Small Larmor Radius Research Articles

Convergence analysis of asymptotic preserving schemes for strongly magnetized plasmas

The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes (Filbet and Rodrigues in SIAM J. Numer. Anal. 54(2):1120–1146, 2016) for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-in-Cell methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time and space) is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization, stiffness parameter, initial data and time.

Collisionless particle dynamics in trap sections with helical corrugation

Collisionless particle dynamics in a helical magnetic mirror is investigated by methods of the Hamilton dynamics. In the limit of small Larmor radius, the Hamiltonian is fully integrable and particle trajectories can be divided into three types, differing in the direction of the drift velocity. The structure of the trajectories of resonant particles and the width of the resonance are studied. The influence of weak axi-symmetrical corrugation on the particle dynamics and efficiency of plasma flow suppression through a helical mirror are discussed.

Numerical simulation of the plasma acceleration process in a magnetically enhanced micro-cathode vacuum arc thruster

A particle-in-cell simulation is conducted to investigate the plasma acceleration process in a micro-cathode vacuum arc thruster. A coaxial electrode structure thruster with an applied magnetic field configuration is used to investigate the effects of the distribution of the magnetic field on the acceleration process and the mechanism of electrons and ions. The modeling results show that due to the small Larmor radius of electrons, they are magnetized and bound by the magnetic field lines to form a narrow electron channel. Heavy ions with a large Larmor radius take a long time to keep up with the electron movement. The presence of a magnetic field strengthens the charge separation phenomenon. The electric field caused by the charge separation is mainly responsible for the ion acceleration downstream of the computation. The impact of variations in the distribution of the magnetic field on the acceleration of the plasma is also investigated in this study, and it is found that the position of the magnetic coil relative to the thruster exit has an important impact on the acceleration of ions. In order to increase the axial velocity of heavy ions, the design should be considered to reduce the confinement of the magnetic field on the electrons in the downstream divergent part of the applied magnetic field.

Short wavelength ion temperature gradient mode in tokamak plasmas with hollow density profiles

By the gyrokinetic integral eigenmode equation, we numerically investigate the short-wavelength ion-temperature-gradient (SWITG) mode in tokamak plasmas with hollow density profiles. This paper finds that the critical ion temperature gradient (ITG) R/LTiC exists for the hollow density profile. Over such a gradient, the SWITG mode is unstable. The R/LTiC for a hollow density profile is slightly lower than that for a peaked density profile, though far away from the threshold, the mode in the latter case is more unstable. Furthermore, the scaling of the ITG threshold increasing with ion-to-electron temperature ratio Ti/Te has been calculated, suggesting that the SWITG mode is more difficult to excite in hot ion plasmas as compared with that in hot electron plasmas. For the slightly hollow density profiles, it is shown that in the case of flat (peaked) electron temperature profiles, the SWITG modes are destabilized (stabilized) by the trapped electrons. For the steep hollow density profiles, however, the trapped electrons reduce the growth rate substantially, and thus, the SWITG stability threshold is raised significantly. In addition, the dependence of the threshold on local plasma parameters is analyzed in detail. It is found that the SWITG modes become more unstable for the high safety factor, small Larmor radius, weak magnetic shear, and high electron-to-ion temperature ratio.

Analysis of the plasma blob formation/transport and its effects on impurity transport in the scrape‐off layer regions

The purpose of this study was to clarify the effects of a plasma blob on the impurity transport in the scrape‐off layer (SOL) and divertor region of fusion devices. As a first step, plasma blob formation and transport were simulated by a two‐dimensional interchange turbulence model. Its effects on impurity transport were studied by the impurity equations of motion in the trace impurity limit. The results show that a blob is produced by the interchange instability and transported radially towards the wall by the E × B drift due to the polarization electric field by the charge separation between ions and electrons inside the blob due to their magnetic drifts. The polarization electric field also induces the impurity radial E × B drift and the resultant impurity radial transport at least for impurities with a small Larmor radius in comparison with the blob size. These first results provide us with a robust basis for a more systematic study (e.g., dependence of impurity mass and charge state) and for a more quantitative evaluation of the effects of the blob on the impurity transport in the SOL and divertor region.

Neoclassical, semi-collisional tearing mode theory in an axisymmetric torus

A set of layer equations for determining the stability of semi-collisional tearing modes in an axisymmetric torus, incorporating neoclassical physics, in the small ion Larmor radius limit, is provided. These can be used as an inner layer module for inclusion in numerical codes that asymptotically match the layer to toroidal calculations of the tearing mode stability index, $\unicode[STIX]{x1D6E5}^{\prime }$. They are more complete than in earlier work and comprise equations for the perturbed electron density and temperature, the ion temperature, Ampère’s law and the vorticity equation, amounting to a twelvth-order set of radial differential equations. While the toroidal geometry is kept quite general when treating the classical and Pfirsch–Schlüter transport, parallel bootstrap current and semi-collisional physics, it is assumed that the fraction of trapped particles is small for the banana regime contribution. This is to justify the use of a model collision term when acting on the localised (in velocity space) solutions that remain after the Spitzer solutions have been exploited to account for the bulk of the passing distributions. In this respect, unlike standard neoclassical transport theory, the calculation involves the second Spitzer solution connected with a parallel temperature gradient, because this stability problem involves parallel temperature gradients that cannot occur in equilibrium toroidal transport theory. Furthermore, a calculation of the linearised neoclassical radial transport of toroidal momentum for general geometry is required to complete the vorticity equation. The solutions of the resulting set of equations do not match properly to the ideal magnetohydrodynamic (MHD) equations at large distances from the layer, and a further, intermediate layer involving ion corrections to the electrical conductivity and ion parallel thermal transport is invoked to achieve this matching and allow one to correctly calculate the layer $\unicode[STIX]{x1D6E5}^{\prime }$.

Quasilinear diffusion coefficients in a finite Larmor radius expansion for ion cyclotron heated plasmas

In this paper, a reduced model of quasilinear velocity diffusion by a small Larmor radius approximation is derived to couple the Maxwell's equations and the Fokker Planck equation self-consistently for the ion cyclotron range of frequency waves in a tokamak. The reduced model ensures the important properties of the full model by Kennel-Engelmann diffusion, such as diffusion directions, wave polarizations, and H-theorem. The kinetic energy change (Ẇ) is used to derive the reduced model diffusion coefficients for the fundamental damping (n = 1) and the second harmonic damping (n = 2) to the lowest order of the finite Larmor radius expansion. The quasilinear diffusion coefficients are implemented in a coupled code (TORIC-CQL3D) with the equivalent reduced model of the dielectric tensor. We also present the simulations of the ITER minority heating scenario, in which the reduced model is verified within the allowable errors from the full model results.

A new quasilinear formulation for ICRF plasmas in a toroidal geometry

We present a new formulation for quasilinear velocity space diffusion for ICRF plasmas that considers two different aspects: (1) finite Larmor radius approximation and (2) includes the effect of toroidal geometry and constructs a positive definite form. In the first aspect, the Kennel-Engelmann (K-E) quasilinear diffusion coefficients are successfully approximated in a small Larmor radius limit and implemented for the numerical codes (TORIC-CQL3D). In the second aspect, the quasilinear diffusion is reformulated in a toroidal geometry in order to include the parallel dynamics in the inhomogeneous plasmas and magnetic fields. We use these two quasilinear formulations to simulate ITER plasmas with ICRF injection for minority fundamental heating and Tritium second harmonic cyclotron heating.

Transport and entry of plasma clouds/jets across transverse magnetic discontinuities: Three‐dimensional electromagnetic particle‐in‐cell simulations

AbstractIn this paper we use three‐dimensional electromagnetic particle‐in‐cell simulations to investigate the interaction of a small Larmor radius plasma cloud/jet with a transverse nonuniform magnetic field typical to a tangential discontinuity in a parallel geometry. The simulation setup corresponds to an idealized, yet relevant, magnetospheric configuration likely to be observed at the magnetopause during northward orientation of the interplanetary magnetic field. The numerical simulations are adapted to study the kinetic effects and their role on the transport and entry of localized plasma jets similar to those identified inside the Earth's magnetosheath propagating toward the magnetopause. The simulations reveal the formation of a perpendicular polarization electric field inside the main bulk of the plasma cloud that enables its forward transport and entry across the transverse magnetic field. The jet is able to penetrate the transition region when the height of the magnetic barrier does not exceed a certain critical threshold. Otherwise, the forward transport along the injection direction is stopped before full penetration of the magnetopause. Moreover, the jet is pushed back and simultaneously deflected in the perpendicular plane to the magnetic field. Our simulations evidence physical processes advocated previously by the theoretical model of impulsive penetration and revealed in laboratory experiments.

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.

Velocity boundary conditions at a tokamak resistive wall

Velocity boundary conditions appropriate for magnetohydrodynamic simulations have been controversial recently. A comparison of numerical simulations of sideways wall force in disruptions is presented for Dirichlet, Neumann, Robin, and DEBS boundary conditions. It is shown that all the boundary conditions give qualitatively similar results. It is shown that Dirichlet boundary conditions are valid in the small Larmor radius limit of electromagnetic sheath boundary conditions.

Thermal force drift wave

A drift instability of a collisional magnetized plasma, unstable due to the Braginskii thermal force but not requiring any direct dissipation such as resistivity or electron inertia, is examined. Unlike conventional drift-modes, the maximum growth rate of the thermal force drift wave (TFDW) is of order the drift frequency, making for a strongly turbulent nonlinear state. A 3D, magnetized two-fluid code is developed to allow the study of both ideal MHD modes as well as lower frequency drift modes. The governing equations are essentially the ideal MHD equations with the inclusion of Hall and thermal force terms in Ohm’s law. This set of equations is reduced in a finite β, long parallel wavelength, and small but significant Larmor radius ordering and tested for shear Alfven waves, parallel sound waves, and drift modes. The code is employed to recover the TFDW instability, to verify the code against the mode’s analytic linear characteristics, and to study the nonlinear behavior of the TFDW. The TFDW growth is strongly suppressed by parallel thermal conduction and thus this mode is more likely to be observed in low temperature plasmas.

COSMIC-RAY TRANSPORT THEORY IN PARTIALLY TURBULENT SPACE PLASMAS WITH COMPRESSIBLE MAGNETIC TURBULENCE

Recently a new transport theory of cosmic rays in magnetized space plasmas extending the quasilinear approximation to the particle orbit has been developed for the case of an axisymmetric incompressible magnetic turbulence. Here we generalize the approach to the important physical case of a compressible plasma. As previously obtained in the case of an incompressible plasma we allow arbitrary gyrophase deviations from the unperturbed spiral orbits in the uniform magnetic field. For the case of quasi-stationary and spatially homogeneous magnetic turbulence we derive in the small Larmor radius approximation gyro-phase averaged cosmic ray Fokker-Planck coefficients. Upper limits for the perpendicular and pitch-angle Fokker-Planck coefficients and for the perpendicular and parallel spatial diffusion coefficients are presented.

Investigation of tearing instability using GeFi particle simulation model

The gyrokinetic (GK) electron and fully kinetic ion (GeFi) simulation model of Lin et al. [Plasmas Phys. Controlled Fusion 53, 054013 (2011)] has been thoroughly benchmarked and validated for a two-dimensional (2D) Harris current sheet with a finite guide field. First, a gyrokinetic eigenmode theory for the collisionless tearing mode in the small Larmor radius limit is presented. The linear eigenmode structure and growth rate of the tearing mode obtained from the GeFi simulation are benchmarked against those from the GK eigenmode analysis in the limit of L≫ρi>ρe, where L is the current sheet half-width, ρi is ion Larmor radius, and ρe is electron Larmor radius. Second, to valid the GeFi model, both the linear and nonlinear tearing instabilities obtained from the GeFi simulations are compared with the Darwin particle-in-cell (PIC) simulation. The validation of the GeFi model for laboratory and space plasmas is also discussed. Meanwhile, the GeFi simulation is carried out to investigate both the linear and nonlinear tearing instabilities for cases with a broad range of L and guide magnetic field BG. It is found that in a wide current sheet with L > 4.5ρeK, the nonlinear saturation level of the island half-width is ws ≃ 3ρeK, where ρeK = ρeB0/Bx0, B0 is the strength of the asymptotic magnetic field, and Bx0 is the antiparallel field. On the other hand, in a thin current sheet with L < 2.5ρeK, ws ≃ 2.2 L. In addition, a high frequency electrostatic drift mode is found to coexist with the tearing mode.

A NEW COSMIC RAY TRANSPORT THEORY IN PARTIALLY TURBULENT SPACE PLASMAS: EXTENDING THE QUASILINEAR APPROACH

A new transport theory of cosmic rays in magnetized space plasmas with axisymmetric incompressible magnetic turbulence is developed extending the quasilinear approximation to the particle orbit. Arbitrary gyrophase deviations from the unperturbed spiral orbits in the uniform magnetic field are allowed. For quasi-stationary and spatially homogeneous magnetic turbulence, we derive the small Larmor radius approximation gyrophase-averaged cosmic ray Fokker-Planck coefficients. The generalized Fokker-Planck coefficients correctly reduce to their known quasilinear values in the corresponding limit. New forms of the quasilinear Fokker-Planck coefficients in axisymmetric turbulence are derived which no longer involve infinite sums of products of Bessel functions, which facilitate their numerical computation for specified turbulence field correlation tensors. The Fokker-Planck coefficients for arbitrary phase orbits of the cosmic ray particles provide strict upper limits for the perpendicular and pitch-angle Fokker-Planck coefficients, which in turn yield strict upper and lower limits for the perpendicular and parallel spatial diffusion coefficients, respectively, describing the spatial diffusion of the isotropic part of the cosmic ray phase space density. For the associated mean free paths, we find for this general case that the product of the minimum parallel mean free path with the sum of the maximum perpendicular mean free paths equals R 2 L , where RL denotes the cosmic ray gyroradius.

Characterization of a class of stellarator steady states

A stellarator steady state is obtained for a specific class of magnetic fields by a formal expansion in the small Larmor radius parameters of the coupled ion-electron Fokker–Planck equations. A system of relatively simple ordinary differential equations is given to determine the plasma profile functions, the number density, the temperature, and the electrostatic potential. A particular low collisionality ordering is used. The magnetic field is assumed to have stellarator symmetry of N periods in the toroidal direction and is approximated by a closed magnetic line configuration with rotational transform N/R. The magnetic field is nearly quasisymmetric. The chosen magnetic field also includes a small additional component leading to a configuration without closed lines or closed flux surfaces. The theoretical logic behind this choice of magnetic fields is also presented.

Transport of electrons in the tunnel of an ion sensitive probe

Ion sensitive probes serve to measure the ion temperature in magnetized plasma. Such a probe typically consists of a collector submerged inside a hollow tube, which is oriented perpendicularly to the magnetic field. The principle of the probe is based on geometrical shielding of the ion collector from plasma electrons. According to the basic theory, when the collector is retracted in the tube, electrons with their small Larmor radii should not be able to reach it and the collector becomes sensitive to ions. However, experimental results show that the electron shielding is in general inefficient, it only works in the case when the potential of the collector is the same as the potential of the inside surface of the tube.This problem is investigated using a full 3D particle-in-cell Cartesian code with a fast multigrid Poisson solver. We simulate the plasma behaviour in the vicinity of a model of the ion sensitive probe. A positive potential structure is formed at the entrance of the tube due to the space charge of ions that gyrate inside. This structure produces E × B drifts, which push electrons into the shielded space. A stream of electrons hitting the collector is observed for various potentials of the collector. Simulations revealed that electrons can penetrate inside the geometrical shadow in all studied cases; however, they do not reach the collector when the potential of the collector is equal to the potential of the tube.

Flute-Mode Stability of Quadrupole-Anchored Tandem Mirror Plasmas

We study the flute-mode stability of quadrupole-anchored tandem mirror plasmas. The present analysis is based on Newcomb's Lagrangian density with an assumption of small Larmor radius of ions for paraxial approximation. A radial eigenmode equation for flute perturbations is derived without an eikonal approximation in flux coordinates, where effects of E × B plasma rotation due to an ambipolar electric field are considered. The obtained eigenmode equation is applicable to a mode with an arbitrary azimuthal mode number and can describe interchange, E × B rotational, and Kelvin-Helmholtz modes driven by the shear effect of E × B plasma rotation. A quadratic dispersion equation in perturbation frequency ω is derived based on a simplified cylindrical plasma model and is used to discuss the flute stability of quadrupole-anchored GAMMA 10 tandem mirror plasmas, where the anchor beta required for plasma stability is calculated for the m = 1 and 2 modes for different values of central-cell ambipolar potential.

Weakly relativistic dielectric tensor for arbitrary wavenumbers

The validity of Shkarofsky’s dielectric tensor is extended by taking the strictly weakly relativistic limit and removing, when possible, assumptions on the wavenumbers along and across the ambient magnetic field, k∥ and k⊥. An approximation of the time integral is retained, but is shown to be valid under more benign assumptions than those of quasiperpendicular incidence and small Larmor radius. The increased generality with respect to k∥ permits to handle cases of comparable Doppler and relativistic widths of electron cyclotron resonances. The tensor also suits Bernstein waves, as it captures both their natural large k⊥ and the finite k∥ that is typical of some mode conversions, or acquired as a consequence of the large k⊥ when propagating in curved magnetic fields. Finally, relativistic corrections to the optimal angle for the ordinary-extraordinary Bernstein mode conversion are presented.

Numerical study of global stability of oblate field-reversed configurations

Global stability of the oblate (small elongation, E<1) Field-Reversed Configuration (FRC) has been investigated numerically using both three-dimensional magnetohydrodynamic (MHD) and hybrid (fluid electrons and kinetic ions) simulations. For every nonzero value of the toroidal mode number n, there are three MHD modes that must be stabilized. For n=1, these are the interchange, the tilt and the radial shift; while for n>1 these are the interchange and two co-interchange modes with different polarization. It is shown that the n=1 tilt mode becomes an external mode when E<1, and it can be effectively stabilized by close-fitting conducting shells, even in the small Larmor radii (MHD) regime. The tilt mode stability improves with increasing oblateness, however at sufficiently small elongations the radial shift mode becomes more unstable than the tilt mode. The interchange mode stability is strongly profile dependent, and all n⩾1 interchange modes can be stabilized for a class of pressure profile with separatrix beta larger than 0.035. Our results show that all three n=1 modes can be stabilized in the MHD regime, but the stabilization of the n>1 co-interchange modes still remains an open question.