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VENUS-LEVIS and its spline-Fourier interpolation of 3D toroidal magnetic field representation for guiding-centre and full-orbit simulations of charged energetic particles
... The process, introduced by Cheng and Knorr and solved via the time-splitting method [18][19][20], is employed. By following the concept of the splitting method, the formal implicit solutions for equations (24) and (26) can be derived: ...
... 1. Perform a half-time step along the x-axis ( , , + finite difference methods [22,22], spline interpolation [23,24], and others. In this study, we adopted the fluid flux balance method proposed by Eric Fijalkow to address the ionelectron dynamics equation in plasma discharge [25]. ...
Considering that the regulating physical mechanism of the magnetic field on electrolytic plasma is insufficient, this study introduces a plasma discharge model to demonstrate its effect on material removal in magnetic-electrolytic plasma polishing. Firstly, we establish a mathematical model to describe plasma discharge induced by electron collision. The Boltzmann kinematic equations are employed to elucidate the physical dynamics of electrons and their momentum distribution function during ionization collisions. Secondly, we derive the surface material removal equation based on the instantaneous high-temperature melting of plasma discharge and establish the corresponding boundary conditions for simulation calculations. Subsequently, we conduct electromagnetic plasma polishing experiments under varying magnetic field intensities using TA1 titanium alloy as the test material. Our simulation results demonstrate that the magnetic field enhances the spatial electric field formed by the uneven distribution of ion electrons, expediting the plasma discharge process towards the anode and generating interference currents to counterbalance the required material removal current. Experimental data is provided to validate the model, consistently revealing a positive correlation between circuit current and material removal rate. The research demonstrates the rationality and reliability of the material removal model for plasma discharge, offering a fundamental understanding of electromagnetic plasma polishing.
... Although most of the heating in a burning plasma should be provided by fusion-born alpha particles, auxiliary heating will still be of importance before the plasma ignition is achieved and also later for discharge control. In our project, we employ the SCENIC framework [48], which describes ICRH plasma heating and energetic-ion generation via a combination of three codes: LEMan [49] solving the hot-plasma wave-propagation equations, VENUS-LEVIS [50] tracing energetic-particle orbits and using Monte Carlo kick operators [51] for collisions and interaction with the wave field, and ANIMEC [52] solving for a plasma equilibrium with an anisotropic energetic-ion pressure tensor. The role of SCENIC in the TSVV Task 10 is to provide an energetic-particle distribution function needed as input by the stability codes. ...
... In stellarator plasmas, numerical tools for computing ICRH heating and the resulting energetic-particle distribution function have been further developed. The SCENIC code package [48] integrates the ANIMEC [52] (3D MHD equilibrium solver), LEMan [49] (linear full-wave global solver), and the fast-particle code VENUS-LEVIS [50]. Predictive simulations of radio-frequency heating and fast-ion generation have been carried out [86,48,87] in Wendelstein 7-X using the SCENIC framework. ...
The software stack under development within a European coordinated effort on tools for burning plasma modelling is presented. The project is organised as a Task (TSVV Task 10) under the new E-TASC initiative (X. Litaudon et al, Plasma Phys. Contr. Fusion, 64 , 034005 (2022)). This is a continued effort within the EUROfusion inheriting from the earlier European coordination projects as well as research projects based at various European laboratories. The ongoing work of the TSVV Tasks is supported by the Advanced Computing Hubs. Major projects requiring the High Perfromance Computing (HPC) resources are global gyrokinetic codes and global hybrid particle-Magnetohydrodynamics (MHD) codes. Also applications using the integrated modeling tools, such as the Energetic-Particle Workflow, based on the ITER Integrated Modelling & Analysis Suite (IMAS), or the code package for modelling radio-frequency heating and fast-ion generation may require intensive computation and a substantial memory footprint. The continual development of these codes both on the physics side and on the HPC side allows us to tackle frontier problems, such as the interaction of turbulence with MHD-type modes in the presence of fast particles. One of the important mandated outcomes of the E-TASC project is the IMAS-enabling of EUROfusion codes and release of the software stack to the EUROfusion community.
... Resistive effects would however soften the sharp transition in generating magnetic islands and a different approach for generating the equilibrium should be considered for the resonant q case. The smooth representation of the abrupt transition between helical core and axisymmetric mantle was greatly facilitated by the Spline-Fourier interpolation scheme [15]. ...
... which yields Euler-Lagrange equation that coincides with (15) b ...
To identify under what conditions guiding-centre or full-orbit tracing should
be used, an estimation of the spatial variation of the magnetic field is
proposed, not only taking into account gradient and curvature terms but also
the local shear of the field-lines. The criterion is derived for general
three-dimensional magnetic equilibria including stellarator plasmas. Details
are provided on how to implement it in cylindrical coordinates and in flux
coordinates that rely on the geometric toroidal angle. A means of switching
between guiding-centre and full-orbit equations at first order in Larmor radius
with minimal discrepancy is shown. Techniques are applied to a MAST (Mega Amp
Spherical Tokamak) helical core equilibrium in which the inner kinked
flux-surfaces are tightly compressed against the outer axisymmetric mantle and
where the parallel current peaks at the nearly rational surface. This is put in
relation to the simpler situation $\vec{B}(x,y,z) = B_0 [\sin(kx) \vec{e_y} +
\cos(kx)\vec{e_z}]$, for which full orbits and lowest order drifts are obtained
analytically. In the kinked equilibrium, the full orbits of NBI fast ions are
solved numerically and shown to follow helical drift surfaces. This result
partially explains the off-axis redistribution of NBI fast particles in the
presence of MAST Long-Lived Modes (LLM).
... A related method for axisymmetric geometry has been realized in the code K2D (Kasilov et al., 2016). Apart from this quasigeometric approach there exists a variety of codes for tracing guiding-center drift motion including ORBIT (White & Chance, 1984), DCOM (Wakasa et al., 2001), MOCA (Tribaldos, 2001), FORTEC-3D (Satake et al., 2005), VENUS (Isaev et al., 2006), NEO-MC (Allmaier et al., 2008), XGC (Ku et al., 2009), ANTS (Drevlak, 2009), ASCOT (Kurki-Suonio et al., 2009), VENUS-LEVIS (Pfefferlé et al., 2014), SIMPLE etc; see also (Beidler et al., 2011). Most of these codes solve the guiding-center equations primarily in the plasma core of toroidal fusion devices. ...
... It aims to self-consistently compute the magnetic equilibrium, the RF wave field and the fast ion distribution function. For this purpose the following codes are used respectively: VMEC (Hirshman & Whitson 1983) (or its anisotropic pressure version ANIMEC Cooper et al. 2009), LEMan (Popovich, Cooper & Villard 2006) and VENUS-LEVIS (Pfefferlé et al. 2014). These three are iterated until a converged self-consistent state is reached. ...
Good fast ion confinement in quasi-isodynamic stellarators still has to be demonstrated experimentally. In the absence of fusion alphas, auxiliary heating systems can be used to generate fast ions. For this purpose, W7-X has neutral beam injection and its newly installed ion cyclotron heating system. In practice, both systems can run simultaneously, the synergetic running of both is the focus of the current modelling development. With the upgrades to the SCENIC code described here, we find that, compared with pure neutral beam injection (NBI), the fast ion tail is significantly enhanced when radio frequency (RF) heating is enabled. But the effectiveness of heating and fast ion tail generation strongly depends on the plasma parameters and antenna frequency, which calls for further study, both numerical and experimental.
... Other interpolation methods, not constrained to the divergencefree condition, include the scheme presented in Ref. 10 based on Fourier reconstruction in the toroidal and poloidal directions and cubic spline in the radial direction of flux coordinate systems. The method proposed in Ref. 11 assumes a local quadratic expansion of the magnetic field and uses least squares to find the local expansion coefficients assuming a regular grid without imposing a divergencefree constraint. ...
An interpolation method to evaluate magnetic fields, given its unstructured and scattered magnetic data, is presented. The method is based on the reconstruction of the global magnetic field using a superposition of orthogonal functions. The coefficients of the expansion are obtained by minimizing a cost function defined as the L ² norm of the difference between the ground truth and the reconstructed magnetic field evaluated on the training data. The divergence-free condition is incorporated as a constraint in the cost function, allowing the method to achieve arbitrarily small errors in the magnetic field divergence. An exponential decay of the approximation error is observed and compared with the less favorable algebraic decay of local splines. Compared to local methods involving computationally expensive search algorithms, the proposed method exhibits a significant reduction of the computational complexity of the field evaluation, while maintaining a small error in the divergence even in the presence of magnetic islands and stochasticity. Applications to the computation of Poincaré sections using data obtained from numerical solutions of the magnetohydrodynamic equations in toroidal geometry are presented and compared with local methods currently in use.
... Motivated by the upcoming operation phase in Wendelstein 7-X (W7-X), in which the new ICRH antenna [1] will start its first operation at the end of this year, the modelling of ICRH physics for the expected scenarios has been started. For this we use the SCENIC code package [2] which is comprised of the equilibrium code ANIMEC (anisotropic-pressure version of VMEC) [3], the full-wave code LEMan [4], and the particle-following code VENUS-LEVIS [5] as well as some smaller tools and scripts linking the codes. Fig. 1 schematically shows how the codes are linked. ...
The next scientific operation phase of Wendelstein 7-X (W7-X) is scheduled to begin in late autumn of 2022 and will, for the first time, include experiments in which the ICRH (ion-cyclotron-resonance heating) antenna will be used.
In addition to heating the plasma, this system will generate fast ions and thus offers a new way to assess fast-ion confinement in a stellarator such as W7-X. The first plasmas that will be used for the upcoming ICRH operation will be Helium-4 plasmas with a small Hydrogen minority on the order of about 10%. In tokamaks such plasmas typically offer good power absorption and are thus considered a safe way for gaining first experiences with the new antenna in W7-X.
This assessment is confirmed by the SCENIC simulations carried out in this contribution that use profiles foreseen for the upcoming campaign as input. The simulations are carried out in the standard configuration of W7-X in low-beta (0.3% ≲ 〈 β 〉 ≲ 1%) plasmas. However, also scans over minority concentration and background-plasma density are performed. We find that the power absorbed by the Hydrogen minority directly from the radio-frequency wave is typically (provided that the minority concentration is not too high) on the order of about 90% with the rest going to the electrons. Very little power goes to the Helium-4 ions. Under the present simulation conditions only fast-ion energies up to about E ≈ 50 keV can be reached.
Combining SCENIC and ASCOT simulations enables us to track lost particles through the scrape-off-layer to the 3D wall of W7-X and to compute wall loads caused by ICRH. The results show that the wall loads that can be expected from ICRH under the first operating conditions are benign.
... Many orbit-following codes, such as ORBIT, [22] GYCAVA, [15,23] OFMC, [24] ASCOT, [25] SPIRAL, [26] and VENUS-LEVIS, [27] have been developed to study loss and transport of fast ions. These codes solve the guiding-center orbit or full orbit of particles, or both of them, such as AS-COT, SOFT, and VENUS-LEVIS. ...
This paper uses the implicit Monte-Carlo full-orbit-following parallel program ISSDE to calculate the prompt loss and slowing down process of neutral beam injection (NBI) generated fast ions due to Coulomb collisions in the equilibrium configuration of Experimental Advanced Superconducting Tokamak (EAST). This program is based on the weak equivalence of the Fokker-Planck equation under Rosenbluth MacDonald Judd (RMJ) potential and Stratonovich stochastic differential equation (SDE). The prompt loss with the LCFS boundary and the first wall (FW) boundary of the two co-current neutral injection beams are studied. Simulation results indicate that the loss behavior of fast ions using the FW boundary is very different from that of the LCFS boundary, especially for fast ions with a large gyration radius. According to our calculations, about 5.11% of fast ions generated by perpendicular injection drift out of the LCFS and then return inside the LCFS to be captured by the magnetic field. The prompt loss ratio of fast ions and the ratio of orbital types depend on the initial distribution of fast ions in the $P_{\zeta}-\Lambda$ space. Under the effect of Coulomb collisions, the pitch-angle scattering and stochastic diffusion happens, which will cause more fast ion loss. For short time scales, among the particles lost due to collisions, the fraction of banana ions reaches 92.31% in the perpendicular beam and 58.65% in the tangential beam when the fraction of banana ions in the tangential beam is 3.4% of the total ions, which means that the effect of Coulomb collisions on banana fast ions is more significant. For long time scales, the additional fast ion loss caused by Coulomb collisions of tangential and perpendicular beams accounted for 16.21% and 25.05% of the total particles, respectively. We have also investigated the slowing down process of NBI fast ions.
... However, the strength of LEMan is its ability to model 3D geometries such as tokamaks with a kink mode perturbation and stellarators, which need not be stellarator-symmetric (analogous to up-down asymmetry in tokamaks). In addition, LEMan is integrated into the SCENIC code package (comprised of ANIMEC [17], LEMan and VENUS-LEVIS [18]), which allows a self consistent computation of the magnetic equilibrium, the RF wave field, and the minority distribution function. This is not the first coupled code package, many others have been developed over the years, with varying levels of model reduction, see table 1. ...
The generation of energetic trapped ions is important for experiments investigating their confinement in 3D magnetic fields, for plasma heating, for studies into unwanted drive of instabilities, and improved transport regimes. An effective way to generate such energetic ions is with ion cyclotron resonance heating. SCENIC is a tool built to self consistently model the magnetic equilibrium, the radio frequency wave, and the minority distribution function in steady state. In this paper the impact of higher order finite Larmor radius corrections in the dielectric tensor will be described. The RF electric field and the power deposition in the new hot model are compared against the previously used warm model for several JET plasmas. Considerable differences are found in some of the scenarios. The new version of the wave code LEMan also supports the direct use of particle-in-cell marker data to compute the dielectric tensor. An expression for the dielectric tensor is derived, and it is applied to a test case in JET. The power deposition profile agrees very well with that of a Maxwellian reference case, which is promising for future applications. Moreover, a full SCENIC run shows a significantly enhanced fast ion tail. In a demonstration of the novel features of LEMan, it is also applied to minority heating in the intrinsically 3D plasma of W7-X.
... In the last few years, test particle codes with full orbit option have been raised, such as SPIRAL, [20] or-BIT (CUEBIT), [21] VENUS-LEVIS. [22] For CFETRlike devices, the analysis of particle confinement is necessary for the design. Under this background, the particle orbit tracing code (PTC) has been developed. ...
Fusion born α particle confinement is one of the most important issues in burning plasmas, such as ITER and CFETR. However, it is extremely complex due to the nonequilibrium characteristics, and multiple temporal and spatial scales coupling with background plasma. A numerical code using particle orbit tracing method (PTC) has been developed to study energetic particle confinement in tokamak plasmas. Both full orbit and drift orbit solvers are implemented to analyze the Larmor radius effects on α particle confinement. The elastic collisions between alpha particles and thermal plasma are calculated by a Monte Carlo method. A triangle mesh in poloidal section is generated for electromagnetic fields expression. Benchmark between PTC and ORBIT has been accomplished for verification. For CFETR burning plasmas, PTC code is used for α particle source and slowing down process calculation in 2D equilibrium. In future work, 3D field like toroidal field ripples, Alfvén and magnetohydrodynamics instabilities perturbation inducing α particle transport will be analyzed.
... Global kinetic computations of quasi-steady plasma parameters in 3D toroidal fusion devices utilize the evaluation of the distribution function and/or its moments by direct modeling of particle orbits. This includes Monte Carlo transport simulations in given external fields [1][2][3][4][5][6][7][8][9] as well as self-consistent turbulence models with particle codes. [10][11][12][13] Kinetic modeling of 3D plasma equilibria 14 or edge plasmas 15 puts specific requirements on solving the guidingcenter equations. ...
A geometric 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. When applied to collisionless guiding-center orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator configuration, the method demonstrates correct long-term orbit dynamics. Within Monte Carlo evaluation of transport coefficients, the computational efficiency of geometric integration is an order of magnitude higher than with a standard adaptive Runge-Kutta integrator.
... Out in the field, deep in the trenches, several numerical codes [1][2][3][4][5][6][7][8][9][10] use the equations above -or their guiding-center counterparts -to perform FMC simulations of various fast ion (or electron) signals such as the energy flux onto certain wall tiles or the distribution of fast ions recorded by a fast-ion-loss detector (FILD). In practice, each code implements the following algorithm: (i) Assign a marker particle with initial coordinates Z 0 = (X 0 , V 0 ). ...
With Wendelstein 7-X now up and running, and the construction of ITER proceeding, predicting fast-ion losses to sensitive plasma-facing components and detectors is gaining significant interest. A common recipe to perform such studies is to push a large population of marker particles along their equations of motion, the trajectories randomized with Monte Carlo operators accounting for Coulomb collisions, and to record possible intersections of the marker trajectories with synthetic detectors or areas of interest in the first wall. While straightforward to implement and easy to parallelize, this Forward Monte Carlo (FMC) approach tends to suffer from poor statistics and error estimation as the detector domain is often small: it is difficult to guess how to set up the initial weights and locations of the markers for them to remain representative of the source distribution, yet record enough hits to the detector for good statistics. As an alternative, the FMC method can be replaced with a so-called Backward Monte Carlo (BMC) algorithm. Instead of starting with a given initial marker population, one starts from the end condition at the detector and records how the hit probability evolves backwards in time. The scheme eliminates the statistics issue present in the FMC scheme and may provide more accurate and efficient simulations of fast-ion loss signals. The purpose of this paper is to explain the BMC recipe in the fast-ion setting and to discuss the associated nuances, especially how to negate artificial diffusion. For illustration purposes, our numerical example considers a 1-D stochastic harmonic oscillator as a mock-up of a charged particle.
... Such studies require long and highly accurate particle simulations. To obtain the required accuracy, the VENUS-LEVIS guiding center code [13] for example exploits the Fourier decomposition of the magnetic field for orbit calculations. ...
It is shown that free boundary 3D equilibrium calculations in tokamak geometry are capable of capturing the physics of non-linearly saturated external kink modes for monotonic current and q profiles typical of standard (baseline) plasma scenarios. The VMEC ideal MHD equilibrium model exhibits strong flux surface corrugations of the plasma vacuum boundary, driven by the core current profile. A method is presented which conveniently extracts the amplitude of the corrugation in terms of Fourier components in straight field line coordinates. The Fourier spectrum, and condition for non-linear corrugation agrees well with linear simulations, and the saturated amplitude agrees well with non-linear analytic calculations.
... For studying the behavior of the energetic particles in the tokamak magnetized plasmas, many orbit following codes have been developed, such as ORBIT, 20,21 OFMC, 22,23 ASCOT4, 24 VENUS-LEVIS, 25 GCMP. 26 Recently, we have developed the orbit code GYCAVA 27,28 on the basis of the exact canonical variables and the Lie-transform perturbation method, which is widely used in the gyrokinetic theories 29 and simulations. ...
The effects of resonant magnetic perturbations (RMPs) and the magnetic drift on the loss and redistribution of passing ions are investigated numerically by the upgrade version of the orbit following code GYCAVA. The drift island structures of passing ion orbits induced by RMPs and the magnetic drift are consistent with the given toroidal and poloidal mode numbers of RMPs in the orbit simulations. The redistribution and loss of energetic and thermal passing ions with and without RMPs are numerically studied and compared with each other. The redistribution near the edge is due to the loss induced by RMPs and the magnetic drift. The extra loss of passing ions induced by RMPs is related to the drift island structure induced by RMPs and the magnetic drift and the stochasticity induced by overlap of magnetic islands. The loss of passing ions with n = 1 RMPs is larger than that with n = 4 RMPs for the same perturbation amplitude parameter, which is due to the fact that the magnetic islands produced by n = 1 RMPs are larger than those produced by n = 4 RMPs. The loss fraction of passing ions with n = 1 RMPs increases with the amplitude of RMPs and the radial profile parameter of the ion density. The pitch and energy scan for the loss of energetic passing ions with n = 1 RMPs is simulated. For counter-passing ions, the loss fraction of low-pitch ions is larger than that of high-pitch ions. The dependence of the loss of energetic ions with RMPs on energy is related to the pitch angle.
... Two opposing approaches are compared, one where the symmetry breaking field calculated in absence of the plasma is added to an axisymmetric MHD equilibrium calculation (henceforth called the "2D +ripple" approach), while the other where a full 3D free boundary MHD equilibrium calculation naturally includes the plasma response within the 3D deformation of its flux-surfaces (henceforth called the "3D equilibrium" approach). Analyzing the fast particle trajectories for the two descriptions of the magnetic field for DEMO in the orbit code VENUS-LEVIS [37], it was found that the guiding center approximation was adequate for the study, essentially because the scale length of the magnetic field variation is much larger that the Larmor radius of 3.5MeV alpha particles. In addition, unlike in the n=3 RMP study investigated previously [36], the magnetic ripple in DEMO (which has mode number n=18) does not cause a significant plasma response. ...
For several reasons the challenge to keep the loads to the first wall within engineering limits is substantially higher in DEMO compared to ITER. Therefore the pre-conceptual design development for DEMO that is currently ongoing in Europe needs to be based on load estimates that are derived employing the most recent plasma edge physics knowledge. An initial assessment of the static wall heat load limit in DEMO infers that the steady state peak heat flux limit on the majority of the DEMO first wall should not be assumed to be higher than 1.0 MW m⁻². This compares to an average wall heat load of 0.29 MW m⁻² for the design assuming a perfect homogeneous distribution. The main part of this publication concentrates on the development of first DEMO estimates for charged particle, radiation, fast particle (all static) and disruption heat loads. Employing an initial engineering wall design with clear optimization potential in combination with parameters for the flat-top phase (x-point configuration), loads up to 7 MW m⁻² (penalty factor for tolerances etc not applied) have been calculated. Assuming a fraction of power radiated from the x-point region between 1/5 and 1/3, peaks of the total power flux density due to radiation of 0.6-0.8 MW m⁻² are found in the outer baffle region. This first review of wall loads, and the associated limits in DEMO clearly underlines a significant challenge that necessitates substantial engineering efforts as well as a considerable consolidation of the associated physics basis.
... The predicted reduction in the confinement of fast ions due to the 3D fields has been compared favourably with experimental measurements. In simulations of this type, the 3D fields in the orbitequation solver must be represented very accurately, because the fast ions must be followed until they are thermalized, that is, over a macroscopic timescale, and numerical errors would propagate, leading to unphysical results 119 . A significant remaining challenge for modelling energetic-ion confinement in static fields relates to identifying the most accurate model for representing the plasma response to fields that break axial symmetry 122,123 . ...
Magnetic-fusion plasmas are complex self-organized systems with an extremely wide range of spatial and temporal scales, from the electron-orbit scales (similar to 10(-11) s, similar to 10(-5) m) to the diffusion time of electrical current through the plasma (similar to 10(2) s) and the distance along the magnetic field between two solid surfaces in the region that determines the plasma-wall interactions (similar to 100 m). The description of the individual phenomena and of the nonlinear coupling between them involves a hierarchy of models, which, when applied to realistic configurations, require the most advanced numerical techniques and algorithms and the use of state-of-the-art high-performance computers. The common thread of such models resides in the fact that the plasma components are at the same time sources of electromagnetic fields, via the charge and current densities that they generate, and subject to the action of electromagnetic fields. This leads to a wide variety of plasma modes of oscillations that resonate with the particle or fluid motion and makes the plasma dynamics much richer than that of conventional, neutral fluids.
Recent developments in the design of magnetic confinement fusion devices have allowed the construction of exceptionally optimized stellarator configurations. The near-axis expansion in particular has been proven to enable the construction of magnetic configurations with good confinement properties while taking only a fraction of the usual computation time to generate optimized magnetic equilibria. However, not much is known about the overall features of fast-particle orbits computed in such analytical, yet simplified, equilibria when compared with those originating from accurate equilibrium solutions. This work aims to assess and demonstrate the potential of the near-axis expansion to provide accurate information on particle orbits and to compute loss fractions in moderate to high aspect ratios. The configurations used here are all scaled to fusion-relevant parameters and approximate quasi-symmetry to various degrees. This allows us to understand how deviations from quasi-symmetry affect particle orbits and what are their effects on the estimation of the loss fraction. Guiding-centre trajectories of fusion-born alpha particles are traced using gyronimo and SIMPLE codes under the NEAT framework, showing good numerical agreement. Discrepancies between near-axis and magnetohydrodynamic fields have minor effects on passing particles but significant effects on trapped particles, especially in quasi-helically symmetric magnetic fields. Effective expressions were found for estimating orbit widths and passing–trapped separatrix in quasi-symmetric near-axis fields. Loss fractions agree in the prompt losses regime but diverge afterwards.
A neutral beam current drive on the EAST tokamak is studied by using Monte Carlo test particle code TGCO. The phase-space structure of the steady-state fast ion distribution is examined and visualized. We find that trapped ions carry co-current current near the edge and countercurrent current near the core. However, the magnitude of the trapped ion current is one order smaller than that of the passing ions. Therefore, their contribution to the fast ion current is negligible (1% of the fast ion current). We examine the dependence of the fast ion current on two basic plasma parameters: the plasma current Ip and plasma density ne. The results indicate that the dependence of fast ion current on Ip is not monotonic: with Ip increasing, the fast ion current first increases and then decreases. This dependence can be explained by the change of trapped fraction and drift-orbit width with Ip. The fast ion current decreases with the increase in plasma density ne. This dependence is related to the variation of the slowing-down time with ne, which is already well known and is confirmed in our specific situation. The electron shielding effect to the fast ion current is taken into account by using a fitting formula applicable to general tokamak equilibria and arbitrary collisionality regime. The dependence of the net current on the plasma current and density follows the same trend as that of the fast ion current.
An interpolation method to evaluate magnetic fields given unstructured, scattered magnetic data is presented. The method is based on the reconstruction of the global magnetic field using a superposition of orthogonal functions. The coefficients of the expansion are obtained by minimizing a cost function defined as the L^2 norm of the difference between the ground truth and the reconstructed magnetic field evaluated on the training data. The divergence-free condition is incorporated as a constrain in the cost function allowing the method to achieve arbitrarily small errors in the magnetic field divergence. An exponential decay of the approximation error is observed and compared with the less favorable algebraic decay of local splines. Compared to local methods involving computationally expensive search algorithms, the proposed method exhibits a significant reduction of the computational complexity of the field evaluation, while maintaining a small error in the divergence even in the presence of magnetic islands and stochasticity. Applications to the computation of Poincar\'e sections using data obtained from numerical solutions of the magnetohydrodynamic equations in toroidal geometry are presented and compared with local methods currently in use.
The effects of resonant magnetic perturbations (RMPs) on tangential neutral beam heating in the EAST tokamak are studied numerically. RMPs with linear resistive magnetohydrodynamics response are used in the modeling. A variety of representing configurations of RMP coil currents are examined, and their effects on the neutral beam injection (NBI) heating efficiency are compared, in order to find a parameter window where deleterious effects of RMPs on NBI heating efficiency are minimized. It is found that the internal redistribution of fast ions by RMPs induces local accumulation of fast ions, resulting in higher local fast ion pressure than the case without RMPs. It is also found that the toroidal phasing of the RMP with respect to the fast ion source has slight effects on the steady-state radial profile of fast ions. The dependence of fast ion loss fraction on the RMP up-down phase difference shows a similar behavior as the dependence of the radial width of chaotic magnetic field on the phase difference. A statistical method of identifying resonances between RMPs and lost fast ions is proposed, and the results indicate that some resonances between RMPs and lost passing particles may be of non-integer fractional order, rather than the usual integer order.
Strongly peaked tungsten accumulation is a common feature of high performance plasma scenarios in JET with the ITER-like wall, particularly during MHD activity induced by m⁄n = 1⁄1 continuous modes. This study investigates the effect of 1⁄1 long living internal kink modes on heavy impurity transport in the presence of strong flows and NTV ambipolar electric field. A novel formulation which includes these effects is presented and applied in the VENUS-LEVIS code in order to follow tungsten ions in a saturated JET-like 1⁄1 internal kinked toroidally rotating plasma configuration. The synergy between 3D magnetic fields, strong flows and NTV is seen to cause tungsten accumulation in contrast to what is observed in similar axisymmetric configurations. Rapid inward transport of impurities in JET plasmas following the triggering of continuous 1⁄1 modes is explained by the work presented here, and we use the same theory to postulate why outward transport can occur in kinked ASDEX-U plasmas.
The orbit code VENUS-LEVIS is adapted to follow particles in plasma equilibria with discontinuous fields generated by the Stepped Pressure Equilibrium Code (SPEC). The latter is an implementation of the Multi-Region relaxed MHD model, which efficiently computes Taylor states in a series of nested toroidal volumes and supports the formation of magnetic islands and chaotic field regions. To adapt VENUS-LEVIS, an event location procedure is implemented in the existing numerical integrator, which ensures the particle sees the correct field along its trajectory, regardless of the discontinuities present in the Stepped Pressure Equilibrium model. The algorithm is tested in the case where the magnetic field is uniform in the upper and lower half-spaces but has a discontinuity in its direction (shear) on the plane z=0. Particle drifts due to the discontinuity are studied. The convergence properties of the numerical scheme are highlighted by the numerical accuracy, and conservation of the system's invariants, such as energy and momentum. Simulations and convergence studies using the SPEC-LEVIS interface in axisymmetric geometry are then presented. Finally, illustrative particle drifts due to the discontinuity are studied and explained: we examine drifts associated with the change in Larmor radius of passing particles with small excursion from flux surfaces.
In this paper, we report a comparative study of the orbit islands formed by the guiding center drift motion of test energetic particles (EPs), including both high-energy deuterium ions and fusion-born α-particles, with respect to the magnetic islands formed by tracing the field lines, for an ITER plasma representing the 15 MA baseline scenario. The particle drift orbit is modified by the presence of static resonant magnetic perturbations (RMPs) of different toroidal mode numbers n ( n = 1 and 3 in this study), forming orbit islands in the Poincaré plane by passing EPs. The key findings are as follows. (i) With an n = 1 RMP field, the size of the orbit islands combined with the total RMP field, including the plasma response, is about three times smaller than that obtained by assuming the corresponding vacuum field. (ii) The orbit island size is not sensitive to the EP energy, and is comparable to those of the magnetic islands. (iii) Passing EPs with outward radial orbital drift form orbit islands that shift inward with respect to the corresponding magnetic islands, and vice versa. This radial shift, ΔΨ, measured in the normalized equilibrium poloidal flux, is quantified as ΔΨ ≃ 0.03 X for the ITER baseline plasma and assuming n = 1 RMPs, with X ≡ ( M / M p ) 1 / 2 ( E [ M e V ] ) 1 / 2 Z − 1 , where M is the particle mass, M p is the proton mass, E is the particle energy, and Z is the particle charge number. (iv) Trapped EPs do not form drift orbit islands, even in the presence of the 3D fields produced by the 90 kAt ELM control coil current in ITER, and thus possess good confinement properties. Nevertheless, the deformation of the trapped particle drift orbits in the Poincaré plane shows that the canonical toroidal angular momentum of EPs is no longer conserved in the presence of the RMP 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.
Monte Carlo orbit-following simulations based on a phase-space coordinate transform method are presented in this paper. Guiding-center orbits are computed by using this method, that is, using the phase-space coordinate transform and solving the gyrokinetic equations of motion. This method has been adopted in a Monte Carlo orbit-following code GYCAVA for studying the behavior of fast ions in a tokamak. The finite Larmor radius effect can be included in the guiding-center orbit computation. The cylindrical coordinates are used for avoiding singularity of the safety factor at X points of the divertor configuration. An n-point discrete sum method is used to compute the gyro-average, which is related to the finite Larmor radius effect. The GYCAVA code has been integrated with other equilibrium, magnetohydrodynamic and neutral beam injection codes for studying losses and wall loads of fast ions in the presence of perturbation fields. The GYCAVA code has been verified by tests of the orbit computation and the Monte Carlo Coulomb collision. Simulations of fast ion losses in the presence of resonant magnetic perturbations in the EAST tokamak have been performed.
High-energy ions, such as fusion alphas and ions from external heating, can be very sensitive to any non-axisymmetric features in the confining magnetic field due to their collisionless nature. Since understanding the confinement properties of these ions is crucial for ITER (the first fusion reactor currently under construction in Cadarache, France) and beyond, it is of ultimate importance that the predictive simulations are accurate and free of numerical distortions. Adding the third dimension comes at substantial computational cost, calling for new kinds of approaches and computational platforms. In this contribution we discuss what new features, even new physics, the non-axisymmetry brings with it and how one could cope with the ever-increasing demands on both memory and CPU resources. In the end, a few simulation examples with a varying level of non-axisymmetry are given.
Abstract A first principle implicit simulation program, Implicit Stratonovich Stochastic Differential Equations (ISSDE), is constructed for solving stochastic differential equations which can describe plasmas with Coulomb collision. The basic ides of the program is the stochastic equivalence between Fokker-Planck equation and Stratonovich SDE. The use of the implicit discrete method can guarantees the numerical stability and the conservation of kinetic energy. ISSDE is built with C++ language, and is designed to possess standard interfaces and extendible modules. The slowing-down processes of electron beams in both unmagnetized and magnetized plasmas are studied by using ISSDE, which shows the correctness of ISSDE. ISSDE can thus provide a powful tool for studies of collisional plasmas.
The performance of the auxiliary heating systems ion cyclotron resonance heating and neutral beam injection is calculated in three different magnetic mirror configurations foreseen to be used in future experiments in the Wendelstein 7-X stellarator: low, standard and high mirror. This numerical work is implemented with the SCENIC code package, which is designed to model three-dimensional magnetic equilibria whilst retaining effects such as anisotropy and the influence of including a finite orbit width of the particles. The ability to simulate NBI deposition in three-dimensional equilibria, the implementation of the realistic beam injector geometry, and the modification of the SCENIC package to permit the investigation of the 3-ion species heating scheme, are recent developments. Using these modifications, an assessment of the advantages and disadvantages of these two fast-ion producing auxiliary heating systems is made in the three different magnetic mirror equilibria. For NBI heating, the high mirror configuration displays the best global confinement properties, resulting in a larger collisional power transfer to the background plasma. The standard mirror has the best particle confinement in the core region, but the worst towards the edge of the plasma. The low mirror has the largest lost power and thus the lowest total collisional power. For ICRH, the displacement of the RF-resonant surface significantly impacts the heating performance. Due to the large toroidal magnetic mirror in the high mirror equilibrium, resonant particles easily become trapped and cannot remain in resonance, generating only small energetic particle populations. Despite this, global confinement is still the strongest in this equilibrium. The low mirror is the only equilibrium to produce peaked on-axis collisional power deposition, with associated peaked on-axis fast ion pressure profiles. A highly energetic particle population is then produced but this results in larger lost power as this equilibrium is not sufficiently optimised for fast ion confinement. A comparison between the two heating methods concludes that NBI produces a smaller fraction of lost to input power, and a reduced sensitivity of the performance to variations of the toroidal magnetic mirror. The main limit of NBI which does not apply to ICRH is the production of highly energetic particle populations, with predictions of energetic particles of E ∼ 0.45 MeV.
In future tokamaks like ITER with tungsten walls, it is imperative to control tungsten accumulation in the core of operational plasmas, especially since tungsten accumulation can lead to radiative collapse and disruption. We investigate the behavior of tungsten trace impurities in a JET-like hybrid scenario with both axisymmetric and saturated 1/1 ideal helical core in the presence of strong plasma rotation. For this purpose, we obtain the equilibria from VMEC and use VENUS-LEVIS, a guiding-center orbit-following code, to follow heavy impurity particles. In this work, VENUS-LEVIS has been modified to account for strong plasma flows with associated neoclassical effects arising from such flows. We find that the combination of helical core and plasma rotation augments the standard neoclassical inward pinch compared to axisymmetry, and leads to a strong inward pinch of impurities towards the magnetic axis despite the strong outward diffusion provided by the centrifugal force, as frequently observed in experiments.
Absorption of ion-cyclotron range of frequencies waves at the fundamental resonance is an efficient source of plasma heating and fast ion generation in tokamaks and stellarators. This heating method is planned to be exploited as a fast ion source in the Wendelstein 7-X stellarator. The work presented here assesses the possibility of using the newly developed three-ion species scheme (Kazakov et al (2015) Nucl. Fusion 55 032001) in tokamak and stellarator plasmas, which could offer the capability of generating more energetic ions than the traditional minority heating scheme with moderate input power. Using the SCENIC code, it is found that fast ions in the MeV range of energy can be produced in JET-like plasmas. The RF-induced particle pinch is seen to strongly impact the fast ion pressure profile in particular. Our results show that in typical high-density W7-X plasmas, the three-ion species scheme generates more energetic ions than the more traditional minority heating scheme, which makes three-ion scenario promising for fast-ion confinement studies in W7-X.
An improved set of guiding-centre equations, expanded to one order higher in Larmor radius than usually written for guiding-centre codes, are derived for curvilinear flux coordinates and implemented into the orbit following code VENUS-LEVIS. Aside from greatly improving the correspondence between guiding-centre and full particle trajectories, the most important effect of the additional Larmor radius corrections is to modify the definition of the guiding-centre's parallel velocity via the so-called Banos drift. The correct treatment of the guiding-centre push-forward with the Banos term leads to an anisotropic shift in the phase-space distribution of guiding-centres, consistent with the well-known magnetization term. The consequence of these higher order terms are quantified in three cases where energetic ions are usually followed with standard guiding-centre equations: (1) neutral beam injection in a MAST-like low aspect-ratio spherical equilibrium where the fast ion driven current is significantly larger with respect to previous calculations, (2) fast ion losses due to resonant magnetic perturbations where a lower lost fraction and a better confinement is confirmed, (3) alpha particles in the ripple field of the European DEMO where the effect is found to be marginal.
The Accurate Particle Tracer (APT) code is designed for large-scale particle simulations on dynamical systems. Based on a large variety of advanced geometric algorithms, APT possesses long-term numerical accuracy and stability, which are critical for solving multi-scale and non-linear problems. Under the well-designed integrated and modularized framework, APT serves as a universal platform for researchers from different fields, such as plasma physics, accelerator physics, space science, fusion energy research, computational mathematics, software engineering, and high-performance computation. The APT code consists of seven main modules, including the I/O module, the initialization module, the particle pusher module, the parallelization module, the field configuration module, the external force-field module, and the extendible module. The I/O module, supported by Lua and Hdf5 projects, provides a user-friendly interface for both numerical simulation and data analysis. A series of new geometric numerical methods and key physical problems, such as runaway electrons in tokamaks and energetic particles in Van Allen belt, have been studied using APT. As an important realization, the APT-SW version has been successfully distributed on the world's fastest computer, the Sunway TaihuLight supercomputer, by supporting master-slave architecture of Sunway many-core processors.
An assessment of alpha particle confinement is performed in the European DEMO reference design. 3D MHD equilibria with nested flux-surfaces and single magnetic axis are obtained with the VMEC free-boundary code, thereby including the plasma response to the magnetic ripple created by the finite number of TF coils. Populations of fusion alphas that are consistent with the equilibrium profiles are evolved until slowing-down with the VENUS-LEVIS orbit code in the guiding-centre approximation. Fast ion losses through the last-closed flux-surface are numerically evaluated with two ripple models: (1) using the 3D equilibrium and (2) algebraically adding the non-axisymmetric ripple perturbation to the 2D equilibrium. By virtue of the small ripple field and its non-resonant nature, both models quantitatively agree. Differences are however noted in the toroidal location of particles losses on the last-closed flux-surface, which in the first case is 3D and in the second not. Superbanana transport, i.e. ripple-well trapping and separatrix crossing, is expected to be the dominant loss mechanism, the strongest effect on alphas being between 100–200 KeV. Above this, stochastic ripple diffusion is responsible for a rather weak loss rate, as the stochastisation threshold is observed numerically to be higher than analytic estimates. The level of ripple in the current 18 TF coil design of the European DEMO is not found to be detrimental to fusion alpha confinement.
Fast ions in W7-X will be produced either by neutral beam injection (NBI) or by ion-cyclotron resonant heating (ICRH). The latter presents the advantage of depositing power locally and does not suffer from core accessibility issues (Drevlak et al 2014 Nucl. Fusion 54 073002). This work assesses the possibility of using ICRH as a fast ion source in W7-X relevant conditions. The SCENIC package is used to resolve the full wave propagation and absorption in a three-dimensional plasma equilibrium. The source of the ion-cyclotron range of frequency (ICRF) wave is modelled in this work by an antenna formulation allowing its localisation in both the poloidal and toroidal directions. The actual antenna dimension and localization is therefore approximated with good agreement. The local wave deposition breaks the five-fold periodicity of W7-X. It appears that generation of fast ions is hindered by high collisionality and significant particle losses. The particle trapping mechanism induced by ICRH is found to enhance drift induced losses caused by the finite orbit width of trapped particles. The inclusion of a neoclassically resolved radial electric field is also investigated and shows a significant reduction of particle losses.
We present SCENIC simulations of a W7X 4He plasma with 1% H minority and with an antenna model close to the design foreseen for the W7X ICRF antenna [1, 2]. A high mirror and a standard equilibrium are considered. The injected wave frequency is fixed at 33.8 MHz and 39.6MHz respectively and only fundamental minority heating is considered. Included in this calculation is a new realistic model of the antenna, where it is found that the localization of the antenna geometry tends to break the five-fold periodicity of the system. We assess the heat transfer through the toroidal periods via Coulomb collisions.
In the following theoretical and numerically oriented work, a number of findings have been assembled. The newly devised VENUS-LEVIS code, designed to accurately solve the motion of energetic particles in the presence of 3D magnetic fields, relies on a non-canonical general coordinate Lagrangian formulation of the guiding-centre and full-orbit equations of motion. VENUS-LEVIS can switch between guiding-centre and full-orbit equations with minimal discrepancy at first order in Larmor radius by verifying the perpendicular variation of magnetic vector field, not only including gradients and curvature terms but also parallel currents and the shearing of field-lines. By virtue of a Fourier representation of the fields in poloidal and toroidal coordinates and a cubic spline in the radial variable, the order of the Runge-Kutta integrating scheme is preserved and convergence of Hamiltonian properties is obtained. This interpolation scheme is crucial to compute orbits over slowing-down times, as well as to mitigate the singularity of the magnetic axis in toroidal flux coordinate systems. Three-dimensional saturated MHD states are associated with many tokamak phenomena including snakes and LLMs in spherical or more conventional tokamaks, and are inherent to stellarator devices. The VMEC equilibrium code conveniently reproduces such 3D magnetic configurations. Slowing-down simulations of energetic ions from NBI predict off-axis deposition of particles during LLM MHD activity in hybrid-like plasmas of the MAST. Co-passing particles helically align in the opposite side of the plasma deformation, whereas counter-passing and trapped particles are less affected by the presence of a helical core. Qualitative agreement is found against experimental measurements of the neutron emission. Two opposing approaches to include RMPs in fast ion simulations are compared, one where the vacuum field caused by the RMP current coils is added to the axisymmetric MHD equilibrium, the other where the MHD equilibrium includes the plasma response within the 3D deformation of its flux-surfaces. The first model admits large regions of stochastic field-lines that penetrate the plasma without alteration. The second assumes nested flux-surfaces with a single magnetic axis, embedding the RMPs in a 3D saturated ideal MHD state but excluding stochastic field-lines within the last closed flux-surface. Simulations of fast ion populations from NBI are applied to MAST n=3 RMP coil configuration with 4 different activation patterns. At low beam energies, particle losses are dominated by parallel transport due to the stochasticity of the field-lines, whereas at higher energies, losses are accredited to the 3D structure of the perturbed plasma as well as drift resonances.
Two opposing approaches to include resonant magnetic perturbations (RMPs) in fast ion simulations are compared, one where the vacuum field caused by the RMP current coils is added to the axisymmetric MHD equilibrium, the other where the MHD equilibrium includes the plasma response within the 3D deformation of its flux-surfaces. The first model admits large regions of stochastic field-lines that penetrate the plasma without alteration. The second assumes nested flux-surfaces with a single magnetic axis, which excludes stochastic field-lines, and embeds the RMPs within a 3D saturated ideal MHD state. The two descriptions of RMPs have been implemented in the VENUS-LEVIS guiding-centre orbit code. Simulations of fast ion populations resulting from MAST neutral beam injection have been applied to MAST n = 3 RMP coil configuration. At low beam energies, particle losses are dominated by parallel transport due to the stochasticity of the field-lines (vacuum-RMP model), whereas at higher energies, losses are accredited to the 3D structure of the perturbed plasma and the resulting drifts (equilibrium-RMP model).
Energetic ions are found to be transported strongly from the core of MAST hybrid-like plasmas during long-lived mode (LLM) magnetohydrodynamic activity. The resulting impact on the neutral beam ion deposition and concurrent current drive is modelled using the guiding-centre approximation in the internal kinked magnetic topology. General coordinate guiding-centre equations are extended for this purpose. It is found that the kinked core spirals around the position of strongest ionization, which remains geometrically centred, so that a large fraction of the population is deposited in the high shear external region where the plasma is almost axisymmetric. Those particles ionized in the low shear region exhibit exotic drift motion due to the strongly non-axisymmetric equilibrium, periodically passing near the magnetic axis and then reflected by the boundary of the kinked equilibrium, which in this respect acts as a confining pinch. Broad agreement is found against experimental measurement of fast ion particle confinement degradation as the MAST LLM amplitude varies.
The impact of edge localized modes (ELMs) and externally applied resonant and non-resonant magnetic perturbations (MPs) on fast-ion confinement/transport have been investigated in the ASDEX Upgrade (AUG), DIII-D and KSTAR tokamaks. Two phases with respect to the ELM cycle can be clearly distinguished in ELM-induced fast-ion losses. Inter-ELM losses are characterized by a coherent modulation of the plasma density around the separatrix while intra-ELM losses appear as well-defined bursts. In high collisionality plasmas with mitigated ELMs, externally applied MPs have little effect on kinetic profiles, including fast-ions, while a strong impact on kinetic profiles is observed in low-collisionality, low q95 plasmas with resonant and non-resonant MPs. In low-collisionality H-mode plasmas, the large fast-ion filaments observed during ELMs are replaced by a loss of fast-ions with a broad-band frequency and an amplitude of up to an order of magnitude higher than the neutral beam injection prompt loss signal without MPs. A clear synergy in the overall fast-ion transport is observed between MPs and neoclassical tearing modes. Measured fast-ion losses are typically on banana orbits that explore the entire pedestal/scrape-off layer. The fast-ion response to externally applied MPs presented here may be of general interest for the community to better understand the MP field penetration and overall plasma response.
Guiding center simulations are an important means of predicting the
effect of resistive and ideal magnetohydrodynamic instabilities on
particle distributions in toroidal magnetically confined thermonuclear
fusion research devices. Because saturated instabilities typically have
amplitudes of δB/B of a few times 10-4 numerical
accuracy is of concern in discovering the effect of mode particle
resonances. We develop a means of following guiding center orbits which
is greatly superior to the methods currently in use. For full
implementation, the method requires some breaking of axisymmetry, either
through toroidal field ripple or magnetohydrodynamic instabilities. In
the presence of ripple or time dependent magnetic perturbations both
energy and canonical momentum are conserved in a time step to better
than one part in 1014, an improvement of nine orders of
magnitude over standard Runge-Kutta integration, and the relation
between changes in canonical momentum and energy is also conserved to
very high order.
Guiding-center theory provides the reduced dynamical equations for the motion of charged particles in slowly varying electromagnetic fields, when the fields have weak variations over a gyration radius (or gyroradius) in space and a gyration period (or gyroperiod) in time. Canonical and noncanonical Hamiltonian formulations of guiding-center motion offer improvements over non-Hamiltonian formulations: Hamiltonian formulations possess Noether's theorem (hence invariants follow from symmetries), and they preserve the Poincare invariants (so that spurious attractors are prevented from appearing in simulations of guiding-center dynamics). Hamiltonian guiding-center theory is guaranteed to have an energy conservation law for time-independent fields--something that is not true of non-Hamiltonian guiding-center theories. The use of the phase-space Lagrangian approach facilitates this development, as there is no need to transform a priori to canonical coordinates, such as flux coordinates, which have less physical meaning. The theory of Hamiltonian dynamics is reviewed, and is used to derive the noncanonical Hamiltonian theory of guiding-center motion. This theory is further explored within the context of magnetic flux coordinates, including the generic form along with those applicable to systems in which the magnetic fields lie on nested tori. It is shown how to return to canonical coordinates to arbitrary accuracy by the Hazeltine-Meiss method and by a perturbation theory applied to the phase-space Lagrangian. This noncanonical Hamiltonian theory is used to derive the higher-order corrections to the magnetic moment adiabatic invariant and to compute the longitudinal adiabatic invariant. Noncanonical guiding-center theory is also developed for relativistic dynamics, where covariant and noncovariant results are presented. The latter is important for computations in which it is convenient to use the ordinary time as the independent variable rather than the proper time. The final section uses noncanonical guiding-center theory to discuss the dynamics of particles in systems in which the magnetic-field lines lie on nested toroidal flux surfaces. A hierarchy in the extent to which particles move off of flux surfaces is established. This hierarchy extends from no motion off flux surfaces for any particle to no average motion off flux surfaces for particular types of particles. Future work in magnetically confined plasmas may make use of this hierarchy in designing systems that minimize transport losses.
A Monte Carlo method for the collisional guiding-center Fokker-Planck kinetic equation is derived in the five-dimensional guiding-center phase space, where the effects of magnetic drifts due to the background magnetic field nonuniformity are included. It is shown that, in the limit of a homogeneous magnetic field, our guiding-center Monte Carlo collision operator reduces to the guiding-center Monte Carlo Coulomb operator previously derived by Xu and Rosenbluth [Phys. Fluids B 3, 627 (1991)]. Applications of the present work will focus on the collisional transport of energetic ions in complex nonuniform magnetized plasmas in the large mean-free-path (collisionless) limit, where magnetic drifts must be retained.
A Hamiltonian/Lagrangian theory to describe guiding centre orbit drift motion which is canonical in the Boozer coordinate frame has been extended to include full electromagnetic perturbed fields in anisotropic pressure 3D equilibria with nested magnetic flux surfaces. A redefinition of the guiding centre velocity to eliminate the motion due to finite equilibrium radial magnetic fields and the choice of a gauge condition that sets the radial component of the electromagnetic vector potential to zero are invoked to guarantee that the Boozer angular coordinates retain the canonical structure. The canonical momenta are identified and the guiding centre particle radial drift motion and parallel gyroradius evolution are derived. The particle coordinate position is linearly modified by wave–particle interactions. All the nonlinear wave–wave interactions appear explicitly only in the evolution of the parallel gyroradius. The radial variation of the electrostatic potential is related to the binormal component of the displacement vector for MHD-type perturbations. The electromagnetic vector potential projections can then be determined from the electrostatic potential and the radial component of the MHD displacement vector.
We present the interface between a gyrokinetic code and a guiding centre code dedicated to the study of fast ion turbulent transport. A set of velocity space-dependent (kinetic) transport quantities, representing the link between the two codes, is presented. The code suite is applied to DEMO and TCV plasmas. While negligible alpha particle transport is observed for both tokamaks, important beam ion redistribution is obtained for simulations of DEMO. Results for the TCV tokamak demonstrate that the influence of turbulent fields on fast ion transport strongly depends on the plasma scenario. (Some figures may appear in colour only in the online journal)
We apply the code package SCENIC to a two field-period quasi-axisymmetric stellarator. Ion cyclotron resonance heating (ICRH) is applied both on the high- and low-field side to a 1% 3He minority in a deuterium plasma. It is shown that due to toroidal variations, the results are considerably different from similar tokamak studies. In particular, toroidal variations in power deposition and pressure are created and accentuated during radio frequency heating, such that modifications to the magnetic equilibrium depend on toroidal angle. We demonstrate that due to enhanced particle loss, low-field side heating is significantly less efficient than high-field side heating, and that toroidally trapped particles impose upper power limits for efficient radio frequency injection.
Magnetohydrodynamic equilibrium states with a three-dimensional helical core are computed to model the MAST spherical tokamak and the RFX-mod reversed field pinch. The boundary is fixed as axisymmetric. The MAST equilibrium state has the appearance of an internal kink mode and is obtained under conditions of weak reversed central shear. The RFX-mod equilibrium state has seven-fold periodicity. An ideal magnetohydrodynamic stability analysis reveals that the reversal of the core magnetic shear can stabilize a periodicity-breaking mode that is dominantly m/n = 1/8 strongly coupled to a m/n = 2/15 component, as long as the central rotational transform does not exceed the value of 8.
A full orbit test-particle approach is used to study the collisional transport of impurity (carbon) ions in spherical tokamak (ST) plasmas with transonic and subsonic toroidal flows. The efficacy of this approach is demonstrated by reproducing the results of classical transport theory in the large aspect ratio limit. The equilibrium parameters used in the ST modelling are similar to those of plasmas in the MAST experiment. The effects on impurity ion confinement of both counter-current and co-current rotation are determined. Various majority ion density and temperature profiles, approximating measured profiles in rotating and non-rotating MAST plasmas, are used in the modelling. It is shown that transonic rotation (both counter-current and co-current) has the effect of reducing substantially the confinement time of the impurity ions. This effect arises primarily because the impurity ions, displaced by the centrifugal force to the low field region of the tokamak, are subject to a collisional diffusivity that is greater than the flux surface-averaged value of this quantity (Helander 1998 Phys. Plasmas5 1209). For a given set of plasma profiles, the ions are found to be significantly less well confined in co-rotating plasmas than in counter-rotating plasmas. The poloidal distribution of losses exhibits a pronounced up/down asymmetry that is consistent with the direction of the net vertical drift of the impurity ions.
A stochastic magnetic boundary, produced by an applied edge resonant magnetic perturbation, is used to suppress most large edge-localized modes (ELMs) in high confinement (H-mode) plasmas. The resulting H mode displays rapid, small oscillations with a bursty character modulated by a coherent 130 Hz envelope. The H mode transport barrier and core confinement are unaffected by the stochastic boundary, despite a threefold drop in the toroidal rotation. These results demonstrate that stochastic boundaries are compatible with H modes and may be attractive for ELM control in next-step fusion tokamaks.
The penetration dynamics of the resonant magnetic perturbation (RMP) field is simulated in the full toroidal geometry, under realistic plasma conditions in MAST experiments. The physics associated with several aspects of the RMP penetration-the plasma response and rotational screening, the resonant and non-resonant torques and the toroidal momentum balance-are highlighted. In particular, the plasma response is found to significantly amplify the non-resonant component of the RMP field for some of the MAST plasmas. A fast rotating plasma, in response to static external magnetic fields, experiences a more distributed electromagnetic torque due to the resonance with continuum waves in the plasma. At fast plasma flow (such as for the MAST plasma), the electromagnetic torque is normally dominant over the neoclassical toroidal viscous (NTV) torque. However, at sufficiently slow plasma flow, the NTV torque can play a significant role in the toroidal momentum balance, thanks to the precession drift resonance enhanced, so-called superbanana plateau regime.
Magnetohydrodynamical instabilities such as Alfvén Eigenmodes and Neoclassical tearing modes redistribute energetic particles and, thus, potentially endanger the confinement of, e.g., fusion born alphas in Tokamaks. The orbit-following studies so far have been restricted either to time-independent approximation of the rotating modes, or to an axisymmetric magnetic field, which is an assumption severely compromised in ITER. In this paper we extend the previous work to accommodate time-dependent modes in non-axisymmetric magnetic fields.
A method is given for expressing the magnetic field strength in magnetic coordinates for a given field. This expression is central to the study of equilibrium, stability, and transport in asymmetric plasmas.
Using a ‘‘Monte Carlo interpretation’’ of particle simulations, a general description of low-noise techniques, such as the δf method, is developed in terms well-known Monte Carlo variance reduction methods. Some of these techniques then are applied to linear and nonlinear studies of pure electron plasmas in cylindrical geometry, with emphasis on the generation and nonlinear evolution of electron vortices. Long-lived l=1 and l=2 vortices, and others produced by unstable diocotron modes in hollow profiles, are studied. It is shown that low-noise techniques make it possible to follow the linear evolution and saturation of even the very weakly unstable resonant diocotron modes.
A Hamiltonian guiding center drift orbit formalism is developed which permits the efficient calculation of particle trajectories in magnetic field configurations of arbitrary cross section with arbitrary plasma ..beta... The magnetic field is assumed to be a small perturbation from a zero-order ''equilibrium'' field possessing magnetic surfaces. The equilibrium field, possessing helical or toroidal symmetry, can be modeled analytically or obtained numerically from equilibrium codes. The formalism is used to study trapped particle precession. Finite banana width corrections to the toroidal precession rate are derived, and the bounce averaged trapped particle motion is expressed in Hamiltonian form. Particle drift-pumping associated with the ''fishbone'' oscillation is investigated. A numerical code based on the formalism is used to study particle orbits in circular and bean-shaped tokamak configurations.
Ideal magnetohydrodynamic theory and its application to magnetic fusion systems are reviewed. The review begins with a description and derivation of the model as well as a discussion of the region of validity. Next, the general properties are derived which are valid for arbitrary geometry and demonstrate the inherently sound physical foundation of the model. The equilibrium behavior of the currently most promising toroidal magnetic fusion concepts are then discussed in detail. Finally, the stability of such equilibria is investigated. Included are discussions of the general stability properties of arbitrary magnetic geometries and of detailed applications to those concepts of current fusion interest.
An energy principle to obtain the solution of the magnetohydrodynamic (MHD) equilibrium equation J Vector x B Vector - del p = 0 for nested magnetic flux surfaces that are expressed in the inverse coordinate representation x Vector = x Vector (rho, theta, zeta) was used. Here, theta and zeta are poloidal and toroidal flux coordinate angles, respectively, and p = p(rho) labels a magnetic surface. Ordinary differential equations in rho are obtained for the Fourier amplitudes (moments) in the doubly periodic spectral decomposition of x Vector. A steepest descent iteration is developed for efficiently solving these nonlinear, coupled moment equations. The existence of a positive definite energy functional guarantees the monotonic convergence of this iteration toward an equilibrium solution (in the absence of magnetic island formation). A renormalization parameter lambda is introduced to ensure the rapid convergence of the Fourier series for x Vector, while simultaneously satisfying the MHD requirement that magnetic field lines are straight in flux coordinates. A descent iteration is also developed for determining the self consistent value for lambda.
A new charged particle orbit following code HECTOR is described. The code simulates the behaviour of thermal particles and high energy particles, such as those resulting from the ICRF wave field interactions or from thermonuclear reactions within the confining magnetic fields of non-circular axisymmetric tokamak plasmas. The particle trajectories are traced using a new, fast, and efficient hybrid orbit-following scheme, based upon the drift equations in the guiding centre approximation and the constants of motion. The Monte Carlo technique is used to describe the Coulomb scattering processes of dynamical friction, pitch angle scattering, energy diffusion, and the ICRF interaction processes. The code is specifically designed to operate within the experimental environment.
The adiabatic theory of charged-particle motion is developed ; systematically. The general expressions for guiding-center motion and particle ; energy change are given, with application to the Van Allen radiation and to Fermi ; acceleration. It is shown that Fermi acceleration and betatron acceleration ; should not be regarded as distinct processes. Modifications of the ; nonrelativistic theory that are necessary when the particle is relativistic are ; discussed. Proofs are given of the invariance to lowest order of the first and ; second adiabatic invariants for the case of static fields. Finally, applications ; are made to the theory of plasmas. (auth)
The equations for particle drift orbits are given in a new magnetic coordinate system. This form of the equations separates the fast motion along the magnetic field lines from the slow motion across the lines. In addition, less information is required about the magnetic field structure than in alternative forms of the drift equations.
A Hamiltonian formulation of the guiding-center drift in arbitrary, steady-state, magnetic and electric fields is given. The canonical variables of this formulation are simply related to the magnetic coordinates. The modifications required to treat a large class of ergodic magnetic fields using magnetic coordinates are explicitly given in the Hamiltonian formulation.
Guiding center equations for particle motion in a toroidal magnetic configuration are derived using general magnetic coordinates. For the case of axisymmetry, the explicit transformation to exact Hamiltonian canonical variables is presented for the first time. Approximate canonical coordinates are introduced also for three-dimensional configurations with strong toroidal magnetic field. Previous derivations made use of so-called Boozer equilibrium coordinates, which are highly nonuniform and are canonical only in the exceptional case of low beta, up–down symmetric configurations. The present formalism is valid for arbitrary, spatially well distributed magnetic coordinates, greatly increasing the accuracy of calculations. Magnetostatic equilibrium is not assumed in the present formalism, the analysis holds for any configuration with nested flux surfaces. © 2003 American Institute of Physics.
The traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented in a way which clearly displays their differential‐geometric foundations. These are then generalized to the case of noncanonical in phase space. In the new method the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p dq−Hdt. The perturbation analysis is applied to this entire form, in all of its components.
An elementary but rigorous derivation is given for a variational principle for guiding centre motion. The equations of motion resulting from the variational principle (the drift equations) possess exact conservation laws for phase volume, energy (for time-independent systems), and angular momentum (for azimuthally symmetric systems). The results of carrying the variational principle to higher order in the adiabatic parameter are displayed. The behaviour of guiding centre motion in azimuthally symmetric fields is discussed, and the role of angular momentum is clarified. The application of variational principles in the derivation and solution of gyrokinetic equations is discussed.
Orbits are considered in conventional stellarators (i.e. with helical coils) using Boozer co-ordinates. The Advanced Toroidal Facility (ATF) in Oak Ridge, Tennessee, will be used as an example to study the effects of its configurational flexibility on orbit topology. It is shown that the symplectic integration technique yields superior results for single particle orbits. These orbits will be compared with predictions using the J* invariant. J* conservation allows examination and understanding of the global stellarator topology, both with and without radial electric fields
The effects of edge-localized mode (ELM) mitigation coils (ELM coils) on the loss of NBI-produced fast ions and fusion-produced alpha particles are investigated using an orbit following Monte Carlo code. The ELM mitigation coil field (EMC field) may cause a significant loss of fast ions produced by NBI on the order of 16.0–17.0% for a 9 MA steady-state ITER scenario. A significant transit-particle loss occurs in the case of the toroidal mode number n = 4 in which magnetic surfaces are ergodic near the plasma periphery. When the number of ELM coils in each toroidal row is nine, the main toroidal mode n = 4 is accompanied by a complementary mode nc = 5. Concerning the resonance of fast-ion trajectories, the anti-resonant surfaces of n = 4 are very close to the resonant surfaces of nc = 5 and vice versa. Since the effect of resonance on fast-ion trajectories dominates that of anti-resonance, a synergy effect of the main and complementary modes effectively enlarges the resonant regions. In a single n-mode EMC field, the resonant and anti-resonant regions are well separated. The peak heat load due to the loss of NB-produced fast ions near the upper ELM coils is as high as 1.0–1.5 MW m−2, which exceeds the allowable level in ITER. Rotation of the EMC field is essential for ITER to alleviate the local peak heat load. Most loss particles hit the inner side of the torus of the dome in the ITER divertor. The loss of alpha particles is also increased by the effect of the EMC field. The loss is still acceptably low at less than 1.0%.
A new set of exact canonical variables of guiding-center motion in an axisymmetric torus has been systematically identified. And an action-angle variable formalism has been established. In this new canonical Hamiltonian theory, the time scale of poloidal motion and the time scale of toroidal drift (precession) of the guiding-centers are well separated.
The problem of modelling the self-consistent interaction of an energetic particle ensemble with a wave spectrum specific to magnetically confined plasmas in a torus is discussed. Particle motion in a magnetic field coordinate system, whose surfaces are perturbed by a spectrum of finite amplitude magnetohydrodynamical (MHD) waves, is described using a Hamiltonian formulation. Employing the δƒ method enables the simulation particles to only represent the change in the total particle distribution function and consequently possesses significant computational advantages over standard techniques. Changes to the particle distribution function subsequently affect the wave spectrum through wave-particle interactions. The model is validated using large aspect-ratio asymptotic limits as well as through a comparison with other numerical work. A consideration of the Kinetic Toroidal Alfvén Eigenmode instability driven by fusion born α-particles in a D-T JET plasma illustrates a use of the code and demonstrates nonlinear saturation of the instability, together with the resultant redistribution of particles both in energy and across the plasma cross section.
A new linear MHD stability code MINERVA is developed for investigating a toroidal rotation effect on the stability of ideal MHD modes in tokamak plasmas. This code solves the Frieman–Rotenberg equation as not only the generalized eigenvalue problem but also the initial value problem. The parallel computing method used in this code realizes the stability analysis of both long and short wavelength MHD modes in short time. The results of some benchmarking tests show the validity of this MINERVA code. The numerical study with MINERVA about the toroidal rotation effect on the edge MHD stability shows that the rotation shear destabilizes the intermediate wavelength modes but stabilizes the short wavelength edge localized MHD modes, though the rotation frequency destabilizes both the long and the short wavelength MHD modes.
Free boundary three-dimensional anisotropic pressure magnetohydrodynamic equilibria with nested magnetic flux surfaces are computed through the minimisation of the plasma energy functional $W={\int}_{V}{d^3}x\left[{B^2}/(2\mu_0)+p_{||}/(\Gamma-1)\right]$. The plasma–vacuum interface is varied to guarantee the continuity of the total pressure $\left[{p}_{\perp}+{B^2}/(2\mu_0)\right]$ across it and the vacuum magnetic field must satisfy the Neumann boundary condition that its component normal to this interface surface vanishes. The vacuum magnetic field corresponds to that driven by the plasma current and external coils plus the gradient of a potential function whose solution is obtained using a Green's function method. The energetic particle contributions to the pressure are evaluated analytically from the moments of the variant of a bi-Maxwellian distribution function that satisfies the constraint ${\bf B\cdot\nabla}{\cal F}_h=0$. Applications to demonstrate the versatility and reliability of the numerical method employed have concentrated on high-β off-axis energetic particle deposition with large parallel and perpendicular pressure anisotropies in a 2-field period quasiaxisymmetric stellarator reactor system. For large perpendicular pressure anisotropy, the hot particle component of the pperpendicular distribution localises in the regions where the energetic particles are deposited. For large parallel pressure anisotropy, the pressures are more uniform around the flux surfaces.
A magnetohydrodynamic (MHD) model is applied to the problem of the stability of magnetically confined ther monuclear plasmas of interest in the pursuit of fusion power. Previous studies limited to two-dimensional con figurations are here generalized to three-dimensional toroidal plasmas. Using finite Fourier representations in the angle coordinates and finite hybrid elements in the radial direction, we solve the discretized Euler-Lagrange equations to determine the linear stability properties of the plasma.
The classical adiabatic motion of charged particles in strong magnetic fields, otherwise known as guiding center motion, exhibits an anholonomic phase similar to Berry's phase and the phase discovered in a general context by Hannay. Analysis of this effect reveals that there often is no best way to define the gyrophase when magnetic field lines are curved. Instead, a change in the definition of the phase is a kind of gauge transformation, the study of which leads to new insight into guiding center theory. The path-dependent phase that occurs in this problem is coupled with the metrical structure of physical space, giving rise to a transport process for vectors and frames which is similar to parallel transport in non-Euclidean geometry. Strong analogies with Fermi-Walker transport and Thomas precession in special relativity are pointed out.
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