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The classical finite-difference time-domain (FDTD) approach to the numerical solution of the time-dependent Maxwell's equations is based on the second order, in space and time, Yee algorithm. However, for an increasing number of applications this algorithm has insufficient accuracy. We replace it by a compact implicit 4th order accuracy scheme that uses the same stencil, but doesn't have drawbacks of the Yee algorithm. A major difficulty with high order methods is the treatment of the dielectric coefficient which is discontinuous across the interface. So we also study the asymptotic and numerical behavior of the solution of the Maxwell equations and the wave equation with discontinuous coefficients in one dimension in both time and frequency space. We present a method for the treatment of the discontinuity that preserves a high order of accuracy for the numerical scheme.

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Recent advances in the direct kinetic simulation of fusion plasma turbulence now lay the groundwork and provide an enormous impetus for kinetic closure (using kinetic simulation) of MHD computational models. The topic is lively and is still very much an open research area. This is because efficient and practical nonlinear MHD and kinetic methods require subtle underlying orderings, equations and numerical methods (gyrokinetics, gyro-Landau fluid, semi-implicit, finite-element, particle-in-cell, drift-ordering, etc.) all of which must properly meld together into one grand simulation. Even on a particular and well-defined MHD problem, (e.g. internal kink instability, edge- localized modes, tearing modes) knowing whether it is better to use kinetic closure of MHD or solve the problem directly using kinetics is very much unanswered at this point. In this talk we will discuss the possible ways to close MHD equations using kinetics, as well as more direct MHD-like kinetic models. This talk will highlight some recent successes in kinetic-MHD, including modeling of energetic particle effects in fusion plasmas. We will also discuss recent kinetic and kinetic-MHD models of tearing mode behavior.

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Magnetic self-organization via Taylor relaxation in a driven plasma underlies the guiding principle of laboratory helicity injection experiments that form spheromaks and the reversed field pinch, and provide non-inductive current drive in a spherical torus. It is also thought to explain the large scale astrophysical magnetic field, for example, in a radio-lobe powered by the accretion disk of a black hole. The critical concept in Taylor relaxation is a linear resonance effect that provides flux amplification. We will first explain the nature of this resonance and its fate when plasma departs from Taylor state. The second part of the talk approaches the same problem by following the dynamics that lead to relaxation. In particular, the so-called instability cascade route to relaxation will be illustrated. Laboratory and astrophysical examples will be drawn upon to appreciate the physical consequences of our analysis.

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We analyze single ion motion in a model field-reversed configuration. A two-dimensional effective potential is derived and shown to possess a potential trough as well as isolated critical points. Sufficient conditions for Lyapunov stability are derived for these equilibria and shown to allow large populations of energetically trapped orbits, which can be regular or chaotic. Among these the classical guiding center orbits gyrating about closed field lines form a small minority. Indeed, for moderate field elongation the great majority of trapped orbits appear to be chaotic, with significant populations of regular orbits librating about stable periodic orbits. For larger conserved angular momentum the potential trough disappears and ions are energetically trapped in a larger convex potential well. The dynamics in this regime is very sensitive to elongation, with large resonances and chaotic regions for particular integer values of the inverse elongation. These theoretical results are well confirmed by numerical orbits, Poincare' sections, and Lyapunov exponents. The abundance of periodic orbits and paucity of guiding center orbits suggests that the frequency of the imposed rotating magnetic field in RMP experiments should be chosen close to the libration frequencies of the dominant periodic orbits rather than the cyclotron frequency.

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There are two holy grails regarding the production of charged-particle beams: high brightness and high average current. Both are within reach using the latest accelerator technology. However, conventional design codes based on controlling global moments of the beam provide grossly insufficient intelligence as to the whereabouts of these grails. Consequently, the Beam Physics and Astrophysics Group at Northern Illinois University has been delving into the fundamental physics of space charge. Major topics of investigation have included: chaotic orbits in both time-independent and time-dependent beams, phase mixing and rapid collisionless relaxation, the validity of the Vlasov-Poisson limit, halo formation, and the importance of noise. In short, the lesson learned is that details do matter: the phenomenology of space charge is intricate and involves multiscale dynamics. This talk will present illustrative examples and point to future directions for beam simulation codes.

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Modeling a magnetized plasma using single-fluid MHD is inadequate to describe many phenomena. The next simplest model describes the plasma as two fluids, ions and electrons. While this two-fluid model still omits most important kinetic physics, an efficient and accurate numerical treatment of a two-fluid plasma forms the basis for many additional kinetic extensions. Additionally, some examples are given, including the field-reversed configuration and Harris sheet reconnection, where two-fluid calculations would be (or have already been) extremely useful. Next, the problem of efficient numerical solution using time-implicit methods is discussed and contrasted to the single-fluid situation. A uniform, two-fluid plasma supports only real frequency waves, and we seek difference approximations which preserve this feature. The required time differencing is developed and implemented in the NIMROD code, a fusion community-wide finite element code previously applied to single fluid modeling of tokamak and other toroidal plasmas. Results of dispersion tests are discussed and future applications discussed.

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The evolution of collisionless and semi-collisional tearing mode instabilities is studied using an electromagnetic gyrokinetic $\delta f$ particle-in-cell simulation model. Drift-kinetic electrons are used. Linear eigenmode analysis is presented for the case of fixed ions and there is excellent agreement with simulation. A double peaked eigenmode structure is seen indicative of a positive $\Delta^\prime$. Nonlinear evolution of a magnetic island is studied and the results compare well with existing theory in terms of saturation level and electron bounce oscillations. Electron-ion collisions are included to study the semi-collisional regime. The algebraic growth stage is observed and compares favorably with theory. Nonlinear saturation following the Rutherford regime is observed.

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I will present results from DC breakdown experiments designed to imitate (to some extent) breakdown in superconducting microwave resonators. "Before" and "after" pictures demonstrate the dangers of contaminant particles, and post-breakdown surface analyses show the damage caused by the arc around the field emitter, including the extent of ion bombardment. A simple model can explain the initiation of breakdown at a field emitter around which a monolayer of neutral atoms suddenly desorbs; computer simulations using OOPIC show in more detail how breakdown might be thus triggered, and confirm the model`s predictions of a critical current and gas density necessary for breakdown. Although the source of the gas remains unexplained in most cases, I will present a possible explanation for helium processing of superconducting microwave cavities.

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For a radio-frequency sheath, it has been found that the rf sheath dynamics is characterized by the ratio of the rf frequency and the ion transit frequency crossing the sheath.
Based on a one-dimensional fluid model, the sheath dynamics in different frequency regimes have been studied by solving the continuity and momentum equations for electrons and ions and Poisson's equation. In this model, the presheath dynamics is taken into account. If the rf frequency is smaller than the ion transit frequency crossing the presheath, the ions in the presheath respond instantaneously to the rf field. Consequently, the ion current entering the sheath is time-varying which affects the sheath dynamics significantly.
To investigate the ion kinetic effects, the one-dimensional Vlasov equation for ions is solved by using the cubic interpolated propagation scheme (CIP) while the drift-diffusion model is assumed for electrons. It is found that the ion energy distributions (IEDs) of the kinetic model depend on the ionization term. If the ion production rate is significant in the sheath, multiple peaks of the IED will be formed.

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I will discuss the formulation and the properties of the moment implicit particle in cell (PIC) method developed at Los Alamos National Laboratory.
The talk will be divided into three parts.
First, I will discus the challenges of multiple scale problems in plasma physics. Plasmas host a variety of processes, often some are of more interest than others. Often the processes of interest are on long space and time scales. The implicit approach is an excellent way to handle this situation. It focuses on the long scales of interest, with proportionate resolution, without needing to resolve smaller scales accurately. The method implicitly averages over the smaller and faster scales. I will discuss the general properties of the implicit method.
Second, I will discuss how the implicit moment method is designed and turned into a computer code. I will summarize the actual formulation we currently use in our CELESTE3D code. I will spend a little more time discussing the most recent advances in this area: the formulation of the Maxwell's equations and the boundary conditions for them.
Lastly, I will discuss some benchmark calculations meant to illustrate the performance of CELESTE3D.

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The cooling process and the thermodynamics of an electron plasma are investigated in strongly magnetized limit where the gyroradius of the electron is small compared with the mean interparticle spacing. In the limit, the transfer of longitudinal and transverse energy nearly vanishes. For such a plasma there is effectively an extra thermodynamic parameter, as the longitudinal and transverse energies are independently conserved. As a cooling process, we introduce microwave cooling to the strongly magnetized electron plasma. Unlike ion plasmas, an electron plasma which has no internal degree of freedom cannot be cooled down below a heat bath temperature. However, the longitudinal cooling can be achieved by energy transfer from the poorly cooled longitudinal degree of freedom to the well cooled (by synchrotron radiation) transverse degree of freedom. A microwave tuned to a frequency below the gyrofrequency forces electrons moving towards the microwave to absorb a microwave photon. Simultaneously the electrons move up one in Landau state and then lose their longitudinal momentum. In this process, the longitudinal temperature of the electron plasma can be decreased. On the basis that the transverse temperature is below the Landau temperature of the plasma, we set up two level transition equations and then derive a Fokker-Planck equation from the two level equations. With an aid of a finite element method (FEM) code for the equation, the cooling times for several values of the magnetic field, the microwave cavity, and the relative detuning frequency from the gyrofrequency, are calculated. Consequently, the optimal values of microwave cavity and detuning frequency from the gyrofrequency, for longitudinal cooling of a strongly magnetized electron plasma with microwave bath, have been found. By applying the optimal values with an appropriate microwave intensity, the best cooling can be obtained. For the electron plasma magnetized with 10T, the cooling time to the solid state is approximately 2 hours.

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Our goal is to construct a model for the nonlinear saturation of the ETG instability, which is one possible explanation of observed electron transport in tokamaks. We will present a hamiltonian, in slab geometry, for electron dynamics due to E x B drift and E|| acceleration. We will also present preliminary results of an electron resonant with the ETG mode.

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We have measured the heating rate of laser-cooled ions in a Penning trap using Doppler laser spectroscopy and observed evidence of the solid-liquid phase transition. Between 104 and 106, Be+ ions are trapped in a 4.5 Tesla Penning trap and laser-cooled to around 1 mK, where they form a crystalline plasma. This system is a rigorous realization of a one-component plasma. The ion temperature is measured as a function of time after turning off the laser-cooling and a rapid temperature increase is observed as the plasma undergoes the solid-liquid phase transition. We present evidence that this anomalous heating is caused by a sudden release of energy from a non-thermally excited mode of the plasma, presumably the cyclotron mode of heavier-mass ions surrounding the Be+ ions.

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The use of nonlinear focusing in particle accelerators has been proposed in a variety of applications. This work proposes and studies yet another application and analyzes the dynamics associated with nonlinear focusing. To begin with, it is proposed that beam halos can be controlled by combining nonlinear focusing and collimation, which is verified by numerical simulations. The study relies on a one dimensional, continuous focusing Particle-in-Cell (PIC) model and a Particle-Core model. Results from the PIC simulations establish the importance of reducing the mismatch of the beam in order to reduce halo formation. It is then shown that nonlinear focusing leads to damping of the beam oscillations thereby reducing the mismatch. This damping is accompanied by emittance growth causing the beam to spread in phase space. To compensate for this, the beam is collimated and further evolution of the beam shows that the halo is not generated. The use of the idealized, one-dimensional, continuous focusing model is justified by analyzing nonlinear alternate gradient focusing systems. The Lie Transform perturbation theory is used to derive an equivalent continuous focusing system for the alternate gradient focusing channel by canonically averaging over the lattice or fast oscillating time scale. The analysis shows the existence of a condition in which the system is azimuthally symmetric in the canonically transformed, slowly oscillating frame. Numerical results show that this condition leads to reduced chaos and improved confinement in the charged particle motion. The Lie Transform analysis is then extended to include space charge effects which enables one to calculate a near equilibrium distribution function which is azimuthally symmetric in the nonlinear lattice.

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Using recent theoretical developments concerning the fluid moments of the gyro-averaged Vlasov equation, we have implemented a high-resolution computational model of Alfvén wave propagation. We use full electron and ion gyrofluid equations. This model includes electron inertia and ion pressure terms. We include realistic variations in plasma density, temperature and magnetic field. The resulting profiles feature variations of several orders of magnitude. We study how Alfvén waves propagate from the Earth's magnetosphere into the aurora cusp region of the ionosphere. This model includes a finite electric field parallel to magnetic field lines, and can therefore be used to study the contribution of the Alfvénic disturbances to the acceleration of charged particles. We reproduce the physical phenomena of ionospheric resonance, dispersive electron acceleration, and cold electron burst acceleration. We also study the propagation of Alfvénic disturbances originating from the Io torus into the Jovian ionosphere. The Jovian ionospheric resonator is shown as a possible generator of the observed S-burst radiation emitted from the Io magnetic footprint.

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Global studies of magnetically confined plasmas have, up to now, relied primarily on the MHD plasma model to study the macroscopic stability and nonlinear dynamics of large plasma systems. It has long been known that basic two-fluid processes, i.e., allowing the electrons and ions to move independently while keeping the set of velocity-moment equations, have important effects on plasma instabilities. Recent developments in computational power and numerics allow two-fluid and other extended MHD models to be used for nonlinear simulation. Results are presented to show that two-fluid effects change the steady state picture and beta limits of a helical-toroidal fusion plasma (the proposed high beta stellarator NCSX), compared to MHD. Magnetic reconnection is enhanced and becomes an important limiting factor in two-fluids. The results explain a number of puzzling experimental observations in stellarators.

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The nonlinear dynamics of single ions inside the magnetic field reversed configuration (FRC) were investigated. Due to the high nonlinearity in the equations of motion, the behavior of the system is extremely complex, showing different regimes, depending on the values of the conserved azimuthal angular momentum and the geometry of the fusion vessel. The averaged Hamiltonian was used to study the structure of phase space and find the location of major resonances in the nonlinear regime. The condition for the onset of strong chaos was obtained using Chirikov island overlap criteria. A linear regime was found at higher values of azimuthal angular momenta, where the unperturbed Hamiltonian has a form of two uncoupled simple harmonic oscillators.

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