Guiding Center Equations Research Articles

Hamiltonian structure of the guiding-center Vlasov–Maxwell equations

The Hamiltonian structure of the guiding-center Vlasov-Maxwell equations is presented in terms of a Hamiltonian functional and a guiding-center Vlasov-Maxwell bracket. The bracket, which is shown to satisfy the Jacobi identity exactly, is used to show that the guiding-center momentum and angular-momentum conservation laws can also be expressed in Hamiltonian form.

Spatially dependent modeling and simulation of runaway electron mitigation in DIII-D

New simulations with the Kinetic Orbit Runaway electron (RE) Code KORC show RE deconfinement losses to the wall during plasma scrape off are the primary current dissipation mechanism in DIII-D experiments with high-Z impurity injection, and not collisional slowing down. The majority of simulations also exhibit an increase in the RE beam energy due to acceleration by the induced toroidal electric field, even while the RE beam current is decreasing. In this study, KORC integrates RE orbits using the relativistic guiding center equations of motion, and incorporates time-sequenced, experimental reconstructions of the magnetic and electric fields and line integrated electron density to construct spatiotemporal models of electron and partially-ionized impurity transport in the companion plasma. Comparisons of experimental current evolution and KORC results demonstrate the importance of including Coulomb collisions with partially-ionized impurity physics, initial RE energy, pitch angle, and spatial distributions, and spatiotemporal electron and partially-ionized impurity transport. This research provides an initial quantification of the efficacy of RE mitigation via injected impurities, and identification of the critical role played by loss of confinement due to plasma scrape off on the inner wall as compared to the relatively slow collisional damping.

Modelling of nanometer scale dust grains in tokamak

AbstractDust poses a serious threat to tokamak operation and safety. It is important to study the behaviour of dust grains under tokamak's discharge conditions, which depends heavily on their size and charge. Existing simulations mainly address issues on dust grains with radii larger than 1 μm, in which case, the drift effect due to electromagnetic fields can be safely ignored. For nanometer scale dust grains, however, the drift effect becomes significant and a new model based on guiding‐centre system needs to be established. In this work, the NDS has been done under BOUT++ framework. The simulation contains two parts. Part one, NDS evaluates the charging and ablation processes of the dust grains. In the second part, the guiding‐centre orbits of dust particles are tracked in tokamak plasmas, whose parameters are obtained from BOUT++, a highly desirable C++ code package for performing parallel plasma fluid simulations with an arbitrary number of equations in 3D curvilinear coordinates. The orbit of nanodust dynamics is described by guiding centre equations for simplicity, and these equations are numerically solved by conventional fourth‐order Runge Kutta method. Simulations provide results such as trajectories and evolutions of dust particles with different sizes and velocities for different tokamak geometries. Results show tungsten dust grains with a radius of a few nanometers launched from outer midplane will oscillate before totally ablated in C‐Mod. The oscillation in this case is driven by the ion drag force. Larger Nanodust with a radius of 100 nm, on the contrary, cannot be completely constrained by the electromagnetic field. The high plasma temperature and density in the seperatrix region causes severe dust ablation, resulting in total ablation within several ms.

A computational model for exploring particle acceleration during reconnection in macroscale systems

A new computational model suitable for exploring the self-consistent production of energetic electrons during magnetic reconnection in macroscale systems is presented. The equations are based on the recent discovery that parallel electric fields are ineffective drivers of energetic particles during reconnection so that the kinetic scales which control the development of such fields can be ordered out of the equations. The resulting equations consist of a magnetohydrodynamic (MHD) backbone with the energetic component represented by macro-particles described by the guiding center equations. Crucially, the energetic component feeds back on the MHD equations so that the total energy of the MHD fluid and the energetic particles is conserved. The equations correctly describe the firehose instability, whose dynamics plays a key role in throttling reconnection and in controlling the spectra of energetic particles. The results of early tests of the model, including the propagation of Alfvén waves in a system with pressure anisotropy and the growth of firehose modes, establish that the basic algorithm is stable and produces reliable physics results in preparation for further benchmarking with particle-in-cell models of reconnection.

Eulerian variational formulations and momentum conservation laws for kinetic plasma systems

The Eulerian variational principle for the Vlasov-Poisson-Ampère system of equations in a general coordinate system is presented. The invariance of the action integral under an arbitrary spatial coordinate transformation is used to obtain the momentum conservation law and the symmetric pressure in a more direct way than using the translational and rotational symmetries of the system. Next, the Eulerian variational principle is given for the collisionless drift kinetic equation, where particles' phase-space trajectories in given electromagnetic fields are described by Littlejohn's guiding center equations [R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983)]. Then, it is shown that, in comparison with the conventional moment method, the invariance under a general spatial coordinate transformation yields a more convenient way to obtain the momentum balance as a three-dimensional vector equation in which the symmetric pressure tensor, the Lorentz force, and the magnetization current are properly expressed. Furthermore, the Eulerian formulation is presented for the extended drift kinetic system, for which, in addition to the drift kinetic equations for the distribution functions of all particle species, the quasineutrality condition and Ampère's law to determine the self-consistent electromagnetic fields are given. Again, the momentum conservation law for the extended system is derived from the invariance under the general spatial coordinate transformation. Besides, the momentum balances are investigated for the cases where the collision and/or external source terms are added to the Vlasov and drift kinetic equations.

Electron acceleration in a JET disruption simulation

Runaways are suprathermal electrons having sufficiently high energy to be continuously accelerated up to tens of MeV by a driving electric field (Connor and Hastie 1975 Nucl. Fusion 15 415). Highly energetic runaway electron (RE) beams capable of damaging the tokamak first wall can be observed after a plasma disruption (Reux et al 2015 Nucl. Fusion 55 129501). Therefore, it is of primary importance to fully understand their generation mechanisms in order to design mitigation systems able to guarantee safe tokamak operations. In a previous work, Sommariva et al (2018 Nucl. Fusion 58), a test particle tracker was introduced in the JOREK 3D non-linear MHD code and used for studying the electron confinement during a simulated JET-like disruption. It was found in Sommariva et al (2018 Nucl. Fusion 58) that relativistic electrons are not completely deconfined by the stochastic magnetic field taking place during the disruption thermal quench (TQ). This is due to the reformation of closed magnetic surfaces at the beginning of the current quench (CQ). This result was obtained neglecting the inductive electric field in order to avoid the unrealistic particle acceleration which otherwise would have happened due to the absence of collision effects. The present paper extends (Sommariva et al 2018 Nucl. Fusion 58) analysing test electron dynamics in the same simulated JET-like disruption using the complete electric field. For doing so, a simplified collision model is introduced in the particle tracker guiding center equations. We show that electrons at thermal energies can become RE during or promptly after the TQ due to a combination of three phenomena: a first REs acceleration during the TQ due to the presence of a complex MHD-induced electric field, particle reconfinement caused by the fast reformation of closed magnetic surfaces after the TQ and a secondary acceleration induced by the CQ electric field.

Conservative magnetic moment of runaway electrons and collisionless pitch-angle scattering

Recently, the validity of the guiding-center approach to model relativistic runaway electrons in tokamaks has been challenged by full-orbit simulations that demonstrate the breakdown of the standard magnetic moment conservation. In this paper, we derive a new expression for the magnetic moment of relativistic runaway electrons with , which is conserved significantly better than the standard one. The new result includes one of the second-order corrections in the standard guiding-center theory which, in case of runaway electrons with , can peculiarly be of the same order as the lowest-order term. The better conservation of the new magnetic moment also explains the collisionless pitch-angle-scattering effect observed in full-orbit simulations since it allows momentum transfer between the perpendicular and parallel directions when the runaway electron is accelerated by an electric field. While the derivation of the second-order correction to the magnetic moment in general case would require the full extent of the relativistic second-order guiding-center theory, we exploit the Lie-perturbation method at the limit which simplifies the computations significantly. Consequently, we present the corresponding guiding-center equations applicable to runaway electrons.

Centrifugal particle confinement in mirror geometry

The use of supersonic rotation of a plasma in mirror geometry has distinct advantages for thermonuclear fusion. The device is steady state, there are no disruptions, the loss cone is almost closed, sheared rotation stabilizes magnetohydrodynamic instabilities as well as plasma turbulence, there are no runaway electrons, and the coil configuration is simple. In this work, we examine the effect of rotation on mirror confinement using a full cyclotron orbit code. The full cyclotron simulations give a much more complete description of the particle energy distribution and losses than the use of guiding center equations. Both collisionless loss as a function of rotation and the effect of collisions are investigated. Although the cross field diffusion is classical, we find that the local rotating Maxwellian is increased to higher energy, increasing the fusion rate and also enhancing the radial diffusion. We find a loss channel not envisioned with a guiding center treatment, but a design can be chosen that can satisfy the Lawson criterion for ions. Of course, the rotation has a minimal effect on the alpha particle birth distribution, so there is initially loss through the usual loss cone, just as in a mirror with no rotation. However after this loss, the alphas slow down on the electrons with little pitch angle scattering until reaching low energy, so over half of the initial alpha energy is transferred to the electrons. The important problem of energy confinement, with losses primarily through the electron channel, is not addressed in this work. We also discuss the use of rotating mirror geometry to produce an ion thruster.

Toroidal regularization of the guiding center Lagrangian

In the Lagrangian theory of guiding center motion, an effective magnetic field B*=B+(m/e)v∥∇× b appears prominently in the equations of motion. Because the parallel component of this field can vanish, there is a range of parallel velocities where the Lagrangian guiding center equations of motion are either ill-defined or very badly behaved. Moreover, the velocity dependence of B* greatly complicates the identification of canonical variables and therefore the formulation of symplectic integrators for guiding center dynamics. This letter introduces a simple coordinate transformation that alleviates both these problems simultaneously. In the new coordinates, the Liouville volume element is equal to the toroidal contravariant component of the magnetic field. Consequently, the large-velocity singularity is completely eliminated. Moreover, passing from the new coordinate system to canonical coordinates is extremely simple, even if the magnetic field is devoid of flux surfaces. We demonstrate the utility of this approach in regularizing the guiding center Lagrangian by presenting a new and stable one-step variational integrator for guiding centers moving in arbitrary time-dependent electromagnetic fields.

Higher order Larmor radius corrections to guiding-centre equations and application to fast ion equilibrium distributions

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 Baños drift. The correct treatment of the guiding-centre push-forward with the Baños 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.

Simulation of runaway electrons, transport affected by J-TEXT resonant magnetic perturbation

The topology of a magnetic field and transport properties of runaway electrons can be changed by a resonant magnetic perturbation field. The J-TEXT magnetic topology can be effectively altered via static resonant magnetic perturbation (SRMP) and dynamic resonant magnetic perturbation (DRMP). This paper studies the effect of resonant magnetic perturbation (RMP) on the confinement of runaway electrons via simulating their drift orbits in the magnetic perturbation field and calculating the orbit losses for different runaway initial energies and different runaway electrons, initial locations. The model adopted is based on Hamiltonian guiding center equations for runaway electrons, and the J-TEXT magnetic turbulences and RMP are taken into account. The simulation indicates that the loss rate of runaway electrons is sensitive to the radial position of electrons. The loss of energetic runaway beam is dominated by the shrinkage of the confinement region. Outside the shrinkage region of the runaway electrons are lost rapidly. Inside the shrinkage region the runaway beam is confined very well and is less sensitive to the magnetic perturbation. The experimental result on the response of runaway transport to the application RMP indicates that the loss of runaway electrons is dominated by the shrinkage of the confinement region, other than the external magnetic perturbation.

Variational formulations of guiding-center Vlasov-Maxwell theory

The variational formulations of guiding-center Vlasov-Maxwell theory based on Lagrange, Euler, and Euler-Poincaré variational principles are presented. Each variational principle yields a different approach to deriving guiding-center polarization and magnetization effects into the guiding-center Maxwell equations. The conservation laws of energy, momentum, and angular momentum are also derived by Noether method, where the guiding-center stress tensor is now shown to be explicitly symmetric.

Asymptotically Stable Particle-In-Cell Methods for the Vlasov--Poisson System with a Strong External Magnetic Field

This paper deals with the numerical resolution of the Vlasov--Poisson system with a strong external magnetic field by particle-in-cell (PIC) methods. In this regime, classical PIC methods are subject to stability constraints on the time and space steps related to the small Larmor radius and plasma frequency. Here we propose an asymptotic-preserving PIC scheme which is not subjected to these limitations. Our approach is based on first- and higher-order semi-implicit numerical schemes already validated on dissipative systems [S. Boscarino, F. Filbet, and G. Russo, J. Sci. Comput., 2016, doi:10.1007/s10915-016-0168-y]. Additionally, when the magnitude of the external magnetic field becomes large, this method provides a consistent PIC discretization of the guiding-center equation, that is, an incompressible Euler equation in vorticity form. We propose several numerical experiments which provide a solid validation of the method and its underlying concepts.

Particle transport induced by electrostatic wave fluctuations

Particle transport driven by electrostatic waves at the plasma edge is numerically investigated, for large aspect ratio tokamaks, by considering a kinetic model derived from guiding-center equations of motion. Initially, the transport is estimated for trajectories obtained from differential equations for a wave spectrum generated by a dominant spatial mode and three time modes. Then, in case of infinite time modes, the differential equations of motion are used to introduce a symplectic map that allows to analyze the particle transport. The particle transport barriers are observed for spatial localized dominant perturbation and infinite modes. In presence of infinite spatial modes, periodic islands arise in between chaotic trajectories at the plasma edge.

Development of variational guiding center algorithms for parallel calculations in experimental magnetic equilibria

Structure-preserving algorithms obtained via discrete variational principles exhibit strong promise for the calculation of guiding center test particle trajectories. The non-canonical Hamiltonian structure of the guiding center equations forms a novel and challenging context for geometric integration. To demonstrate the practical relevance of these methods, a prototypical variational midpoint algorithm is applied to an experimental magnetic equilibrium. The stability characteristics, conservation properties and implementation requirements associated with the variational algorithms are addressed. Furthermore, computational run time is reduced for large numbers of particles by parallelizing the calculation to use general-purpose graphics processing unit hardware.

Particle energisation in a collapsing magnetic trap model: the relativistic regime

In solar flares, a large number of charged particles is accelerated to high energies. By which physical processes this is achieved is one of the main open problems in solar physics. It has been suggested that during a flare, regions of the rapidly relaxing magnetic field can form a collapsing magnetic trap (CMT) and that this trap may contribute to particle energisation.} In this Research Note we focus on a particular analytical CMT model based on kinematic magnetohydrodynamics. Previous investigations of particle acceleration for this CMT model focused on the non-relativistic energy regime. It is the specific aim of this Research Note to extend the previous work to relativistic particle energies. Particle orbits were calculated numerically using the relativistic guiding centre equations. We also calculated particle orbits using the non-relativistic guiding centre equations for comparison. For mildly relativistic energies the relativistic and non-relativistic particle orbits mainly agree well, but clear deviations are seen for higher energies. In particular, the final particle energies obtained from the relativistic calculations are systematically lower than the energies reached from the corresponding non-relativistic calculations, and the mirror points of the relativistic orbits are systematically higher than for the corresponding non-relativistic orbits. While the overall behaviour of particle orbits in CMTs does not differ qualitatively when using the relativistic guiding centre equations, there are a few systematic quantitative differences between relativistic and non-relativistic particle dynamics.

Guiding Center Orbit Following Calculation of Edge Particle and Heat Transport in Stochastic Magnetic Field

AbstractParticle and heat transport in tokamak edge is investigated by guiding center orbit following calculation. The guiding center equation is solved for both electrons and ions in the presence of magnetic perturbation. It is suggested that the remnants of the magnetic islands play a role in characterizing the radial transport. The transport coefficient is estimated which also demonstrate local structure in the vicinity of magnetic islands. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

NBI fast ion confinement in the helical core of MAST hybrid-like plasmas

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.

Guiding center equations of high accuracy

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.

Energy conservation tests of a coupled kinetic plasma–kinetic neutral transport code

An approach to coupling two kinetic particle codes for the simulation of neutral–plasma interactions in magnetic fusion devices is described. The behavior of the neutral atoms and molecules is modeled with a Monte Carlo code. The plasma species are simulated with a particle-in-cell code that integrates the guiding center equations of motion and computes a self-consistent electric field. The coupling algorithm is designed to conserve mass in the neutral–plasma exchanges to statistical accuracy. Although energy is not fully conserved due to the velocity dependence of the charge exchange cross section and the kinetic character of both species, the impact of this non-conservation on the overall simulation is negligible.