**
Numerical Challenges for Turbulence Computation: Statistical
Equipartition and the Method of Spectral Reduction**

John C. Bowman, B. A. Shadwick, and P. J. Morrison

**Abstract**

Numerical issues in the implementation of spectral reduction, a new method for the computation of statistical moments of homogeneous turbulence, are examined. The method implements a coarse graining in Fourier space and exploits the fact that statistical moments are much smoother functions of wave number than the underlying fluctuating velocities. A notable feature of this turbulence model is the existence of a control parameter (bin size) that can be varied to increase the accuracy of the approximation. The inviscid version of spectral reduction satisfies a Liouville theorem and yields statistical equipartition solutions. However, if the wavenumber bins are of nonuniform size (as is desirable for efficiency), an additional bin-dependent rescaling of time by the relative bin area must be introduced to obtain the correct equipartition. This rescaling of the time derivative term drastically increases the stiffness of the spectrally reduced equations. The prospect of developing an implicit nonlinear integrator for this highly stiffened convection problem is examined.

**
Coulomb Scattering in a Strong Magnetic Field
**

B. Hu, W. Horton, C. Chiu, and T. Petrovsky

**Abstract**

The presence of a strong magnetic field intrinsically changes the orbit of
electrons in their Coulomb interactions with ions for a range of
parameters. The characteristic scale for the orbit modification is
r_{0} = (Zm_{e}/4pe_{0}B^{2})^{1/3} where
Z_{e} is the ion charge, m_{e} the electron mass and B the
magnetic field strength. The scale length is comparable to the
inter-particle spacing when n_{e}^{1/3} r_{0} =
(w_{pe}/w_{ce})^{2/3} (Z/4p)^{1/3}~ 1 where w_{pe} / w_{ce} are the electron plasma and cyclotron
frequencies, respectively. For large angle scattering events we show
complex chaotic scattering interaction events for low-energy electrons. The
scattering angle has a fractal dependence on the impact parameter in the
chaotic scattering intervals. The process is thought to be important in
strongly magnetized, low-temperature plasmas, but the overall macroscopic
effects remain to be determined. Test particle simulations are presented
that show probability distributions are required to describe the outgoing
states of the events with fixed impact parameter and energy which specify a
unique Rutherford scattering angle.

**
Finite Larmor-radius theory of magnetic island evolution
**

F. L. Waelbroeck, J. W. Connor and H. R. Wilson

**Abstract**

Gyro-kinetic theory is used to investigate the effect of the polarization drift on magnetic island evolution. Three regimes are found. For island phase velocities between the ion and electric drift-velocities the polarization current is shown to be stabilizing. For phase velocities between the electric and electron drift-velocities, the island emits drift waves. This results in a radiative drag force. For all other phase velocities the polarization current is destabilizing, in agreement with the fluid limit.

**
Temporal Evolution of Drift Alfvén Waves in an Inhomogeneous Plasma
with Homogeneous Shear Flow
**

Vladimir S. Mikhailenko, Martin F. Heyn, and Swadesh M. Mahajan

**Abstract**

The temporal evolution of drift-Alfvén waves in a plasma flow with homogeneous shear is studied as an initial value problem without the use of spectral expansion in time. It is shown that the conventional modal structure of the drift-Alfvén instability (as well as drift and Alfvén waves) pertains only in the initial stage of its evolution. For larger times, non-modal effects due to the velocity shear impede the development of the drift-Alfvén instability; the frequency increase caused by the shear flow brings the Alfvén wave phase speed close to the electron thermal speed where strong electron Landau damping occurs. At this stage, a kinetic approach for the description of these waves becomes necessary. For the observed magnitudes of the velocity shear in the enhanced confinement regime in the edge layer of present-day tokamaks, non-modal effects may actually control the temporal evolution of initial perturbations on time scales less than the inverse growth rate of the modal drift-Alfvén instability and, thus, may prevent the development of this instability in the edge layer.

**
A Laser Electron Accelerator System for Radiation Therapy
(M.A. Thesis)
**

Frank Raischel

**Abstract**

A new compact electron accelerator for radiation therapy, based on laser wake field acceleration, is presented. To ensure its suitability for medical application, an achromatic magnet system to broaden and collimate the electron beam is introduced, and the radiological quantity of dose is calculated. Background information on dose and its practical computation, on the acceleration mechanism, and on the design of magnet systems is provided. The system is found to achieve operation parameters comparable to present-day linear accelerators, while its dimensions are anticipated to be smaller. Further research and scrutiny is required before practical use can be conceived.

** Ion kinetics in a magnetized plasma source **

Boris N. Breizman and Alexey V. Arefiev

**Abstract**

There are regimes of plasma source operation in which the ion motion is controlled by the ambipolar electric field and ion-atom collisions. This paper presents a derivation of the corresponding ion distribution function and plasma density profiles in the source for a given electron temperature and a given gas ionization rate. Conditions are discussed under which the ion flux is predominantly radial or predominantly axial. It is found that the presence of plasma can substantially change the neutral gas pressure due to production of fast charge-exchange neutrals.

**
Theoretical Interpretation of Alfvén Cascades in Tokamaks
with Non-monotonic q-profiles
**

H. L. Berk, D. N. Borba, B. N. Breizman, S. D. Pinches, and S. E. Sharapov

**Abstract**

Alfvén spectra in a reversed-shear tokamak plasma with a population of energetic ions exhibit a quasiperiodic pattern of primarily upward frequency sweeping (Alfvén Cascade). Presented here is an explanation for such asymmetric sweeping behavior which involves finding a new energetic particle mode localized around the point of zero magnetic shear.

**
Frequency Sweeping of Phase Space Structures
**

D. Yu. Eremin and H. L. Berk

**Abstract**

Holes and clumps, spontaneously formed in the nonlinear development of a single mode driven by a weakly destabilizing kinetic distribution function in a dissipative medium, were expected to persist for an appropriate collisional time scale. However, Fokker-Planck calculations for the nonlinear system abruptly break down in the course of the calculation for some initial conditions. We find that this is because an adiabatic description of phase space structures at zero collisionality does not necessarily lead to continual adiabatic frequency sweeping. Criteria are found for a class of initial distribution functions that determine whether adiabatic frequency sweeping continues indefinitely,å or suddenly terminates. Passing particle contribution is found to be unimportant in the dynamics of the phase space structures in the framework of adiabatic description. An unresolved bifurcation problem is also observed under certain conditions.

**
Fluid Description of Relativistic, Magnetized Plasma
**

R. D . Hazeltine and S. M. Mahajan

**Abstract**

Many astrophysical plasmas and some laboratory plasmas are relativistic, in that either the thermal speed or the local flow speed in a convenient reference frame approaches the speed of light. Conventional fluid theories of magnetized plasma are not consistent with special relativity; indeed the usual definition of what constitutes a magnetized plasma must be reformulated in the relativistic case.å Beginning with exact moments of the kinetic equation, we derive a closed set of Lorentz--covariant fluid equations.å The system allows for anisotropy of the pressure tensor as well as heat flow along the magnetic field.å When anisotropy and heat flow are suppressed the closed set of fluid equations becomes a manifestly covariant expression of relativistic MHD.

**
Predictive Modelling and Simulations of
Internal Transport Barriers in Tokamaks
(Ph.D. Thesis)
**

Ping Zhu

**Abstract**

An Internal Transport Barrier (ITB) is a localized region inside a (tokamak) plasma where a steep temperature and/or density gradient forms due to much lower thermal and/or particle transport than in the surrounding regions. Internal transport barriers have now been observed in all large tokamaks after they were first discovered in the Japan Atomic Energy Research Institute Tokamak-60 Upgrade (JT-60U) in 1993. While suggesting a promising practical approach to the realization of fusion ignition conditions, this high-performance regime poses a great challenge to our understanding of tokamak anomalous transport in physics.

**
Magneto-Fluid Coupling - Eruptive Events in the Solar Corona
**

Shuichi Ohsaki, Nana L. Shatashvili, Zensho Yoshida and Swadesh M. Mahajan

**Abstract**

By modeling the coronal structures by "slowly" evolving Double-Beltrami two-fluid equilibria (created by the interaction of the magnetic and velocity fields), the conditions for catastrophic transformations of the original state are derived. It is shown that, at the transition, much of the magnetic energy of the original state is converted to the flow kinetic energy.

**
IGNITOR Physics Assessment and Confinement Projections
**

W. Horton, F. Porcelli, P. Zhu, A. Aydemir, Y. Kishimoto, and T. Tajima

**Abstract**

An independent assessment of the physics of Ignitor is presented. Ignitor is a physics demonstration experiment, with the main goal of achieving thermonuclear ignition, defined as the regime where fusion alpha heating compensates for all forms of energy losses. Simulations show that a pulse of a particle power up to 10-20 MW is produced for period over a few seconds. Crucial issues are the production of peaked density profiles on a time-scale of several energy confinement times, the control of current penetration for the optimization of ohmic heating and sawtooth avoidance. The presence of a 10-20 MW Ion Cyclotron Radio frequency system and the operation of a high-speed pellet injector are considered essential to provide added flexibility in order to counter unexpected, adverse plasma behavior.

**
Self-trapping of strong electromagnetic beams in relativistic plasmas
**

V. I. Berezhiani, S. M. Mahajan, Z. Yoshida, and M. Ohhashi

**Abstract**

Interaction of an intense electromagnetic (EM) beam with hot relativistic
plasma is investigated. It is shown that the thermal pressure brings about
a fundamental change in the dynamics --- localized, high amplitude, EM
field structures, not accessible to a cold (but relativistic) plasma, can
now be formed under well-defined conditions. Examples of the trapping of EM
beams in self-guiding regimes to form stable 2D solitonic structures in a
pure *e-p* plasma are worked out.

**
Dynamics of self-trapped singular beams in underdense plasma
**

V. I. Berezhiani, S. M. Mahajan, Z. Yoshida, and M. Pekker

**Abstract**

Dynamics of an intense short laser pulse with a phase singularity,
propagating in an underdense cold plasma, is investigated. Such a pulse can
propagate as a vortex soliton in a self-created channel. It is shown that
vortices with the topological charge *m=*1 (and a corresponding angular
momentum) are unstable against symmetry-breaking perturbations: the breakup
of the original vortex leads to the formation of stable spatial solitons
which steadily fly away tangentially from the initial ring of vortex
distribution.

**
Noise Effects, Emittance Control, and Luminosity Issues¬in Laser
Wakefield Accelerators
(Ph.D. THESIS)
**

Sergey Valeriev Cheshkov

**Abstract**

To reach the new high energy frontiers (higher than a TeV center of mass energy) new acceleration methods seem to be needed. The plasma-based wakefield accelerator is one possible candidate which can provide an ultra high gradient acceleration and thus make the total acceleration distance reasonable. However, the final energy is not the only requirement. The accelerator should maintain an excellent beam quality to meet the luminosity requirements at the Interaction Point (IP). One of the most important figures of merit which describes the quality of the beam is its emittance. We study the particle dynamics in laser pulse-driven wakefields over multi-stages in a several TeV range center-of-mass energy e+e- collider. The approach is based on a map of phase space dynamics over a stage of wakefield acceleration induced by a laser pulse (or electron beam). The entire system of the collider is generated with a product of multiple maps of wakefields, drifts, and magnets, etc. This systems map may include offsets of various elements of the accelerator, representing noise and errors arising from the operation of such a complex device. We find that an unmitigated strong focusing of the wakefield coupled with the alignment errors of the position (or laser beam aiming) of each wakefield stage and the unavoidable dispersion in individual particle betatron frequencies leads to a phase space mixing and causes a transverse emittance degradation. The rate of the emittance increase in the limit of constant energy is proportional to the number of stages, the energy of the particles, the betatron frequency, the square of the misalignment amplitude, and the square of the betatron phase shift over a single stage. The accelerator with a weakened focusing force in a channel can, therefore,largely suppress the emittance degradation. To improve the emittance we introduce several methods: a mitigated wakefield focusing by working with a plasma channel, an approximately synchronous acceleration in a superunit setup, the ``horn'' model based on exactly synchronous acceleration achieved through plasma density variation and lastly an algorithm based on minimization of the final beam emittance to actively control the stage displacement of such an accelerator.

We analyze the IP Physics luminosity and background issues in a high beamstrahlung parameter regime using the Yokoya's Monte Carlo code ``CAIN''. The possibility for delivering polarized electron and positron beams at the collision point as an additional leverage to control the complicated background situation is also investigated. We prove that the initial beam polarization is not degraded significantly by the beam transport and acceleration in the plasma based wakefield accelerator and by the beamstrahlung at IP.

Finally, we propose a beam driven acceleration scheme which can provide an ultra high acceleration gradient (greater than 100 GeV/m). This scheme is based on a reasonable modification of current SLAC beam parameters.

**
Structure Formation through Magnetohydrodynamical Instabilities in
Protoplanetary and Accretion Disks
(Ph.D. THESIS)
**

K. Noguchi

**Abstract**

Structure formation in various astronomical systems through magnetohydrodynamics (MHD) instabilities has been investigated. The effect of magnetic field enhancement in sheared flows is studied as the eigenmode problem in a non-self-adjoint system, and new mathematical and physical aspects of the instability are shown. The mechanism for the faster structure formation in a protoplanetary disk with MHD instabilities is suggested with linear analysis and simulation. Experiments to simulate the plasmas of protoplanetary and active galactic nuclei are also suggested. The coupling of magnetic field enhancement and magnetic buoyancy has been studied and mode coupling of two instabilities are shown.

**
The Twisted Top
**

Jean-Luc Thiffeault and P. J. Morrison

**Abstract**

We describe a new type of top, the twisted top, obtained by appending a cocycle to the Lie--Poisson bracket for the charged heavy top, thus breaking its semidirect product structure. The twisted top has an integrable case that corresponds to the Lagrange (symmetric) top. We give a canonical description of the twisted top in terms of Euler angles. We also show by a numerical calculation of the largest Lyapunov exponent that the Kovalevskaya case of the twisted top is chaotic.

**
Coupling of the Resistive Wall Mode to Liquid Wall Surface Modes
**

H. L. Rappaport

**Abstract**

An analysis of the resistive wall mode of a fusion plasma surrounded by a cylindrically symmetric rotating thin liquid metal shell is given. Coupling between the resistive wall mode and free fluid boundary surface modes is described. Consequences of fluid perturbations induced in the wall, on the wall mode growth rate, are found to be greater when the wall fluid has a vacuum-fluid interface than when the fluid is forced to flow between solid bounding shells. The effect of the fluid perturbations is always found to be destabilizing, except close to the wall mode/ surface mode resonance. Criteria for resonance, an analytic solution to the dispersion relation for the wall mode in the resonance regime, and implications of this work for relevant experiments, are given.

**
IFSR-919**

**
Proton-Boron (p-B ^{11}) Colliding Beam Fusion Reactor
**

H. Vernon Wong, B. N. Breizman, and J. W. Van Dam

**Abstract**

In the proton-boron colliding beam fusion reactor, the power which must be supplied to maintain an optimal colliding beam configuration is estimated to be at least 5.1 times greater than the fusion power. This implies that effective power conversion efficiencies to electrical power in excess of 84% will be required. Furthermore, if the transverse collisional spread of the proton beam is to be limited by electron drag, the boron density is constrained to have magnitudes well below the optimal value at which the fusion power is maximised.

**
IFSR-918**

**
Prediction and Modeling of Magnetospheric Substorms**

R. S. Weigel

**Abstract**

The interaction of the solar-wind plasma with the magnetic field of Earth is a dynamic, multi-scale process that can cause strong variations in near-Earth currents and fields. During magnetic substorms, as much as one hundred Terajoules of energy are dissipated in the ionosphere. Electron energization can cause communication satellite interruptions, and rapid changes in ionospheric currents cause ground induction currents that interfere with the operation of electric power grids.

On a time scale of greater than several minutes, the magnetosphere-ionosphere system behaves as a low-dimensional system in the sense that large-scale fields and currents can be predicted by models that have few degrees of freedom compared with the system as a whole. This thesis begins by comparing several low-dimensional models of the solar-wind driven magnetosphere-ionosphere system. The performance of these models is compared with a neural network predictive system that uses only measurements of the solar wind with a one-hour lead time to predict the maximum westward electrojet currents in the ionosphere. Two physics additions to these low-dimensional models are developed. A nonlinear conductance model that allows the conductance of the ionosphere to depend on the level of electron precipitation during the substorm is developed and tested against a substorm database. The coupled region 2 shielding currents are models, and their role in substorm dynamics is determined by simulation over a substorm database. Using the substorm model WINDMI as a guide, it is shown that most substorms in an isolated substorm database fall into one of three distinct categories. The category depends on the type of trigger initiating the fast unloading of energy stored in the magnetosphere onto the ionosphere.