J. S. Kim

The calculation of Lyapunov spectra

Abstract Linearization around the chaotic orbits of dynamical systems offers good prospects for understanding complex systems. The naturally available information that can be obtained from such studies are the magnitudes and directions in phase space of stretching and shrinking of an initial small sphere of realizations. This information is a property of a time displacement matrix that relates the initial bundle of realizations to its later configuration. In this paper we extract information from this matrix through its singular values, and the associated orthogonal matrices. This given an algebraic foundation to the geometric methods that have been used previously. The equations of the orthogonal principal stretching and contraction directions show that asymptotically, after a long time, they depend only on the location in phase space, and not on the orbit. As an example, we show that these directions sketch out the attractor of the Lorenz model.

Introduction of a metric tensor into linearized evolution equations

Abstract Since particular solutions of nonlinear evolution equations are frequently uniformative, it is desirable to consider ensembles of solutions. The natural ensemble that is used in computational treatments is a tight bunch of realizations that can be calculated by linearizing around a given realization. It is useful to understand the details of linearization to the fullest extent. The evolution of the separation between realizations in phase space is an important result of the solutions of the linearized equations. Measurement of this separation requires the introduction of a metric, but the detailed form of the metric is not dictated by the problem at hand. Here the role of metrics in the linearized equations is thoroughly explored. In particular, the role of metrics in determining the stability of the numerical schemes that are used in the evolution of the linearized equations is discussed. Their stability is independent of the metric.

IONEX: a meshfree ion extraction code based on "particle in cloud of points" concept.

Ion Extraction (IONEX) is an ion extraction modeling code, developed at FAR-TECH, Inc., based on the meshless particle-in-cloud-of-points concept. IONEX self-consistently solves motion equations for ions and Poissons equation for the electrostatic field, assuming a Boltzmann distribution for the electrons. IONEX is capable of handling multiple species and is graphical user interface-driven. The two-dimensional version is benchmarked with IGUN. The basic algorithm and sample runs are presented.

Electron cyclotron resonance charge breeder ion source simulation by MCBC and GEM

Numerical simulation results by the GEM and MCBC codes are presented, along with a comparison with experiments for beam capture dynamics and parameter studies of charge state distribution (CSD) of electron cyclotron resonance charge breeder ion sources. First, steady state plasma profiles are presented by GEM with respect to key experimental parameters such as rf power and gas pressure. As rf power increases, electron density increases by a small amount and electron energy by a large amount. The central electrostatic potential dip also increased. Next, MCBC is used to trace injected beam ions to obtain beam capture profiles. Using the captured ion profiles, GEM obtains a CSD of beam ions. As backscattering can be significant, capturing the ions near the center of the device enhances the CSD. The effect of rf power on the beam CSD is mainly due to different steady states plasmas. Example cases are presented assuming that the beam ions are small enough not to affect the plasma.

Integrated modeling of electron cyclotron resonance ion sources and charge breeders with GEM, MCBC, and IonEx.

A numerical toolset to help in understanding physical processes in the electron cyclotron resonance charge breeder (ECRCB) and further to help optimization and design of current and future machines is presented. The toolset consists of three modules (Monte Carlo charge breeding code, generalized electron cyclotron resonance ion source modeling, and ion extraction), each modeling different processes occurring in the ECRCB from beam injection to extraction. The toolset provides qualitative study, such as parameter studies, and scaling of the operation, and physical understanding in the ECRCB. The methodology and a sample integrated modeling are presented.

Soft x-ray virtual diagnostics for tokamak simulations

The numerical toolset, FAR-TECH Virtual Diagnostic Utility, for generating virtual experimental data based on theoretical models and comparing it with experimental data, has been developed for soft x-ray diagnostics on DIII-D. The virtual (or synthetic) soft x-ray signals for a sample DIII-D discharge are compared with the experimental data. The plasma density and temperature radial profiles needed in the soft x-ray signal modeling are obtained from experimental data, i.e., from Thomson scattering and electron cyclotron emission. The virtual soft x-ray diagnostics for the equilibriums have a good agreement with the experimental data. The virtual diagnostics based on an ideal linear instability also agree reasonably well with the experimental data. The agreements are good enough to justify the methodology presented here for utilizing virtual diagnostics for routine comparison of experimental data. The agreements also motivate further detailed simulations with improved physical models such as the nonideal magnetohydrodynamics contributions (resistivity, viscosity, nonaxisymmetric error fields, etc.) and other nonlinear effects, which can be tested by virtual diagnostics with various stability modeling.

C60-fullerene composite plasma jets formation and acceleration for application to disruption mitigation and magneto-inertial fusion

We present the progress on the development of a new idea of using high-Mach number high-density composite plasma jets from coaxial plasma guns for disruption mitigation in tokamak¹ and magneto-inertial fusion² (MIF). The key element of the idea is the solid state pulsed power source with TiH<inf>2</inf> (or TiDT) grains and C<inf>60</inf> micron size powder³. Very fast injection of the molecular gas mixture provided by hydrogen release and sublimation of C<inf>60</inf> into the plasma gun is achieved by a special filter grid with supersonic Laval nozzles. The estimations based on the physical models of TiH2 grains heating, C<inf>60</inf> powder sublimation, molecular gas injection, mass separation, and plasma slug acceleration will be detailed. For disruption mitigation, our calculations show that the plasma gun is able to provide the required impurity mass¹ (∼1–2 g) and the ram pressure to penetrate the tokamak hot plasma and to overcome the confining magnetic field pressure. Core tokamak plasma penetration can be achieved and impurity mass delivered in less than 1 ms, as required by ITER tokamak. The magnetized target fusion (MTF) plasma for MIF is created by injecting two high-Mach number (M≫5) high-density (≫1017 cm⁻³) plasma jets composed of fuel (D-T) and “pusher” (C<inf>60</inf>/C) along the axis of a pulsed magnetic (∼1–2 T) mirror into a metallic cylindrical liner. The high-density (∼10¹⁸ cm⁻³) cylindrical MTF created by head-on collision and stagnation in the magnetic field is compressed radially by the Z-pinch of the liner and prevented to expand axially by the incoming C<inf>60</inf>/C end-plugs. We estimated that, due to the much longer MTF axial dimension (∼30 cm) as compared to other inertial confinement fusion plasmas, the electron thermal conduction time to the C<inf>60</inf>/C end-plugs is longer than liner implosion time.

3D hybrid meshless adaptive algorithm and code for ion extraction problem

We present progress on the development of a new 3D hybrid electrostatic code, IonEx3D, for simulating ion extraction from plasma. The code is based on the meshless, adaptive Particle-In-Cloud-Of-Points (PICOP)1 approach. The ions are treated fully kinetically whereas electrons are described as a neutralizing fluid obeying the Boltzmann distribution. Steady state ion trajectories are found in the self-consistent electrostatic field and the given magnetic field. The semi-linear Poissons equation is solved on a meshless cloud of points, which is iteratively adapted to the field and the beam structure. The algorithm conserves both the full energy and the angular momentum. 3D effects on the ion beam formation will be presented. An effective algorithm for the code parallelization on petascale computers will be also discussed.