Electromagnetic Vlasov Research Articles

Kinetic instability of whistlers in electron beam-plasma systems

The whistlers in space plasmas and in magnetic fusion experiments are destabilized by beams of fast electrons. While the linear regime of instability is analytically tractable, in most practical cases, the instability operates at the saturated level during the stages of observation and measurement. The saturated states, however, involve nonlinear whistlers, which remain best accessible for analysis by kinetic simulations. Results of electromagnetic Vlasov simulations are presented, analyzing an anisotropic electron beam driven whistler instability. The simulations cover the initially unstable regime followed by a saturated or marginally stable regime. Both regimes are separated by an intermediate nonlinear regime during which the electron distribution undergoes a kinetically self-consistent modification. A linearly obtained generalized marginal stability condition is applied to the stabilized state. The condition obtained in its dispersive version shows the β|| at threshold and, in turn, the residual anisotropy, to be a function of the whistler mode number k.

Whistler heat flux instability governed interaction of anisotropic beam electrons in electromagnetic Vlasov simulations

The kinetic instability of whistlers in a warm plasma, arising from electron temperature anisotropy with respect to directions parallel and perpendicular to the magnetizing field, is studied. Whistlers resonantly interacting with the electron beams, for example, the fast electrons accelerated by strong parallel electric fields and the so-called runaway electrons in a tokamak, are strong players in the schema of thermalization of stellar winds and mitigation of fast electrons in tokamak disruption events. As an evidence of their role in runaway mitigation, most fusion plasma experiments are found to show a threshold magnetic field strength for the generation of runaways. In many of these examples, the faster primary runaways produce a secondary runaway beam having an avalanche-like non-thermal velocity distribution. The electromagnetic Vlasov simulations presented here self-consistently examine the collisionless interaction of anisotropic electron beams, including an avalanche-like beam distribution, with parallel propagating whistlers and dependence of this process on the magnetic field strength. Analysis of the interaction process includes comparison with the simulations done using more analytically accessible anisotropic bulk and beam electron distributions, namely, the bi-Maxwellian and bi-kappa, for the reference.

A parallel low-rank solver for the six-dimensional Vlasov–Maxwell equations

Continuum Vlasov simulations can be utilized for highly accurate modelling of fully kinetic plasmas. Great progress has been made recently regarding the applicability of the method in realistic plasma configurations. However, a reduction of the high computational cost that is inherent to fully kinetic simulations would be desirable, especially at high velocity space resolutions. For this purpose, low-rank approximations can be employed. The so far available low-rank solvers are restricted to either electrostatic systems or low dimensionality and can therefore not be applied to most space, astrophysical and fusion plasmas. In this paper we present a new parallel low-rank solver for the full six-dimensional electromagnetic Vlasov–Maxwell equations that can utilize distributed memory architectures. Special care is taken to ensure the conservation of mass and a good representation of Gauss's law. The low-rank Vlasov solver is applied to standard benchmark problems of plasma turbulence and magnetic reconnection and compared to the full grid method. It yields accurate results at significantly reduced computational cost.

Exact conservation laws for gauge-free electromagnetic gyrokinetic equations

The exact energy and angular momentum conservation laws are derived by the Noether method for the Hamiltonian and symplectic representations of the gauge-free electromagnetic gyrokinetic Vlasov–Maxwell equations. These gyrokinetic equations, which are solely expressed in terms of electromagnetic fields, describe the low-frequency turbulent fluctuations that perturb a time-independent toroidally-axisymmetric magnetized plasma. The explicit proofs presented here provide a complete picture of the transfer of energy and angular momentum between the gyrocentres and the perturbed electromagnetic fields, in which the crucial roles played by gyrocentre polarization and magnetization effects are highlighted. In addition to yielding an exact angular momentum conservation law, the gyrokinetic Noether equation yields an exact momentum transport equation, which might be useful in more general equilibrium magnetic geometries.

Orb5: A global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry

This paper presents the current state of the global gyrokinetic code Orb5 as an update of the previous reference (Jolliet et al., 2007). The Orb5 code solves the electromagnetic Vlasov–Maxwell system of equations using a PIC scheme and also includes collisions and strong flows. The code assumes multiple gyrokinetic ion species at all wavelengths for the polarization density and drift-kinetic electrons. Variants of the physical model can be selected for electrons such as assuming an adiabatic response or a “hybrid” model in which passing electrons are assumed adiabatic and trapped electrons are drift-kinetic. A Fourier filter as well as various control variates and noise reduction techniques enable simulations with good signal-to-noise ratios at a limited numerical cost. They are completed with different momentum and zonal flow-conserving heat sources allowing for temperature-gradient and flux-driven simulations. The code, which runs on both CPUs and GPUs, is well benchmarked against other similar codes and analytical predictions, and shows good scalability up to thousands of nodes.

A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm

For decades, the Vlasov–Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. Here, we explore a fully implicit PIC algorithm for the Vlasov–Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space–time-centered, employing particle sub-cycling and orbit-averaging. The algorithm conserves total energy, local charge, canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. We demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D–3V.

A high-resolution global Vlasov simulation of a small dielectric body with a weak intrinsic magnetic field on the K computer

The interaction between the solar wind and solar system bodies, such as planets, satellites, and asteroids, is one of the fundamental global-scale phenomena in space plasma physics. In the present study, the electromagnetic environment around a small dielectric body with a weak intrinsic magnetic field is studied by means of a first-principle kinetic plasma simulation, which is a challenging task in space plasma physics as well as high-performance computing. Due to several computational limitations, five-dimensional full electromagnetic Vlasov simulations with two configuration space and three velocity space coordinates are performed with two different spatial resolutions. The Debye-scale charge separation is not solved correctly in the simulation run with a low spatial resolution, while all the physical processes in collisionless plasma are included in the simulation run with a high spatial resolution. The direction comparison of electromagnetic fields between the two runs shows that there is small difference in the structure of magnetic field lines. On the other hand, small-scale fine structures of electrostatic fields are enhanced by the electric charge separation and the charge accumulation on the surface of the body in the high-resolution run, while these structures are absent in the low-resolution runs. These results are consistent with the conventional understanding of plasma physics that the structure and dynamics of global magnetic fields, which are generally described by the magneto-hydro-dynamics (MHD) equations, are not affected by electron-scale microphysics.

A finite volume formulation of the multi-moment advection scheme for Vlasov simulations of magnetized plasma

We present a finite volume version of the multi-moment advection scheme (Minoshima et al., 2011, 2013). The scheme advances zeroth to second order piecewise moments at cell centers through their numerical fluxes obtained from one-dimensional high order interpolation. The modification simplifies the scheme without losing its high performance. We apply the scheme to two-dimensional electromagnetic Vlasov simulations of linear wave propagation and nonlinear magnetic reconnection problems. Our Vlasov simulation resolves microscopic structure of the non-Maxwellian plasma velocity distribution around the magnetic reconnection site as well as macroscopic structure of fluid quantities within the energy error of 2%, and is in good agreement with previous studies.

Electromagnetic nonlinear gyrokinetics with polarization drift

A set of new nonlinear electromagnetic gyrokinetic Vlasov equation with polarization drift and gyrokinetic Maxwell equations is systematically derived by using the Lie-transform perturbation method in toroidal geometry. For the first time, we recover the drift-kinetic expression for parallel acceleration [R. M. Kulsrud, in Basic Plasma Physics, edited by A. A. Galeev and R. N. Sudan (North-Holland, Amsterdam, 1983)] from the nonlinear gyrokinetic equations, thereby bridging a gap between the two formulations. This formalism should be useful in addressing nonlinear ion Compton scattering of intermediate-mode-number toroidal Alfvén eigenmodes for which the polarization current nonlinearity [T. S. Hahm and L. Chen, Phys. Rev. Lett. 74, 266 (1995)] and the usual finite Larmor radius effects should compete.

An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov–Darwin particle-in-cell algorithm

A recent proof-of-principle study proposes a nonlinear electrostatic implicit particle-in-cell (PIC) algorithm in one dimension (Chen et al., 2011). The algorithm employs a kinetically enslaved Jacobian-free Newton–Krylov (JFNK) method, and conserves energy and charge to numerical round-off. In this study, we generalize the method to electromagnetic simulations in 1D using the Darwin approximation to Maxwell’s equations, which avoids radiative noise issues by ordering out the light wave. An implicit, orbit-averaged, time–space-centered finite difference scheme is employed in both the 1D Darwin field equations (in potential form) and the 1D-3V particle orbit equations to produce a discrete system that remains exactly charge- and energy-conserving. Furthermore, enabled by the implicit Darwin equations, exact conservation of the canonical momentum per particle in any ignorable direction is enforced via a suitable scattering rule for the magnetic field. We have developed a simple preconditioner that targets electrostatic waves and skin currents, and allows us to employ time steps O(mi/mec/veT) larger than the explicit CFL. Several 1D numerical experiments demonstrate the accuracy, performance, and conservation properties of the algorithm. In particular, the scheme is shown to be second-order accurate, and CPU speedups of more than three orders of magnitude vs. an explicit Vlasov–Maxwell solver are demonstrated in the “cold” plasma regime (where kλD≪1).

Ion kinetic effects on nonlinear processes of the Kelvin–Helmholtz instability

The nonlinear evolution of the Kelvin–Helmholtz (KH) instability at a transverse velocity shear layer in an inhomogeneous space plasma is investigated by means of a four-dimensional (two spatial and two velocity dimensions) electromagnetic Vlasov simulation. When the rotation direction of the primary KH vortex and the direction of ion gyro motion are the same (i.e., the inner product between the vorticity of the primary velocity shear and the magnetic field vector is negative) there exists a strong ion cyclotron damping. In this case, spatial inhomogeneity inside the primary KH vortex is smoothed and the secondary Rayleigh–Taylor/KH instabilities are suppressed. It is also found that another secondary instability on the electron inertial scale is simultaneously generated at secondary shear layers for both cases, but at different locations. The small-scale secondary instability takes place only when the inner product between the vorticity of the secondary shear layer and the magnetic field vector is positive, suggesting the damping of small-scale processes by ion gyro motion. These results indicate that secondary instabilities occurring in the nonlinear stage of the primary KHI show different evolutions depending on the sign of the inner product between the magnetic field and the vorticity of the velocity shear layer.

Entry of solar-wind ions into the wake of a small body with a magnetic anomaly: A global Vlasov simulation

The interaction between a plasma flow and a small dielectric body with a weak intrinsic global magnetic field is studied by means of a five-dimensional full electromagnetic Vlasov simulation with two configuration spaces and three velocity spaces. In the present study, entry processes of ions into the nightside wake tail are examined. The simulation result shows that the bow shock and the magnetopause are formed on the dayside. However, most of solar-wind ions are reflected at the dayside magnetopause and are picked up by the interplanetary magnetic field. Then, a small part of the reflected ions are taken into the deep wake tail near the body by the E×B cycloid motion.

Numerical techniques for parallel dynamics in electromagnetic gyrokinetic Vlasov simulations

Numerical techniques for parallel dynamics in electromagnetic gyrokinetic simulations are introduced to regulate unphysical grid-size oscillations in the field-aligned coordinate. It is found that a fixed boundary condition and the nonlinear mode coupling in the field-aligned coordinate, as well as numerical errors of non-dissipative finite difference methods, produce fluctuations with high parallel wave numbers. The theoretical and numerical analyses demonstrate that an outflow boundary condition and a low-pass filter efficiently remove the numerical oscillations, providing small but acceptable errors of the entropy variables. The new method is advantageous for quantitative evaluation of the entropy balance that is required for obtaining a steady state in gyrokinetic turbulence.

Multi-moment advection scheme in three dimension for Vlasov simulations of magnetized plasma

We present an extension of the multi-moment advection scheme [T. Minoshima, Y. Matsumoto, T. Amano, Multi-moment advection scheme for Vlasov simulations, Journal of Computational Physics 230 (2011) 6800–6823] to the three-dimensional case, for full electromagnetic Vlasov simulations of magnetized plasma. The scheme treats not only point values of a profile but also its zeroth to second order piecewise moments as dependent variables, and advances them on the basis of their governing equations. Similar to the two-dimensional scheme, the three-dimensional scheme can accurately solve the solid body rotation problem of a gaussian profile with little numerical dispersion or diffusion. This is a very important property for Vlasov simulations of magnetized plasma. We apply the scheme to electromagnetic Vlasov simulations. Propagation of linear waves and nonlinear evolution of the electron temperature anisotropy instability are successfully simulated with a good accuracy of the energy conservation.

Effect of ion cyclotron motion on the structure of wakes: A Vlasov simulation

The global structure of wake fields behind an unmagnetized object in the solar wind is studied by means of a 2.5-dimensional full electromagnetic Vlasov simulation. The interaction of a plasma flow with an unmagnetized object is quite different from that with a magnetized object such as the Earth. Due to the absence of the global magnetic field, the unmagnetized object absorbs plasma particles which reach the surface, generating a plasma cavity called a wake on the anti-solar side of the object. The interaction between the solar wind with an out-of-plane interplanetary magnetic field and an unmagnetized object of ion-gyro scale is examined to study the effect of the ion cyclotron motion on the structure of wake fields. It is shown that the wake boundary layer is broadened when the inner product of the electric force and the density gradient of the wake boundary layer is negative. On the other hand, a steep wake boundary layer is formed with a negative inner product. The result suggests that the structures of wake fields become asymmetric due to the in-plane E cross B drift motion of ions.

Multi-moment advection scheme for Vlasov simulations

We present a new numerical scheme for solving the advection equation and its application to Vlasov simulations. The scheme treats not only point values of a profile but also its zeroth to second order piecewise moments as dependent variables, for better conservation of the information entropy. We have developed one-and two-dimensional schemes and show that they provide quite accurate solutions within reasonable usage of computational resources compared to other existing schemes. The two-dimensional scheme can accurately solve the solid body rotation problem of a gaussian profile for more than hundred rotation periods with little numerical diffusion. This is crucially important for Vlasov simulations of magnetized plasmas. Applications of the one- and two-dimensional schemes to electrostatic and electromagnetic Vlasov simulations are presented with some benchmark tests.

Full electromagnetic Vlasov code simulation of the Kelvin–Helmholtz instability

Recent advancement in numerical techniques for Vlasov simulations and their application to cross-scale coupling in the plasma universe are discussed. Magnetohydrodynamic (MHD) simulations are now widely used for numerical modeling of global and macroscopic phenomena. In the framework of the MHD approximation, however, diffusion coefficients such as resistivity and adiabatic index are given from empirical models. Thus there are recent attempts to understand first-principle kinetic processes in macroscopic phenomena, such as magnetic reconnection and the Kelvin–Helmholtz (KH) instability via full kinetic particle-in-cell and Vlasov codes. In the present study, a benchmark test for a new four-dimensional full electromagnetic Vlasov code is performed. First, the computational speed of the Vlasov code is measured and a linear performance scaling is obtained on a massively parallel supercomputer with more than 12 000 cores. Second, a first-principle Vlasov simulation of the KH instability is performed in order to evaluate current status of numerical techniques for Vlasov simulations. The KH instability is usually adopted as a benchmark test problem for guiding-center Vlasov codes, in which a cyclotron motion of charged particles is neglected. There is not any full electromagnetic Vlasov simulation of the KH instability; this is because it is difficult to follow E⃗×B⃗ drift motion accurately without approximations. The present first-principle Vlasov simulation has successfully represented the formation of KH vortices and its secondary instability. These results suggest that Vlasov code simulations would be a powerful approach for studies of cross-scale coupling on future Peta-scale supercomputers.

Structures of diffusion regions in collisionless magnetic reconnection

Detailed structures of diffusion regions in two-dimensional collisionless magnetic reconnection are studied by using an electromagnetic Vlasov simulation. It has been well known that plasma number density decreases near the X-point of the reconnection. However, numerical thermal fluctuations exist in particle-in-cell simulations, and there is a possibility that detailed structures near the X-point diffuse numerically when the number of particles per cell is not enough. In the present study, a high-resolution two-dimensional Vlasov simulation is performed. It is found that electron number density in the electron diffusion region decreases to a hundredth of the initial value. Structures of electron diffusion region are determined by the local electron inertial length.

Analytical and simulation studies of nonlinear effects caused by upper-hybrid waves in plasmas

Nonlinear interactions between large amplitude upper-hybrid waves and low-frequency electromagnetic waves in a magnetized plasma have been considered. Specifically, the decay of upper-hybrid waves into lower-hybrid and magnetosonic waves, as well as the modulational instabilities leading to the formation of density cavities which trap upper-hybrid wave envelopes, are investigated both theoretically and numerically. It is found that the three-wave process between two upper-hybrid waves and the lower-hybrid wave, which is the fastest growing instability of the system, is important for the generation of waves of small wave numbers, via the self-interaction of the generated decay products from the three-wave interaction. A comparison between the theory of the parametric decay, simulations of a generalized Zakharov system of equations modeling the interaction between upper-hybrid, lower-hybrid and magnetosonic (fast Alfvénic) waves is presented. Comparisons are made with a simulation of the fully electromagnetic Vlasov–Maxwell system for ions and electrons where a sudden onset of cavities and long wavelength magnetosonic wave emission could be seen, similar as in the simulation of the Zakharov-type system.

A non-periodic 2D semi-Lagrangian Vlasov code for laser–plasma interaction on parallel computer

For the first time, a 2D electromagnetic and relativistic semi-Lagrangian Vlasov model for a multi-computer environment was developed to study the laser–plasma interaction in an open system. Numerical simulations are presented for situations relevant to the penetration of an ultra-intense laser pulse inside a moderately overdense plasma and the relativistic filamentation instability in the case of an underdense plasma. The Vlasov model revealed a rich variety of phenomena associated with the fast particle dynamics induced by the laser pulse as particle trapping, particle acceleration and relativistic self-induced transparency in overdense plasma. Attention was focused on the efficiency and stability properties on the numerical scheme and implementation facilities on massively parallel computers. Success of the semi-Lagrangian Vlasov model is enhanced by the good conservation of the continuity equation and stability of Maxwell system due to the fine description of the electron distribution function and particularly of the charge density and current density.