Structure-preserving algorithms for guiding center dynamics based on the slow manifold of classical Pauli particle

The classical Pauli particle(CPP) serves as a slow manifold, substituting the conventional guiding center dynamics. Based on the CPP, we utilize the averaged vector field(AVF) method in the computations of drift orbits. Demonstrating significantly higher efficiency, this advanced method is capable of accomplishing the simulation in less than one-third of the time of directly computing the guiding center motion. In contrast to the CPP-based Boris algorithm, this approach inherits the advantages of the AVF method, yielding stable trajectories even achieved with a tenfold time step and reducing the energy error by two orders of magnitude. By comparing these two CPP algorithms with the traditional RK4 method, the numerical results indicate a remarkable performance in terms of both the computational efficiency and error elimination. Moreover, we verify the properties of slow manifold integrators and successfully observe the bounce on both sides of the limiting slow manifold with deliberately chosen perturbed initial conditions. To evaluate the practical value of the methods, we conduct simulations in non-axisymmetric perturbation magnetic fields as part of the experiments,demonstrating that our CPP-based AVF method can handle simulations under complex magnetic field configurations with high accuracy, which the CPP-based Boris algorithm lacks. Through numerical experiments, we demonstrate that the CPP can replace guiding center dynamics in using energy-preserving algorithms for computations, providing a new, efficient, as well as stable approach for applying structure-preserving algorithms in plasma simulations.

High-order field theory and a weak Euler–Lagrange–Barut equation for classical relativistic particle-field systems

In both quantum and classical field systems,conservation laws such as the conservation of energy and momentum are widely regarded as fundamental properties.A broadly accepted approach to deriving conservation laws is built using Noether's method.However,this procedure is still unclear for relativistic particle-field systems where particles are regarded as classical world lines.In the present study,we establish a general manifestly covariant or geometric field theory for classical relativistic particle-field systems.In contrast to quantum systems,where particles are viewed as quantum fields,classical relativistic particle-field systems present specific challenges.These challenges arise from two sides.The first comes from the mass-shell constraint.To deal with the mass-shell constraint,the Euler–Lagrange–Barut(ELB)equation is used to determine the particle's world lines in the four-dimensional(4D)Minkowski space.Besides,the infinitesimal criterion,which is a differential equation in formal field theory,is reconstructed by an integro-differential form.The other difficulty is that fields and particles depend on heterogeneous manifolds.To overcome this challenge,we propose using a weak version of the ELB equation that allows us to connect local conservation laws and continuous symmetries in classical relativistic particle-field systems.By applying a weak ELB equation to classical relativistic particle-field systems,we can systematically derive local conservation laws by examining the underlying symmetries of the system.Our proposed approach provides a new perspective on understanding conservation laws in classical relativistic particle-field systems.

Discovering exact,gauge-invariant,local energy–momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas.However,exact local energy–momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite continuous effort.Without such local conservation laws,energy and momentum can be instantaneously transported across spacetime,which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory.The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic field reside on different manifolds.To overcome this difficulty,we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact,local conservation laws,in particular the energy–momentum conservation laws,for the electromagnetic gyrokinetic system.A weak Euler–Lagrange(EL)equation is established to replace the standard EL equation for the particles.It is discovered that an induced weak EL current enters the local conservation laws,and it is the new physics captured by the high-order field theory on heterogeneous manifolds.A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy–momentum conservation laws electromagnetic gauge-invariant.

Particle simulations on propagation and resonance of lower hybrid wave launched by phased array antenna in linear devices

In this work, we performed first-principles electromagnetic-kinetic simulations to study a phased antenna array and its interaction with deuterium plasmas within the lower hybrid range of frequency. We first gave wave accessibility and resonance results, which agree well with theoretical prediction. In addition, we further investigated the antenna power spectrum with different antenna phases in the presence of the plasma and compared it with that in a vacuum,which directly indicates wave coupling and plasma absorption. Furthermore, for the case with zero phasing difference, our simulation results show that, albeit the launch is away from the accessibility region, tunneling effect and mode conversion occurred, which enhanced coupling and absorption. Moreover, consistent interactions between the injected wave and the plasma concerning various antenna phase differences are shown. We presented the inchoate response of the plasma in terms of the launching directions. Our results could be favorable for the engineering design of wave heating experiments with a tunable phased antenna array in linear devices, such as simple magnetic mirrors or tandem mirrors.

Explicit structure-preserving geometric particle-in-cell algorithm in curvilinear orthogonal coordinate systems and its applications to whole-device 6D kinetic simulations of tokamak physics

Explicit structure-preserving geometric particle-in-cell(PIC)algorithm in curvilinear orthogonal coordinate systems is developed.The work reported represents a further development of the structure-preserving geometric PIC algorithm achieving the goal of practical applications in magnetic fusion research.The algorithm is constructed by discretizing the field theory for the system of charged particles and electromagnetic field using Whitney forms,discrete exterior calculus,and explicit non-canonical symplectic integration.In addition to the truncated infinitely dimensional symplectic structure,the algorithm preserves exactly many important physical symmetries and conservation laws,such as local energy conservation,gauge symmetry and the corresponding local charge conservation.As a result,the algorithm possesses the long-term accuracy and fidelity required for first-principles-based simulations of the multiscale tokamak physics.The algorithm has been implemented in the Sym PIC code,which is designed for highefficiency massively-parallel PIC simulations in modern clusters.The code has been applied to carry out whole-device 6 D kinetic simulation studies of tokamak physics.A self-consistent kinetic steady state for fusion plasma in the tokamak geometry is numerically found with a predominately diagonal and anisotropic pressure tensor.The state also admits a steady-state subsonic ion flow in the range of 10 km s-1,agreeing with experimental observations and analytical calculations Kinetic ballooning instability in the self-consistent kinetic steady state is simulated.It is shown that high-n ballooning modes have larger growth rates than low-n global modes,and in the nonlinear phase the modes saturate approximately in 5 ion transit times at the 2%level by the E×B flow generated by the instability.These results are consistent with early and recent electromagnetic gyrokinetic simulations.

ISSDE:A Monte Carlo implicit simulation code based on Stratonovich SDE approach of Coulomb collision

A Monte Carlo implicit simulation program,Implicit Stratonovich Stochastic Differential Equations(ISSDE),is developed for solving stochastic differential equations(SDEs)that describe plasmas with Coulomb collision.The basic idea of the program is the stochastic equivalence between the Fokker-Planck equation and the Stratonovich SDEs.The splitting method is used to increase the numerical stability of the algorithm for dynamics of charged particles with Coulomb collision.The cases of Lorentzian plasma,Maxwellian plasma and arbitrary distribution function of background plasma have been considered.The adoption of the implicit midpoint method guarantees exactly the energy conservation for the diffusion term and thus improves the numerical stability compared with conventional Runge-Kutta methods.ISSDE is built with C++and has standard interfaces and extensible modules.The slowing down processes of electron beams in unmagnetized plasma and relaxation process in magnetized plasma are studied using the ISSDE,which shows its correctness and reliability.

Structure-preserving geometric particle-in-cell methods for Vlasov-Maxwell systems

Recent development of structure-preserving geometric particle-in-cell （PIC） algorithms for Vlasov-Maxwell systems is summarized. With the arrival of 100 petaflop and exaflop computing power, it is now possible to carry out direct simulations of multi-scale plasma dynamics based on first-principles. However, standard algorithms currently adopted by the plasma physics community do not possess the long-term accuracy and fidelity required for these large-scale simulations. This is because conventional simulation algorithms are based on numerically solving the underpinning differential （or integro-differential） equations, and the algorithms used in general do not preserve the geometric and physical structures of the systems, such as the local energy-momentum conservation law, the symplectic structure, and the gauge symmetry. As a consequence, numerical errors accumulate coherently with time and long-term simulation results are not reliable. To overcome this difficulty and to harness the power of exascale computers, a new generation of structure-preserving geometric PIC algorithms have been developed. This new generation of algorithms utilizes modem mathematical techniques, such as discrete manifolds, interpolating differential forms, and non-canonical symplectic integrators, to ensure gauge symmetry, space-time symmetry and the conservation of charge, energy-momentum, and the symplectic structure. These highly desired properties are difficult to achieve using the conventional PIC algorithms. In addition to summarizing the recent development and demonstrating practical implementations, several new results are also presented, including a structure-preserving geometric relativistic PIC algorithm, the proof of the correspondence between discrete gauge symmetry and discrete charge conservation law, and a reformulation of the explicit non-canonical symplectic algorithm for the discrete Poisson bracket using the variational approach. Numerical examples are given to verify the advantages of the structure- preserving geometric PIC algorithms in comparison with the conventional PIC methods.

Modeling of beam ions loss and slowing down with Coulomb collisions in EAST

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 Mac Donald 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 Pζ–Λspace.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.