Download or read book Plasma Oscillations and Associated Electron Transport Within Hall Thrusters written by Aaron Kombai Knoll and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hall thruster is a type of plasma propulsion system for space vehicle applications. The thrust produced by this device is derived from the momentum of ions, which are accelerated to high exit velocities by the action of an electric field sustained within the plasma. The advantage of the Hall thruster compared to conventional chemical rocket propulsion is a significantly higher exhaust velocity, which leads to better utilization of propellant mass. Since the early days of Hall thruster research, experiments have suggested that the mobility of electrons along the axis of the thruster, perpendicular to an imposed magnetic field, is higher than can be explained by classical collision transfer processes alone. A lack of understanding regarding the mechanism for this enhanced mobility has proved a significant challenge toward the development of reliable simulations capable of predicting the performance of these devices. This thesis examines the role of high frequency plasma oscillations on the electron mobility using a combination of experimental studies on a laboratory Hall thruster, and numerical simulations capable of capturing these oscillations and quantifying their impact on the electron mobility. Two high frequency oscillations were consistently observed in the experiments: a 10MHz mode which appeared strongest in the vicinity of the anode, and a 4.5MHz mode which was strongest in the mid-channel region of the thruster. These were relatively low wave number (long wavelength) oscillations: approximately 6cm for the 4.5MHz oscillation and 3cm for the 10MHz oscillation. The angle of these waves varied considerably depending on the operating conditions of the thruster. They were found to be closely aligned to the axis of the thruster for experiments conducted with Xenon propellant, and were aligned with the circumference of the thruster (in the direction of electron drift) for experiments conducted with Krypton. A Hall thruster simulation, formulated in the axial-azimuthal coordinates of the thruster, was able to capture high frequency oscillations in reasonable agreement with experimental findings: 13MHz near the anode and 5MHz in the mid-channel region of the thruster for 160V discharge conditions. The simulation results demonstrated the crucial role of these oscillations in regulating the electron transport. In the vicinity of these oscillations the electron mobility was increased by a factor of five or more. The central finding of this thesis is that high frequency oscillations in the range 1 - 50MHz can account for the observed discrepancy between classical and experimental electron mobility in the Hall thruster.

Download or read book Plasma Oscillations and Associated Electron Transport Within Hall Thrusters written by A. K. Knoll and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Fundamentals of Electric Propulsion written by Dan M. Goebel and published by John Wiley & Sons. This book was released on 2008-12-22 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: Throughout most of the twentieth century, electric propulsion was considered the technology of the future. Now, the future has arrived. This important new book explains the fundamentals of electric propulsion for spacecraft and describes in detail the physics and characteristics of the two major electric thrusters in use today, ion and Hall thrusters. The authors provide an introduction to plasma physics in order to allow readers to understand the models and derivations used in determining electric thruster performance. They then go on to present detailed explanations of: Thruster principles Ion thruster plasma generators and accelerator grids Hollow cathodes Hall thrusters Ion and Hall thruster plumes Flight ion and Hall thrusters Based largely on research and development performed at the Jet Propulsion Laboratory (JPL) and complemented with scores of tables, figures, homework problems, and references, Fundamentals of Electric Propulsion: Ion and Hall Thrusters is an indispensable textbook for advanced undergraduate and graduate students who are preparing to enter the aerospace industry. It also serves as an equally valuable resource for professional engineers already at work in the field.

Download or read book Field Structure and Electron Transport in the Near field of Coaxial Hall Thrusters written by Andrew Wayne Smith and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hall thruster is an electric propulsion device developed in the former USSR during the Cold War, capable of efficiently providing sustained, low-levels of thrust. Coaxial Hall thrusters are comprised of an annular channel (at the base of which the anode is generally found), and a series of electromagnets that produce a predominantly radial magnetic field near the channel exit. A cathode, located outside the annular channel, injects electrons that serve a dual purpose: they neutralize the ion beam, and they sustain the core discharge. They plasma ions can achieve considerable exhaust velocities, lending the Hall thruster a high specific impulse; however, the propellant flow rate is generally on the order of a few mg/s, keeping the overall thrust low. Despite their desirable high efficiency, the detailed physics of Hall thruster operation is not clearly understood. In particular, the mechanism by which electrons are able to diffuse across the magnetic field lines at a rate in excess of classical predictions is the subject of dispute and ongoing research. Rectifying this deficiency within the near-field region (defined to lie between the exit plane of the annular channel and the external cathode) is the primary motivation for this work. A clear understanding of the mechanisms of electron transport in the near-field can aid the development of more efficient thrusters and provide direction for future experiments. The present study approaches the problem on two fronts. First, an extensive, 3-D map of the plasma potential (in addition to the floating potential and electron temperature) is obtained via a series of time-resolved experiments. These transient measurements are referenced to the periodic oscillation in the discharge current of Hall thrusters (known as the breathing-mode) and provide an unprecedented visualization of the low-frequency field dynamics. Second, the electron transport physics in the near-field is investigated in 3-D, electron-kinetic simulations. These simulations implement the experimentally-observed plasma potential (and, in some cases, fluctuations in the plasma potential). These simulations demonstrate that the 3-D nature of the fields is an important driver of near-field transport; however, collisions with the front-face of the thruster are critical to the anomalous diffusion of electrons across the magnetic field lines in this region. In simulations that considered static fields, up to 35 % of the electrons reached the channel during simulated lifetimes exceeding 1 microsecond, but often yielded very inhomogeneous density distributions. Imposing the measured helical plasma potential fluctuations in the simulations resulted in a dramatic azimuthal homogenization of the electron density distribution, and reduced the fraction of electrons reaching the channel to about 10 %, on par with experimental observations. In every case tested, plasma potential fluctuations (both axial and helical at a variety of frequencies) reduced the electron current reaching the channel. The results further suggest that the location and orientation of the cathode (as well as the properties of the emitted electron beam) have a strong effect on the transport. Gas-phase collisions, even when allowed to occur at a greatly exaggerated rate, are found to have negligible effect on either the channel/beam current ratio or the density distribution in the near-field. These results also suggest that random turbulence in the plasma properties (at least for frequencies less than or equal to 10 MHz) is unlikely to significantly impact the net electron transport (i.e., the channel/beam current ratio or density distribution). Importantly, axisymmetric simulations are found to yield dramatically disparate results (often yielding zero electron-current transport to the channel) compared to the simulations that considered 3-D fields (which introduce azimuthal components in the electric and magnetic fields); a result which questions the validity of pervasive 2-D Hall thruster simulations.

Download or read book Plasma Engineering written by Michael Keidar and published by Academic Press. This book was released on 2018-08-06 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: Plasma Engineering, Second Edition, applies the unique properties of plasmas (ionized gases) to improve processes and performance over many fields, such as materials processing, spacecraft propulsion and nanofabrication. The book considers this rapidly expanding discipline from a unified standpoint, addressing fundamentals of physics and modeling, as well as new and real-word applications in aerospace, nanotechnology and bioengineering. This updated edition covers the fundamentals of plasma physics at a level suitable for students using application examples and contains the widest variety of applications of any text on the market, spanning the areas of aerospace engineering, nanotechnology and nanobioengineering. This is highly useful for courses on plasma engineering or plasma physics in departments of Aerospace Engineering, Electrical Engineering and Physics. It is also useful as an introduction to plasma engineering and its applications for early career researchers and practicing engineers. Features new material relevant to application, including emerging areas of plasma nanotechnology and medicine Contains a new chapter on plasma-based control, as well as a description of RF and microwave-based plasma applications, plasma lighting, reforming and other most recent application areas Provides a technical treatment of the fundamental and engineering principles used in plasma applications

Download or read book Simulation of High Frequency Plasma Oscillations Within Hall Thrusters microform written by Knoll, Aaron Kombai and published by Library and Archives Canada = Bibliothèque et Archives Canada. This book was released on 2005 with total page 430 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Numerical Study of Current Driven Instabilities and Anomalous Electron Transport in Hall effect Thrusters written by Jonathan Tran and published by . This book was released on 2017 with total page 79 pages. Available in PDF, EPUB and Kindle. Book excerpt: Plasma turbulence and the resulting anomalous electron transport due to azimuthal current driven instabilities in Hall-effect thrusters is a promising candidate for developing predictive models for the observed anomalous transport. A theory for anomalous electron transport and current driven instabilities has been recently studied by [Lafluer et al., 2016a]. Due to the extreme cost of fully resolving the Debye length and plasma frequency, hybrid plasma simulations utilizing kinetic ions and quasi-steady state fluid electrons have long been the principle workhorse methodology for Hall-effect thruster modeling. Using a reduced dimension particle in cell simulation implemented in the Thermophysics Universal Research Framework developed by the Air Force Research Lab, we show collective electron-wave scattering due to large amplitude azimuthal fluctuations of the electric field and the plasma density. These high-frequency and short wavelength fluctuations can lead to an effective cross-field mobility many orders of magnitude larger than what is expected from classical electron-neutral momentum collisions in the low neutral density regime. We further adapt the previous study by [Lampe et al., 1971] and [Stringer, 1964] for related current driven instabilities to electric propulsion relevant mass ratios and conditions. Finally, we conduct a preliminary study of resolving this instability with a modified hybrid simulation with the hope of integration with established hybrid Hall-effect thruster simulations.

Download or read book Two dimensional Axial azimuthal Z Simulation of Cross field Electron Transport in a Hall Thruster Plasma Discharge written by Cheryl Meilin Lam and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hall thruster (or Hall effect thruster) is an electric propulsion device used for space flight applications. Despite its use as a deployed production technology, much of the underlying plasma physics which governs thruster behavior and performance is not well understood. Specifically, laboratory experiments indicate an anomalously high electron mobility in the direction perpendicular to the magnetic field, which exceeds that predicted by classical theory. Predicting this so-called anomalous electron transport remains a key research challenge. One possible mechanism for the generation of super-classical electron transport is the interaction of correlated quasi-coherent fluctuations in the plasma properties. Instabilities in the plasma can lead to quasi-coherent wave fluctuations in the electric potential, electron number density, and electron velocities; if these fluctuations are appropriately correlated, they can serve to either enhance or reduce electron transport across the magnetic field. In this work, we use numerical simulations as a tool to characterize axial and azimuthal fluctuations in the plasma discharge properties and study their impact on cross-field electron transport. We employ a two-dimensional axial-azimuthal (z-theta) model to simulate an annular Hall thruster discharge. We use a hybrid fluid-Particle-In-Cell approach in which the positive ion (Xe+) and neutral (Xe) species are modeled using a Particle-In-Cell (PIC) treatment and the electrons are modeled as a fluid continuum. The ion and neutral species are modeled as discrete collisionless superparticles; due to their large mass and consequently large Larmor radius, we neglect the magnetic field effect on the ions. For the electron fluid, we include the first three moments of the Boltzmann equation to obtain 2D continuity and momentum equations, using the drift-diffusion approximation, and a quasi-1D energy equation. The PIC and fluid treatments are coupled by assuming space charge neutrality, or quasineutrality, between the ions and electrons. We chose a simulated thruster model geometry and operating conditions to enable comparisons to experimental measurements of the Stanford Hall Thruster (SHT) laboratory discharge. The simulated thruster channel is 8 cm long, with an outer diameter of 9.4 cm; we include the full azimuth throughout the simulated domain, which includes the entire channel length and the near-plume. Using a non-uniform spatial resolution of 3 mm - 10 mm and maximum time step of 10 ns, we can achieve a simulated time of extent on the order of milliseconds, using a single PC processor core for a wall clock time of several days. We present results for a representative simulated low voltage operating condition. Simulated plasma properties are compared to experimental measurements of the plasma properties and the effective electron mobility. We further analyze the simulated data to characterize predicted axial and azimuthal fluctuations in the electric potential, electron number density, and electron velocities. We consider the simulated wave fluctuations in the context of linearized fluid theory models for specific dispersive propagation modes, as we attempt to characterize their impact on the effective axial electron transport for various axial regions within the thruster discharge. For the simulated time and spatial scales presented here, correlated fluctuations appear to enhance electron transport in some regions of the discharge and inhibit electron transport in others. In the mid-channel region, where we believe gradients and in the electron density and magnetic field may contribute to gradient-driven waves, we observe enhancement of the electron mobility beyond classical mobility values. Near the channel exit plane, however, we observe a distinct electron transport barrier, similar to that observed in experimental measurements. Just upstream of the channel exit plane, correlated fluctuations in the electron number density and the axial electron velocity appear to generate negative current which opposes the positive bulk discharge current; in this region, we believe the axial shear in the electron velocity may play a role in disrupting fluctuations and reducing electron transport. In both cases, it is clear that simulated wave fluctuations impact axial electron transport. Even in regions of observed transport enhancement, however, the simulated fluctuation-driven transport does not fully account for the experimentally-observed super-classical mobility. We believe that an additional transport mechanism -- perhaps electron wall scattering or higher frequency, shorter wavelength fluctuations -- is necessary to account for the experimentally-observed electron mobility. Towards this end, we present results for additional simulations which include an artificially enhanced electron collision frequency; these simulations show improved agreement with experimental results and confirm the need to include additional physical mechanisms for anomalous electron transport. Finally, suggestions for future work are included.

Download or read book Waves and Instabilities in E B Dusty Plasma written by Sukhmander Singh and published by . This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hall thrusters are common examples of E √ó B configuration, where electron trajectory gets trapped along the external magnetic field lines. This significantly increases the residence time of electrons in the plasma discharge channel. Hall thrusters are potential candidates for spacecraft station keeping, rephrasing and orbit topping applications because of its high thrust resolutions and efficiency. The goal of this chapter is to explain the working principle of Hall thrusters and to characterize the resistive instability in hot dusty plasma. The studies of these instabilities are useful to design efficient Hall thrusters and to understand the solar dusty plasma. The large amplitude of these oscillations has an adverse effect on the power processing unit of the devices. This reduces the efficiency and specific impulse and shortens the operating life of the Hall thruster. The theory of linearization of fluid equation for small oscillation has been given. The chapter also discusses the origin of plasma oscillation in a plasma discharge mechanics.

Download or read book Electron Transport and Ion Acceleration in a Low power Cylindrical Hall Thruster written by and published by . This book was released on 2004 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Conventional annular Hall thrusters become inefficient when scaled to low power. Cylindrical Hall thrusters, which have lower surface-to-volume ratio, are therefore more promising for scaling down. They presently exhibit performance comparable with conventional annular Hall thrusters. Electron cross-field transport in a 2.6 cm miniaturized cylindrical Hall thruster (100 W power level) has been studied through the analysis of experimental data and Monte Carlo simulations of electron dynamics in the thruster channel. The numerical model takes into account elastic and inelastic electron collisions with atoms, electron-wall collisions, including secondary electron emission, and Bohm diffusion. We show that in order to explain the observed discharge current, the electron anomalous collision frequency [nu][sub B] has to be on the order of the Bohm value, [nu][sub B] [approx] [omega][sub c]/16. The contribution of electron-wall collisions to cross-field transport is found to be insignificant. The plasma density peak observed at the axis of the 2.6 cm cylindrical Hall thruster is likely to be due to the convergent flux of ions, which are born in the annular part of the channel and accelerated towards the thruster axis.

Download or read book The Influence of Current Density and Magnetic Field Topography in Optimizing the Performance Divergence and Plasma Oscillations of High Specific Impulse Hall Thrusters written by National Aeronautics and Space Administration (NASA) and published by Createspace Independent Publishing Platform. This book was released on 2018-06-20 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent studies of xenon Hall thrusters have shown peak efficiencies at specific impulses of less than 3000 s. This was a consequence of modern Hall thruster magnetic field topographies, which have been optimized for 300 V discharges. On-going research at the NASA Glenn Research Center is investigating this behavior and methods to enhance thruster performance. To conduct these studies, a laboratory model Hall thruster that uses a pair of trim coils to tailor the magnetic field topography for high specific impulse operation has been developed. The thruster-the NASA-173Mv2 was tested to determine how current density and magnetic field topography affect performance, divergence, and plasma oscillations at voltages up to 1000 V. Test results showed there was a minimum current density and optimum magnetic field topography at which efficiency monotonically increased with voltage. At 1000 V, 10 milligrams per second the total specific impulse was 3390 s and the total efficiency was 60.8%. Plume divergence decreased at 400-1000 V, but increased at 300-400 V as the result of plasma oscillations. The dominant oscillation frequency steadily increased with voltage, from 14.5 kHz at 300 V, to 22 kHz at 1000 V. An additional oscillatory mode in the 80-90 kHz frequency range began to appear above 500 V. The use of trim coils to modify the magnetic field improved performance while decreasing plume divergence and the frequency and magnitude of plasma oscillations. Hofer, Richard R. and Jankovsky, Robert S. Glenn Research Center NASA/TM-2003-212605, E-14163, IEPC-2003-142

Download or read book Characterization of Inert Gas Plasma Thrusters written by and published by . This book was released on 2001 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The efficiency of modern Hall thrusters is impacted by the ability to maintain a high ratio of the ion to electron current density, J(sub i)/J(sub e), at the exit of the acceleration channel. In principal, there is no limit to this ratio, since the electrons that are necessary to maintain the current balance at the anode can be generated within the ionization zone upstream of the channel exit. However, in practice, this metric of performance in a typical Hall thruster can range from 1 - 10. This is in part due to the loss of electrons and ions through recombination reactions on the channel wall, and also due to the anomalous migration of electrons from the exit plane through the region of strong magnetic fields. It is believed that this poor confinement of electrons is in part due to turbulent fluctuations in plasma properties, which give rise Bohm-like electron transport in a direction orthogonal to the applied magnetic field. The Stanford Research was aimed at understanding this transport, and any affects that wall scattering may have on also enhancing electron current flow within the discharge channel. The broader long-term objective is to develop passive and/or active control strategies to enhance or suppress electron flow in regions of the channel where such actuation is desirable. Feedback-controlled actuation methods to reduce plasma fluctuations where only marginally successful back in the mid 70's in early versions of Hall thrusters 1,2. Using modern diagnostic and characterization strategies to obtain an improved understanding of the role that fluctuations and wall scattering plays in electron transport, it is anticipated that we can design more effective control and actuation strategies.

Download or read book Dynamic Models of Electron Transport in Hall Thruster Simulations written by Eunsun Cha and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hall-effect thruster (HET) is an electrostatic propulsion device that relies on the Hall effect to generate a dense ExB electron current to ionize the propellant gas. In simulating Hall thrusters, describing electron cross- eld transport has been one of the greatest challenges because the electron transport in a Hall thruster is anomalously higher than that predicted by classical collision theory. Researchers have suggested some explanations of the anomalous transport, but they have failed to establish a reliable physical model for general applications. Establishing a physical model that is applicable to various types of Hall thrusters in various operating conditions is an objective of this work. In this thesis, a 2-D hybrid particle-in-cell (PIC) simulation for the Stanford Hall thruster (SHT) is used to implement the transport (electron mobility) models. Among various attempts, an entropy closure model, as well as a turbulent transport model were successfully implemented and demonstrated results that show reasonable agreement to measured data. The entropy closure model uses a 1-D entropy transport equation in the plasma of a Hall thruster discharge to derive a relation for electron mobility as a function of other plasma properties. The simulated results show a reasonable agreement with experiments. The turbulent transport model seeks for a more straightforward way to incorporate the entropy production mechanism into the simulation. By assuming that the Joule heating is the main source of entropy production, we adopted the turbulent kinetic theory to relate the energy dissipated from the largest eddies with the energy production rate. Through a scaling analysis, electron mobility is expressed as an explicit function of other plasma properties of the simulation. The simulated electron mobility captures the electron transport phenomenon measured experimentally. To test the transportability of the turbulent model, the simulation was modi ed for an SPT-type thruster with a different geometry than the SHT. Also, an alternative propellant, molecular nitrogen, was simulated on the geometry of the SHT using the turbulent model. The dynamic mobility models make it possible to observe the dynamic characteristics of the Hall thruster. The mobility models in this study magnify the capability of Hall thruster simulations to explore design space cost effectively.

Download or read book Electron Transport and Ion Acceleration in a Low power Cylindrical Hall Thruster written by A. I. Smirnov and published by . This book was released on 2004 with total page 10 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Plasma Simulations by Example written by Lubos Brieda and published by CRC Press. This book was released on 2019-12-13 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of plasmas is crucial in improving our understanding of the universe, and they are being increasingly utilised in key technologies such as spacecraft thrusters, plasma medicine, and fusion energy. Providing readers with an easy to follow set of examples that clearly illustrate how simulation codes are written, this book guides readers through how to develop C++ computer codes for simulating plasmas primarily with the kinetic Particle in Cell (PIC) method. This text will be invaluable to advanced undergraduates and graduate students in physics and engineering looking to learn how to put the theory to the test. Features: Provides a step-by-step introduction to plasma simulations with easy to follow examples Discusses the electrostatic and electromagnetic Particle in Cell (PIC) method on structured and unstructured meshes, magnetohydrodynamics (MHD), and Vlasov solvers Covered topics include Direct Simulation Monte Carlo (DSMC) collisions, surface interactions, axisymmetry, and parallelization strategies. Lubos Brieda has over 15 years of experience developing plasma and gas simulation codes for electric propulsion, contamination transport, and plasma-surface interactions. As part of his master’s research work, he developed a 3D ES-PIC electric propulsion plume code, Draco, which is to this date utilized by government labs and private aerospace firms to study plasma thruster plumes. His Ph.D, obtained in 2012 from George Washington University, USA, focused on a multi-scale model for Hall thrusters utilizing fluid-kinetic hybrid PIC codes. He has since then been involved in numerous projects involving development and the use of plasma simulation tools. Since 2014 he has been teaching online courses on plasma simulations through his website: particleincell.com.

Download or read book Modeling Electron Transport Within the Framework of Hydrodynamic Description of Hall Thrusters Preprint written by and published by . This book was released on 2008 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper we consider kinetic effects related to electron transport in the framework on hydrodynamic model of the plasma flow inside the Hall thruster channel. In particular, kinetics of the near wall conductivity (NWC) is analyzed and analytical expression is derived that takes into account the sheath effects. The NWC model is incorporated into the hydrodynamic model. In addition we consider an effect of SEE electron thermalization. It is found that current predicted by the analytical model can provide reasonable solution without any fitting parameters.

Download or read book Introduction to Plasma Physics and Controlled Fusion written by Francis F. Chen and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 427 pages. Available in PDF, EPUB and Kindle. Book excerpt: TO THE SECOND EDITION In the nine years since this book was first written, rapid progress has been made scientifically in nuclear fusion, space physics, and nonlinear plasma theory. At the same time, the energy shortage on the one hand and the exploration of Jupiter and Saturn on the other have increased the national awareness of the important applications of plasma physics to energy production and to the understanding of our space environment. In magnetic confinement fusion, this period has seen the attainment 13 of a Lawson number nTE of 2 x 10 cm -3 sec in the Alcator tokamaks at MIT; neutral-beam heating of the PL T tokamak at Princeton to KTi = 6. 5 keV; increase of average ß to 3%-5% in tokamaks at Oak Ridge and General Atomic; and the stabilization of mirror-confined plasmas at Livermore, together with injection of ion current to near field-reversal conditions in the 2XIIß device. Invention of the tandem mirror has given magnetic confinement a new and exciting dimension. New ideas have emerged, such as the compact torus, surface-field devices, and the EßT mirror-torus hybrid, and some old ideas, such as the stellarator and the reversed-field pinch, have been revived. Radiofrequency heat ing has become a new star with its promise of dc current drive. Perhaps most importantly, great progress has been made in the understanding of the MHD behavior of toroidal plasmas: tearing modes, magnetic Vll Vlll islands, and disruptions.