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.

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.

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.

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.

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.

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.

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.

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.

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.

Using recent theoretical developments concerning the fluid moments of the gyro-averaged Vlasov equation, we have implemented a high-resolution computational model of Alfv?n 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 Alfv?n 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 Alfv?nic 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 Alfv?nic 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.

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.

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.

A growing body of experimental data suggests that non-diffusive radial transport of particles can play a major role in the SOL of tokamaks and other machines. This transport may be associated with the propagation of high density plasma filaments or "blobs" that have been observed in experiments with both fast cameras and probes. Theoretical work has shown that these structures propagate to the wall in the presence of an outwards force F and species- dependent F x B drifts. The transport occurs due to different types of forces in both toroidal and linear machines of greatly varying parameters, so the phenomenon is robust and surprisingly universal. The blob theory is qualitatively consistent with the experimental observations of convective transport, spatial and temporal intermittency, and non-Gaussian statistics in the SOL. Convective transport can be especially important for tokamaks because it reduces the efficiency of the divertor and may be related to the observed density limit on some machines. This talk will discuss the experimental motivation for this work, recent progress in theory and modeling, and some remaining unresolved issues.

This fall, Physics 4150, Introduction to Plasma Physics, is being taught with many of the homework problems assigned in Mathcad. The assigned problems are similar to exercises posted online at the course web site (debye.Colorado.edu/phys4150), thus the student's work will consist of downloading and modifying the posted Mathcad spreadsheets. For example, an assigned problem to find the potential created by an array of conductors might require only changing the boundary conditions in a spreadsheet that solves Laplace's equation. There are 25 posted exercises which 1) solve Laplace's and Poisson's equation by relaxation (illustrating Debye shielding), 2) plot the trajectories of magnetic field lines from straight wires and loops using Runge-Kutta (illustrating the dipole, x-points, shear, and rotational transform), 3) solve the Lorentz equations of motion for inhomogeneous magnetic fields by Runge-Kutta (illustrating the mirror force and drifts), 4) find the roots of complex dispersion relations using root finders including cases where the waves are growing or damped, 5) illustrate the principal value integral in the plasma dispersion function, 6) perform ray tracing in an inhomogeneous dielectric in the WKB approximation (illustrating ionospheric reflection), 7) illustrate diffusion, mobility and resistivity using equations of motion with Monte Carlo collisions, and 8) illustrate the ponderomotive force by modeling the Paul ion trap.