- Seminars at CIPS

A new numerical algorithm that encompasses both the $\delta f$ particle-in-cell (PIC) method and a continuum similar to Denavit's "hybrid'' method has been analyzed using a new interpolation schemes. The basic algorithm is essentially a variant of the delta-f method. Briefly, the algorithm is the following: 1) load particles (or characteristics) on a uniform lattice in phase space. The loading need not be uniform, but it greatly simplifies the algorithm, 2) advance the characteristics M time steps, using the usual delta-f PIC algorithm which involves a grid interpolation, deposition, then field solve all on a spatial grid only, 3) every M time steps, deposit delta-f on a higher-dimensional phase space grid, then reset the particle phase space coordinates back to their initial value on the phase space lattice. Also, reset the particle value of delta-f to the phase space grid value. As M goes to infinity, one recovers the usual delta-f PIC algorithm with a somewhat peculiar uniform loading of particles. For M=1 the algorithm is similar to the Vlasov method of Cheng and Knorr. Any value of M={1,2,3,...} is permissible. A quadratic weighting interpolation scheme is implemented \cite{HE}for a small $(k_\perp \rho_i)^2$ two-dimensional bounded slab model, where the phase space is (x,y,v_parallel). The ion-temperature-gradient instability is studied assuming adiabatic electrons and gyrokinetic ions. We will present results with both linear and quadratic interpolation. We will calculate the effective phase space diffusion by such a repeated interpolation and compare with simulation results.

A new electromagnetic kinetic electron simulation model that uses a generalized split-weight scheme and a parallel canonical momentum formulation has been developed in three-dimensional toroidal flux-tube geometry. The long-standing problem in the simulation of kinetic electrons with finite-beta effects, associated with the electron current of the zero-order distribution (Maxwellian in terms of parallel canonical momentum), is solved by evaluating this current using the same marker particles and the same particle shape as that used for the perturbed distribution. The model also includes electron-ion collisional effects and has been linearly benchmarked with continuum codes. It is found that for H-mode parameters, the nonadiabatic effects of kinetic electrons increase linear growth rates of the Ion-Temperature-Gradien-Driven (ITG) modes, mainly due to trapped-electron drive. The ion heat transport is also increased from that obtained with adiabatic electrons. The linear behavior of the zonal flow is not significantly affected by kinetic electrons. The ion heat transport decreases to below the adiabatic electron level when finite plasma beta is included due to finite-beta stabilization of the ITG modes.

A large part of the expense associated with fusion experiments is due to the uncertainties in the dynamics of the plasma. These dynamics lead to instabilities that can spontaneously erupt and degrade the confinement properties of a plasma and sometimes lead to catastrophic disruptions of the entire plasma itself. These instabilities occur in a broad range of spatial and temporal scales, spanning many orders of magnitude, often resulting from nonlinear interactions. Computational simulations are crucial to understanding these phenomena. This thesis research focuses on the numeric study of kinetic effects on magnetohydrodynamic (MHD) instabilities in fusion plasmas. The significant achievement of this thesis work was the implementation of the $\delta f$ particle-in-cell(PIC) simulation in a general geometry, massively parallel, Lagrange-type finite element based, MHD simulation. This hybrid models captures kinetic effects that are not possible to simulate with a fluid MHD simulation alone. The use of the finite element method(FEM) allow the flexibility of modeling the realistic geometries of fusion devices. However, the irregularity of the simulation grid does not allow for conventional (PIC) coupling to the fluid elements represented by the finite element grid. Particular problems resolved include determining where the particle is in the grid, how to 'gather' the field to the particles, how to 'scatter' the particle effects onto the grid, and implementing in a massively parallel framework compatible with the MHD simulation. The addition of kinetic particle effects captures wave-particle interactions important in the saturation or excitation of various MHD instabilities such as the internal kink mode, sawtooth, and fish bone instabilities. We assume that the kinetic particles are an energetic minority species, i.e. kinetic particle density is small compared to the bulk plasma density but the kinetic particle pressure is comparable to the bulk plasma pressure. The kinetic particles are evolved in the MHD fields using the drift kinetic equations of motion. A pressure tensor is calculated from velocity moments of the kinetic particles. This hot particle pressure tensor is added to the MHD momentum equation forming the hybrid kinetic model. This hybrid kinetic-MHD technique lays the foundations for future work in a kinetic closure to the MHD equations.

A heavy ion beam probe has been used to measure density and electric potential fluctuations and the equilibrium electric potential in the core region (r/a = 0.3 to 0.6) of the Madison Symmetric Torus reversed field pinch. This is the first measurement of potential and potential fluctuations in a hot reversed field pinch plasma. In standard plasmas, the equilibrium potential is positive and 1 to 2 kV above machine ground with electric fields strengths of 0.7 to 3 kV/m. Plasmas with low flow velocities due to locking have much lower potentials and electric fields. The potential is inversely related to the density and the electric field is consistent with expectations from the ion momentum balance equation. The observed fluctuations consist of 3 components, a potential fluctuation with f < 10 kHz that is believed to be due to m=0 electric field fluctuations at the edge, density and potential fluctuations at f ~ 15 kHz that are coherent with the dominant m=1, n=6 magnetic fluctuations, and broadband density and potential fluctuations between 30 and 100 kHz. The ExB particle transport due to these fluctuations is much smaller than the total particle transport in a standard plasma.

This talk will begin with a brief introduction to Lie Transform perturbation theory and its use in averaging over small time scales. This is achieved by canonically transforming to slowly oscillating variables. The theory will be applied to a periodic focusing system, that is, a harmonic oscillator with a rapidly oscillating potential. This will be followed by a comparison with numerical results. After this, the theory will be applied to the dynamics of a charged particle in an accelerator with nonlinear focusing. The analysis yields a condition that improves integrability and minimizes chaos in such a system. This will be confirmed by numerical results.

In the optics of charged particle beams, circular transverse modes can be introduced; they provide an adequate basis for rotation-invariant transformations. A group of these transformations is shown to be identical to a group of the canonical angular momentum preserving mappings.

These mappings and the circular modes are parametrized similar to the Courant-Snyder forms for the conventional uncoupled, or planar, case. The planar-to-circular and reverse transformers (beam adapters) are introduced; their implementation on the basis of skew quadrupole blocks is described. Applications of the planar-to-circular, circular-to-planar and circular-to-circular transformers are discussed. A range of applications includes round beams at the interaction region of circular colliders, flat beams for linear colliders and relativistic electron cooling.

The dynamics of fusion plasmas lead to instabilities that can sponta- neously erupt and degrade confinement and sometimes lead to catastrophic disruptions of the entire plasma itself. These instabilities occur in a broad range of spatial and temporal scales, spanning many orders of magnitude, often resulting from nonlinear interactions. Computational simulations are crucial to understanding these phenomena.

NIMROD(NonIdeal MHD with Rotation - Open Discussion) is a mas- sively parallel three dimensional magnetohydrodynamic simulation utilizing finite elements (FE) to represent the poloidal plane and a fourier decompo- sition in the toroidal direction. The use of finite elements allows flexibility in the representation of the simulation domain. The ability to model ex- perimental shots with NIMROD provides a platform to test new ideas of plasma behavior. To expand the physics capabilities of NIMROD, kinetic effects have been added to NIMROD by the addition of delta-f PIC(Particle in Cell) module. The addition of kinetic particle effects captures essential wave-particle interactions important in the saturation of various MHD insta- bilities such as the internal kink mode, sawtooth and fishbone instabilities, and toroidal Alfven eigenmodes. Particle simulation capabilities in NIMROD can also be extended to simulate various phenomena such as neutral beam injection, ion cyclotron resonance heating, and anomalous los mechanisms. In addition, this hybrid kinetic-MHD technique lays the foundations for a kinetic closure to the MHD equations.

This talk will briefly introduce NIMROD and delta-f PIC in general, then detail the development of PIC in finite elements and their implementation and some preliminary results.

Alfven waves are ubiquitous in space plasma physics, present in field line bending, during magnetic reconnection, and magnetized shock formation. However, due to their typically long wavelength, comparatively few basic laboratory studies have been made of these waves. Experiments must either be long or dense to accommodate several aflv?n wavelengths. At New Mexico Tech we use a high-density helicon generated background plasma to study aflv?n waves under a steady-state current-free conditions. A summary of early studies of alfv?n waves will be presented, as well as our recent results on alfven propagation in plasmas with a varying neutral fraction. favorably.

Stray electrons are suspected of limiting the performance of many of today's ion accelerators. One source of these electrons is the electrons produced when halo beam ions strike the beam pipe walls. I will discuss computer models developed to help understand this process. In particular, I will discuss this effect as it applies to heavy ion fusion experiments at Lawrence Berkeley National Laboratory. I will show that one can expect as many as 1000 electrons to result from each ion collision and what researchers can do to mitigate the problem.

There are a few key obstacles standing in the way of achieving thermonuclear ignition at the National Ignition Facility (NIF). One of them is controlling parametric instabilities, especially Stimulated Raman Scattering (SRS), where the light wave decays into a scattered light wave and an electrostatic plasma wave (EPW). Parametric instabilities can spoil laser power coupling into the target, and can accelerate electrons which preheat the target. The linear-theory thresholds for significant SRS activity are routinely exceeded in experiments on ignition-relevant quasi-homogeneous plasmas. Experimental results from existing lasers make it clear that SRS saturates via non-linear processes. One possible saturation mechanism for SRS is coupling of energy from the SRS daughter EPW to other non-resonant EPWs [Baker et al., PRL 77 (1996) 67]. If the amplitude of the daughter SRS EPW is large enough, it can decay into a counter-propagating EPW and an ion acoustic wave (IAW), i.e., the Langmuir Decay Instability (LDI). Damping of all these waves ultimately saturates SRS. Another possible saturation mechanism is electron trapping by the SRS EPW. On the one hand, the process greatly reduces collisionless EPW damping from classical levels, promoting instability growth above linear theory predictions. On the other hand, the trapped electrons dynamically detune the EPW, and SRS saturates as a result [H.X. Vu et al., Phys. Plasmas 9, 1745 (2002)]. Theoretical considerations indicate that the dominant saturation mechanism should transition from the former to the latter at some value of the Debye length. Whether this theoretical framework is correct, and whether we can quantitatively predict the transition is a key question, and will help lead to a quantitative, predictive understanding of SRS.

Owing to the large areas and high plasma densities found in some recently developed devices, electrostatic theories of plasma resonances and surface wave propagation [1-2] are suspect as the size of the device is much larger than the free space wavelength associated with the peak plasma frequency. Accordingly, an electromagnetic model of surface wave propagation has been developed appropriate for large area plasmas. The predicted wave dispersion of the two models differs for extremely long wavelengths but is degenerate in devices small compared with wavelength. First principles particle-in-cell (PIC) simulations have been conducted which support these results. Given the slow wave character and boundary localized fields of surface waves, a periodic electrode may be used to resonantly excite a strong wave-particle interaction between surface waves and electrons. At saturation, the electron velocity distribution is enhanced above the phase velocity of the applied wave and suppressed below. The use of this technique (''Landau resonant heating'') to selectively heat the electron high energy tail to enhance electron-impact ionization is demonstrated using PIC simulation. A number of techniques to accelerate PIC simulations in this demanding regime were developed; without them, this research would not have been possible. An implicit method of solving the Maxwell equations which allows extremely high mesh Courant numbers (>100) which still retains the effects of displacement current (critical for these waves) was developed. Also, techniques to eliminate memory thrashing inherent in PIC methods were devised. These made it possible to run these large simulations (using ~30M particles) on a Pentium II 400 desktop.

[1] Nickel, Parker, Gould. Phys. Fluids. 7:1489. 1964.

[2] Cooperberg. Phys. Plasmas. 5, No. 4, April 1998.

In this talk I am going to present how to use the OpenDX package to visualize data stored in files of the HDF5 format. Beside giving an introduction to OpenDX and HDF5, two OpenDX modules (extensions) are going to be demonstrated, which have been developed at CIPS. They import HDF5 data about fields and particles into OpenDX. Moreover, I am going to outline both the usage of the two modules in the context of the Vorpal package and the design basics of OpenDX modules.

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.

Notions from information theory are applied to the venerable two-dimensional Ising model. A few intensive quantities are examined, the entropy density, or entropy per spin, and the information density, or information per spin stored by the system at a given instant of time. It is shown that the entropy per spin is observable, in the sense that it can readily be measured from the ensemble of patterns generated by a simulation. Just as the magnetization can be measured from statistics over a single spin, the entropy can be measured by considering only the local statistics over a unit square. This result can be generalized to higher dimensions, and other spin systems. The information per spin is the difference between the entropy of a single spin considered in isolation, and the entropy per spin of the pattern as a whole. This quantity has a sharp maximum at the phase transition. The possible usefulness of these notions to the study of nonlinear PDE's will be discussed. Time permitting, a number of real-time computer simulations of example nonlinear systems will be demonstrated.

In many applications involving intense beams, it is imperative to be able to limit the non-equilibrium fraction of the beam that may reside far from of the core of the distribution known as the beam halo. This aspect of the beam propagation has been notoriously difficult to predict, however recent progress has been made with the development of the so-called particle-core model. Since no direct verification of this model had been undertaken to date, we have carried out a precision experiment to measure halo generation associated with the transport of an intense proton beam through a linear transport channel. The LEDA RFQ was used to inject a 6.7 MeV 10-100 mA beam into a 52-quadrupole channel. Four matching quads at the input of this transport line were used to generate specific mismatch oscillations and the resulting beam profiles were measured at downstream locations over a very wide dynamic range. The results of these experiments tend to support the particle-core model and the significance of controlling mismatch oscillations in minimizing beam halo. However, some anomalous behavior has been observed which has not yet been explained by existing models. An overview of the halo generation process will be given followed by a detailed description of the experimental results.

Remarkably, a magnetized pure electron plasma can behave like an ideal two-dimensional fluid. Recently, such plasmas have been used to study the dynamics of two-dimensional vortices in a cloud of background vorticity. Experiments have shown that background vorticity can cool a chaotic system of intense vortices into a crystal equilibrium. Further experiments have shown that weak vortices tend to migrate to extrema of the background vorticity distribution. New theories have emerged to explain the experimental observations. In this talk, I will summarize the experiments and related theories, and show that they compare favorably.

High performance has been achieved in DIII-D and in many other tokamaks operating in advanced (AT) modes that exhibit high confinement, negative central shear (NCS), and/or internal transport barriers. A critical issue for sustaining high performance NCS discharges is the ability to maintain current distributions with an off axis maximum. Sustaining such hollow current profiles in steady state requires the use of non-inductively driven current sources. On the DIII-D experiment, a combination of neutral beam current drive (NBCD) and bootstrap current have been used to create transient NCS discharges. The electron cyclotron heating (ECH) and current drive (ECCD) system has recently been upgraded from three gyrotrons to six to provide 5MW of power in long-pulse operation to help sustain the required off-axis current drive. To investigate the effectiveness of the EC system and to explore operating scenarios to sustain these discharges, we use time-dependent simulations of the equilibrium, transport and stability. We explore methods to directly alter the safety factor profile, q, through direct current drive or by localized electron heating to modify the bootstrap current profile. Time dependent simulations using a gyro-Bohm-based model for the thermal conductivity indicate the ability to maintain the necessary q profile for several hundred energy confinement times. We will present details of these simulations exploring parametric dependencies of the heating, current drive, and profiles that affect our ability to sustain stable discharges.