of the double-differential emission spectrum of neutrons produced by the reaction and neutron transport calculation. The obtained numerical results are qualitatively consistent in the dependence of neutron emission anisotropy on the neutral-beam-injection direction with the observed experimental data.

Statistical description of collisionless α-particle transport in cases of broken symmetry: from ITER to quasi-toroidally symmetric stellarators

The confinement of α-particles is vital for any future fusion reactor. Unfortunately, the inevitable appearance of inhomogeneities in the magnetic field activates the non-collisional transport by virtue of ripple trapping and ripple induced stochastization. While a large and growing body of literature is devoted to the mitigation of these channels of losses for future reactor designs, far too little attention has been paid to characterize the statistical nature of the underlying stochastic process, which is generally assumed to be diffusive. Here the effect of breaking the toroidal symmetry on collisionless α-particle transport is analyzed numerically with a guiding center orbit following code MOCA for several configurations: a perfectly toroidally symmetric ITER-like tokamak and four stellarators with different levels of quasi-toroidal symmetry. Statistical characterization of banana widths, bouncing times and banana center evolution put into question the classical convection/diffusion approach to adequately describe collisionless α-particle transport as the magnetic configuration departs from toroidal symmetry.

Accelerated methods for direct computation of fusion alpha particle losses within, stellarator optimization

Accelerated statistical computation of collisionless fusion alpha particle losses in stellarator configurations is presented based on direct guiding-centre orbit tracing. The approach relies on the combination of recently developed symplectic integrators in canonicalized magnetic flux coordinates and early classification into regular and chaotic orbit types. Only chaotic orbits have to be traced up to the end, as their behaviour is unpredictable. An implementation of this technique is provided in the code SIMPLE (symplectic integration methods for particle loss estimation, Albert et al., 2020b, doi:10.5281/zenodo.3666820). Reliable results were obtained for an ensemble of 1000 orbits in a quasi-isodynamic, a quasi-helical and a quasi-axisymmetric configuration. Overall, a computational speed up of approximately one order of magnitude is achieved compared to direct integration via adaptive Runge–Kutta methods. This reduces run times to the range of typical magnetic equilibrium computations and makes direct alpha particle loss computation adequate for use within a stellarator optimization loop.

Numerical simulation of fast ions guiding-center orbits in mega ampere spherical tokamak-reactor configuration

Determining the charged particle trajectories in the inhomogeneous electric and magnetic fields is the common problem in the plasma physics. For the tokamak plasma, the orbits of the captured fast ions are affected both by the curvature and gradient of the magnetic field. In the present work, the fast ion trajectories and their guiding-centre orbits in a mega ampere spherical tokamak reactor are found with employing the numerical code, that was elaborated by our team, together with the leap-frog and Runge-Kutta techniques. The magnetic field is calculated by solving the Grad-Shafranov equation at the fixed boundaries on the non-homogeneous mesh grid. The obtained results show that the ions with the energies in the range 20 keV - 80 keV can describe the typical banana orbits as well as the passing trajectories in the poloidal plane. Finally, the numerical simulation scheme stability is tested.

Numerical study of neutral beam injection prompt loss at KSTAR poloidal limiters

We use a newly developed simulation tool to numerically study fast-ion loading on plasma-facing components (PFCs) at the Korea Superconducting Tokamak Advanced Research facility in the high-poloidal-β plasma operation regime. The new code can calculate neutral beam ionization and follow the guiding center orbit of ionized particles. The results of the simulation indicate that fast ions ionized in the high-field side drift out and strike the PFCs as they rotate poloidally. Momentum projection onto a phase space defined by canonical toroidal angular momentum and magnetic moment leads to a simple criterion to avoid fast-ion loading on poloidal limiters (PLs), which are PFCs that are toroidally localized on the low-field side. Control of fast-ion loss is examined by varying the plasma current and plasma boundary. A larger plasma current and inner shifting of the outer plasma boundary is predicted to substantially reduce fast-ion loading on the PLs. The proposed phase-space criterion is qualitatively consistent with the results of the simulation of fast-ion control.