Two-dimensional Velocity Space Research Articles

A Vlasov-Fokker-Planck-Landau code for the simulation of colliding supersonic dense plasma flows

A Vlasov-Fokker-Planck-Landau (VFPL) code is developed for the study of colliding supersonic dense plasma flows, in which the VFPL equations for both electrons and ions are solved by the time splitting strategy in the two-dimensional Cartesian coordinate space and the two-dimensional velocity space (2D2V). To accurately handle two colliding supersonic plasma flows with their velocities even much higher than the thermal velocities of ions, the full Fokker-Planck-Landau operator is employed for the collision terms. Using advanced numerical methods such as the fast spectral method and the asymptotic-preserving scheme, the ion-ion and electron-electron collision terms can be properly solved with a relatively large time step so that both high accuracy and efficiency of the developed code can be achieved. Further, the quantum degeneracy effect due to the high density and relatively low initial temperature of the electrons is included in the model. In addition, both the energy and momentum conservation are well satisfied. The developed code provides a unique numerical tool to study the interactions between high-velocity high-density plasma flows, which may be encountered in some advanced schemes of inertial confinement fusion and laser-driven laboratory astrophysics experiments.

Optimal trajectories for symmetric turns

The problem of determining minimal time trajectories in a plane constrained by an upper bound on the magnitude of the acceleration vector is reexamined. In the previous work [Am. J. Phys. 49(7), 685–688 (1981)], a stationary solution of a functional, applied over curves in two-dimensional velocity space, was used to find explicit expressions for what was claimed to be a minimum turn time trajectory. In this paper, this work is furthered by a formal demonstration that the turn time associated with this trajectory is indeed lower than that corresponding to any other smooth trajectory. Supporting evidence for this claim is provided by numerical procedures, which are developed to allow comparisons between the turn times of competing trajectories across a range of parameter values of the turn width, the initial speed, and the magnitude of the acceleration vector.

An efficient, conservative, time-implicit solver for the fully kinetic arbitrary-species 1D-2V Vlasov-Ampère system

We consider the solution of the fully kinetic (including electrons) Vlasov-Ampère system in a one-dimensional physical space and two-dimensional velocity space (1D-2V) for an arbitrary number of species with a time-implicit Eulerian algorithm. The problem of velocity-space meshing for disparate thermal and bulk velocities is dealt with by an adaptive coordinate transformation of the Vlasov equation for each species, which is then discretized, including the resulting inertial terms. Mass, momentum, and energy are conserved, and Gauss's law is enforced to within the nonlinear convergence tolerance of the iterative solver through a set of nonlinear constraint functions while permitting significant flexibility in choosing discretizations in time, configuration, and velocity space. We mitigate the temporal stiffness introduced by, e.g., the plasma frequency through the use of high-order/low-order (HOLO) acceleration of the iterative implicit solver. We present several numerical results for canonical problems of varying degrees of complexity, including the multiscale ion-acoustic shock wave problem, which demonstrate the efficacy, accuracy, and efficiency of the scheme.

On the Cauchy problem with large data for a space-dependent Boltzmann–Nordheim boson equation

This paper studies a Boltzmann-Nordheim equation in a slab with two-dimensional velocity space and pseudo-Maxwellian forces. Strong solutions are obtained for the Cauchy problem with large initial data in an $ L^1 \cap L^{\infty} $ setting. The main results are existence, uniqueness, and stability of solutions conserving mass, momentum, and energy. The solutions either explode in the $ L^\infty$-norm in finite time, or exist globally in time. They are obtained as limits of solutions to corresponding anyon equations.

Block-structured grids for Eulerian gyrokinetic simulations

In order to predict the turbulent transport in magnetic fusion experiments, global (i.e., full-torus) gyrokinetic simulations are often carried out. In this context, one frequently encounters situations in which the plasma temperature varies by large factors across the radial simulation domain. In grid-based Eulerian codes, this enforces the use of a very large number of grid points in the two-dimensional velocity space, and, thus, an enormous computational effort. To minimize the computational requirements, one may employ block-structured grids, adapted to the radial changes of the temperature. As the block-structured grids rely on a general approach, they can be applied to different Eulerian gyrokinetic implementations. In this paper, we explain the construction and implementation of such grids in the gyrokinetic code GENE, F. Jenko et al. (2000), and present corresponding simulation results.

Hole-Clump Pair Creation in the Evolution of Energetic-Particle-Driven Geodesic Acoustic Modes

Nonlinear frequency chirping of the energetic-particle-driven geodesic acoustic mode (EGAM) is investigated using a hybrid simulation code for energetic particles interacting with a magnetohydrodynamic fluid. It is demonstrated in the simulation result that both frequency chirping up and chirping down take place in the nonlinear evolution of the EGAM. It is found that two hole-clump pairs are formed in the energetic particle distribution function in two-dimensional velocity space of pitch angle variable and energy. One pair is formed in the phase space region that destabilizes the instability, while the other is formed in the stabilizing region. The transit frequency of the hole (clump) in the destabilizing region chirps up (down), while in the stabilizing region the hole (clump) chirps down (up). The transit frequencies of particles in the holes and clumps are in good agreement with the chirping EGAM frequency indicating that the particles are kept resonant with the EGAM during the nonlinear frequency chirping. Continuous energy transfer takes place from the destabilizing phase space region to the stabilizing region during the spontaneous frequency chirping of the wave.

Nonlinear kinetic development of the Weibel instability and the generation of electrostatic coherent structures

The nonlinear evolution of the Weibel instability driven by the anisotropy of the electron distribution function in a collisionless plasma is investigated in a spatially one-dimensional configuration with a Vlasov code in a two-dimensional velocity space. It is found that the electromagnetic fields generated by this instability cause a strong deformation of the electron distribution function in phase space, corresponding to highly filamented magnetic vortices. Eventually, these deformations lead to the generation of short wavelength Langmuir modes that form highly localized electrostatic structures corresponding to jumps of the electrostatic potential.

Velocity-space resolution, entropy production, and upwind dissipation in Eulerian gyrokinetic simulations

Equations which describe the evolution of volume-averaged gyrokinetic entropy are derived and added to GYRO [J. Candy and R.E. Waltz, J. Comput. Phys. 186, 545 (2003)], a Eulerian gyrokinetic turbulence simulation code. In particular, the creation of entropy through spatial upwind dissipation (there is zero velocity-space dissipation in GYRO) and the reduction of entropy via the production of fluctuations are monitored in detail. This new diagnostic has yielded several key confirmations of the validity of the GYRO simulations. First, fluctuations balance dissipation in the ensemble-averaged sense, thus demonstrating that turbulent GYRO simulations achieve a true statistical steady state. Second, at the standard spatial grid size, neither entropy nor energy flux is significantly changed by a 16-fold increase (from 32 to 512 grid points per cell) in the number of grid points in the two-dimensional velocity space. Third, the measured flux is invariant to an eightfold increase in the upwind dissipation coefficients. A notable conclusion is that the lack of change in entropy with grid refinement refutes the familiar but incorrect notion that Eulerian gyrokinetic codes miss important velocity-space structure. The issues of density and energy conservation and their relation to negligible second-order effects such as the parallel nonlinearity are also discussed.

Influence of the non-linearity of the collision operator on ion cyclotron resonance heating

The distribution function of ions heated in the ion cyclotron range of frequencies is obtained by solving the Fokker–Planck equation. For non-minority heating scenarios, the full non-linear Coulomb collision operator must be used in this equation. A method of resolution accounting for the complexity of this operator is presented. We consider a plasma immersed in a homogeneous magnetic field. The adopted method of resolution is based on an expansion of the distribution function in a series of Legendre polynomials. Results of the corresponding code non-linear Fokker–Planck in two-dimensional velocity space (NLFP-2D) are discussed for experimentally relevant JET-like parameters. The convergence of the Legendre expansion has been tested and the importance of the non-linear effects has been shown. Three different regimes, characterized by different modes of indirect ion heating and depending on the absorbed RF power density as well as the minority concentration, were identified. The NLFP-2D code has been used to study the validity of the Maxwellian approximation of the self-collision operator. One finds that this model is only correct in a very limited range of parameters, and leads mostly to an overestimation of the indirect ion heating.

Numerical Simulation of Electron Cyclotron Wave Current Start-Up in Tokamaks

The electron cyclotron wave current start-up in tokamaks has been simulated using Monte-Carlo method in two-dimensional configuration space ( x , y ) and two-dimensional velocity space ( V ∥ , V ⊥ ). The simulation results of the relationship between start-up current and external vertical magnetic field is consistent with experiments, and the simulation results coincides the physical analyses that the start-up current is obtained by canceling the toroidal drift in one toroidal direction by external vertical magnetic field for energetic electrons in low density plasmas.

FPPAC94: A two-dimensional multispecies nonlinear Fokker-Planck package for UNIX systems

FPPAC94 solves the complete nonlinear multispecies Fokker-Planck collison operator for a plasma in two-dimensional velocity space. The operator is expressed in terms of spherical coordinates (speed and pitch angle) under the assumption of azimuthal symmetry. Provision is made for additional physics contributions (e.g. rf heating, electric field acceleration). The charged species, referred to as general species, are assumed to be in the presence of an arbitrary number of fixed Maxwellian species. The electrons may be treated either as one of these Maxwellian species or as a general species. Coulomb interactions among all charged species are considered This program is a new version of FPPAC. FPPAC was last published in Computer Physics Communications in 1988. This new version is identical in scope to the previous version. However, it is written in standard Fortran 77 and is able to execute on a variety of Unix systems. The code has been tested on the Cray-C90, HP-755 and Sun Sparc-1. The answers agree on all platforms where the code has been tested. The test problems are the same as those provided in 1988. This version also corrects a bug in the 1988 version.

Upwind Finite Difference Solution of Boltzmann Equation Applied to Electron Transport in Semiconductor Devices

We present a new numerical method for solving the Boltzmann-Poisson system which describes charge transport in semiconductor devices. The Boltzmann equation is reduced from three dimensions in velocity space to two by taking the electric field parallel to the z axis, which implies invariance of the probability density function under rotation around the z axis. We develop a finite difference discretization of the Boltzmann equation in one spatial dimension and two-dimensional velocity space, coupled to the Poisson equation. The system of equations obtained by taking the first five moments of the Boltzmann equation coupled to the Poisson equation is known as the hydrodynamic model in semiconductor modeling. A comparison of the numerical results from our method and the hydrodynamic model is given. Also a numerical investigation is done with respect to the heat conduction, viscosity, and momentum relaxation terms in the hydrodynamic model.

Anisotropic distribution function of minority tail ions generated by strong ion-cyclotron resonance heating

The highly anisotropic particle distribution function of minority tail ions driven by ion-cyclotron resonance heating at the fundamental harmonic is calculated in a two-dimensional velocity space. It is assumed that the heating is strong enough to drive most of the resonant ions above the ion–electron critical slowing-down energy. Simple analytic expressions for the tail distribution are obtained for the cases when the bounce-averaged quasilinear heating coefficient Db is assumed to be constant in pitch angle and energy, when the Doppler effect is sufficiently large to flatten the sharp pitch angle dependence in Db but the energy dependence is allowed, and when the sharp pitch angle dependence is retained. It is found that a simple constant-Db solution can be used instead of the more complicated velocity-dependent-Db solutions for many analytic purposes.

On suboptimal detection of 3-dimensional moving targets

The author designates matched filters that are completely characterized by the velocity of the target as assumed velocity filters (AVFs). Like most matched filtering techniques where the signal parameters range in a continuum, the AVF must be implemented suboptimally by partitioning the velocity space. The author investigates the possibility of using a signal-to-noise ratio (SNR) loss factor as the criterion for the partition. The loss factor is a measurement of the loss of SNR at the output of the matched filter due to mismatch of filter parameters. In the scenario of detecting a moving satellite from a ground-based sensor, because of the vast sky the sensor has to search, it is important to keep the number of filters minimal. The author shows that, with a fixed loss factor, the number of filters required for coverage increases linearly as the span of the two-dimensional velocity space increases quadratically. The rate of increase is further reduced when the loss factor is made proportional to expected target angular speed. >

FPPAC88: A two-dimensional multispecies nonlinear Fokker-Planck package

FPPAC88 solves the complete nonlinear multispecies Fokker-Planck collision operator for a plasma in two-dimensional velocity space. The operator is expressed in terms of spherical coordinates (speed and pitch angle) under the assumption of azimuthal symmetry. Provision is made for additional physics contributions (e.g. rf heating, electric field acceleration). The charges species, referred to as general species, are assumed to be in the presence of an arbitrary number of fixed Maxwellian species. The electrons may be treated either as one of these Maxwellian species or as a general species. Coulomb interactions among all charged species are considered. This program is a NEW VERSION of FPPAC, which was published in Computer Physics Communications in 1981. The most important new feature is the modification of the spatial differencing to accomodate situations in which advection dominates diffusion. Another important change is the discontinuance of the CDC-7600 version of FPPAC; thus the NEW VERSION, FPPAC88, is applicable to machines with a single fast memory, in particular the Cray-1 with the CFT compiler. Any departures from standard FORTRAN are easily accomodated, so that FPPAC88 may be run on almost any supercomputer with a FORTRAN-77 compiler.

Radial diffusion of energetic tail ions driven by electromagnetic waves of ion-cyclotron range of frequencies in bumpy torus and tokamak geometry

Radial particle and energy transport of the high-energy tail ions created by fundamental resonance heating of ion-cyclotron range of frequency waves is studied for tokamak and bumpy torus geometry. The lowest-order distribution function for the high-energy ions is calculated in a two-dimensional velocity space, and all the diffusion coefficients are explicitly evaluated.

FPPAC: A two-dimensional multispecies nonlinear fokker-planck package

The complete nonlinear multispecies Fokker-Planck collision operator for a plasma in two-dimensional velocity space is solved. The operator is expressed in terms of spherical coordinates (v = speed, theta = angle between velocity and magnetic field directions, phi = azimuthal angle) under the assumption of azimuthal symmetry. Provision is made for additional physics contributions.

Numerical studies of current generation by radio-frequency traveling waves

By injecting radio-frequency traveling waves into a tokamak, continuous toroidal electron currents may be generated. This process is studied by numerically solving the two-dimensional Fokker–Planck equation with an added quasi-linear term. The results are compared with the one-dimensional analytic treatment of Fisch, which predicted a reduced plasma resistivity when high-phase-velocity waves are employed. It is shown that two-dimensional velocity space effects, while retaining the predicted scaling, further reduce the ratio of power dissipated to current generated by about 40%. These effects enhance the attractiveness of steady-state tokamak reactors utilizing this method of current generation.