**The Solar Wind-Driven Magnetosphere-Ionosphere as a
Complex Dynamical System**

W. Horton, J. P. Smith, R. S. Weigel, C. Crabtree, I. Doxas, B. Goode, and J. Cary

**Abstract**

The solar--wind driven magnetosphere--ionosphere system is a classic example of a complex dynamical system (CDS). The defining properties of a CDS are (1)~sensitivity to initial conditions; (2)~multiple space--time scales; (3)~bifurcation sequences with hysteresis in transitions between attractors; and (4)~noncompositional. Noncompositional means that the behavior of the system as a whole is different from the dynamics of its subcomponents taken with passive or no couplings. In particular the dynamics of the geomagnetic tail plasma depends on its coupling to the dissipative ionospheric plasma and on the nature of the solar wind driving electric field over a suitably long (many hours) previous time interval. These complex dynamical system features are shown here in detail using the known WINDMI model for the solar wind driven magnetosphere--ionosphere (MI) system. Numerous features in the bifurcation sequence are identified with known substorm and storm characteristics.

**IFSR-868**

**Laser-Cluster Interaction and its Application in Semi-Conductor
Processing**

Xiaoming Chen

**Abstract**

The cluster plasma is an ionized cluster material in which many new physics still remains unexplored in contrast to gas and solid phase plasmas which have been studied for a long time. We employ a two-dimensional relativistic electromagnetic particle-in-cell code with cluster features and apply it to advance theoretical and computational studies of laser-cluster interaction and its potential applications in next-generation post-optical lithography technology in semiconductor processing.

Our computer simulations have shown that high harmonics generated from Larmor and Bremsstrahlung radiation during the laser-cluster interaction, 3w, 5 w, and up to 13 w harmonics have been found. These high-frequency components constitute the EUV and X-ray production. The harmonics radiation may be developed as a potentially reliable exposure source for post-optical lithography tools. The computational simulation may be utilized for the design of a compact, reliable and affordable EUV and X-ray source.

We survey the optical properties of the "cluster mode" that a transverse electromagnetic wave may propagate with a frequency lower than the cut-off w_p plasma frequency. We in fact see that the "cluster mode" exists and the results are consistent with the previous work.

Applications associated with energetic particle generation from the laser-cluster interaction are discussed by showing detail how electrons are stripped off from clusters during the interaction. The charge separation between ions and electrons is clearly seen in the simulation.

Laser-defect interaction and its application in the semiconductor processing have been also discussed. Laser and high-prediction optics have been used more and more in cutting-edge semiconductor processing with the design rule down to 0.18µm. We have applied laser and high-precision optics principles to study the interaction between laser beams and defects. A calibration methodology has also been established in order to evaluate the detection capability for various commercial available laser scanners. The results from our work further the understanding of the optics, lasers and calibration methodology involved in the inspection technology. These results are also helpful to laser scanner manufacturers, so they can continue to improve and develop a better inspection tool, which can meet a future technology demand.

**IFSR-867**

**Coherent Structures in nonlinear Plasma Dynamics**

Nickolai Petviashvili

**Abstract**

Results of numerical and analytical studies of coherent structures in
plasmas are presented. The systems include interaction of a discrete mode
with particles in a weakly unstable plasma, the turbulent **E**x**B**
drift of the particles in an inhomogeneous magnetic field, radiative condensation
instability at the edge of a tokamak plasma, and non-radiative dynamics
of Langmuir solitons. All these works are united by a common approach to
the problem. This approach is based on the very close interaction of analytical
calculations and numerical simulations. The original results presented
in this dissertation illustrate the advantages of such an approach.

**Hamiltonian Description of Vlasov Dynamics: Action-Angle
Variables for the Continuous Spectrum**

Philip J. Morrison

**Abstract**

The linear Vlasov-Poisson system for homogeneous, stable equilibria is solved by means of a novel invertible integral transform that is a generalization of the Hilbert transform. The integral transform provides a means for describing the dynamics of the continuous spectrum that is well-known to occur in this system. The results are interpreted in the context of Hamiltonian systems theory, where it is shown that the integral transform defines a canonical transformation to action-angle variables. A means for attaching Krein signature to a continuum eigenmode is given.

**IFSR-865**

**Dynamical Range of the WINDMI Model: an Exploration
of Possible Magnetospheric States**

J.P. Smith, J-L. Thiffeault, W. Horton

**Abstract**

This paper explores the dynamical range of the WINDMI model [Horton and Doxas, 1998]. Such low-dimensional models provide us with the tools to understand the relationships of simple physical quantities within the magnetosphere (such as energy deposition, macroscopic currents, cross-tail voltage, etc.) without the necessity of coping with the more complete but unwieldy models (MHD or particle codes, for example). The model is highly versatile: certain regions of the parameter space support stable fixed points, while others contain periodic states that exhibit period doubling, and sometimes chaos. States in each of these regimes (stable, periodic, chaotic) are investigated for their ability to accurately describe the observed properties of the magnetosphere-ionosphere system. A brief discussion of applications of this model to current space physics problems is included.

**The Transition to Continuous Unloading: Differentiating
Storms and Substorms in Magnetosphere Dynamics**

J. P. Smith, W. Horton, and D. N. Baker

**Abstract**

We present a new physical model for the global energy flows in the magnetotail which implies a distinction in the driving of storms and substorms. Using a model that tracks the major energy components of the nightside magnetosphere--ionosphere system, we demonstrate that some properties of both storms and substorms can be obtained from the same dynamical system for different levels of solar wind forcing. To say that storms are made up of substorms would be incorrect. Instead we see that quasiperiodic substorms result for weak forcing, while a steady storm ensues for strong forcing, with a continuous transition between the two. There is an inverse bifurcation from unstable to stable in the state of the system with respect to increasing driving. (Topologically, an attractor of dimension one for weak forcing becomes an attractor of dimension zero for strong forcing). The properties of both states are analyzed for their energy contents and the power flows between energy components. These states relate to two types of unloading of tail flux: storage--unloading and continuous unloading. The rapid storage--unloading process is shown to be equivalent to the substorm expansion onset. The hypothesis is advanced that a magnetic storm is represented by the state of nearly continuous unloading.

**IFSR-863**

**Low-Dimensional Modeling of the earth's Magnetosphere**

J. P. Smith

**Abstract**

The magnetosphere is a multi-scale, complex dynamical plasma system. Unlike a laboratory plasma however, no control experiments can ever be run to help test theories. Modeling of the magnetosphere has proved to be a difficult problem. Moderate success has been achieved with small-scale particle simulations and global MHD models, but both make some severe assumptions, and the results are often difficult to interpret. There is observational evidence that the magnetosphere, as a dynamical system, seems to behave in a low-dimensional manner on large scales. Low-dimensional models provide us with the tools to understand the relationships of simple physical quantities within the magnetosphere (such as energy deposition, macroscopic currents, cross-tail electric fields, or average bulk flow) without the necessity of the more complete but unwieldy models (particle simulations or MHD, for example). The model in this work is a magnetospheric analog to an energy model used to model the Earth's climate. As the solar wind driving, only solar wind convection electric field will be used, and the model basically considers averages over the entire magnetotail of the largest energy components in the plasma system. The dynamical properties of the system are explored. Several particular states of the model are explored in detail, and it is found that this model offers a new interpretation of the relationship between magnetic storms and substorms that agrees well with observational data.

**Multiple Reflections: Cascaded Upshifting of Laser
Pulses by Semiconductors**

S. M. Mahajan, R. Miklaszewski and V. I. Berezhiani

**Abstract**

By studying the detailed dynamics of an ultrashort pulse at the semiconductor--vacuum interfaces, we have shown that the interaction of these pulses with nonstationary semiconductor plasmas can, under appropriate conditions, lead to a variety of interesting phenomena: controlled upshifting of the laser frequency, a possible cascading of upshifting, a trapped pulse bouncing back and forth in the sample, and a ``machine--gun" of increasingly blueshifted pulses.

**Spectral Reduction: A Statistical Description of Turbulence**

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

**Abstract**

A method is described for predicting statistical properties of turbulence. Collections of Fourier amplitudes are represented by nonuniformly spaced modes with enhanced coupling coefficients. The statistics of the full dynamics can be recovered from the time-averaged predictions of the reduced model. A Liouville theorem leads to the expected inviscid equipartition solutions. Excellent agreement is obtained with two-dimensional forced-dissipative pseudospectral simulations. For the two-dimensional enstrophy cascade, logarithmic corrections to the high-order structure functions are observed.

**Magnetospheric Dynamics from a Low-Dimensional Nonlinear
Dynamics Model**

I. Doxas and W. Horton

**Abstract**

A physics based model for the coupled solar WIND-Magnetosphere-Ionosphere system (WINDMI) is described. The model is based on truncated descriptions of the collisionless microscopic energy transfer processes occurring in the quasineutral layer, and includes a thermal flux limit neglected in the Magnetohydrodynamic (MHD) closure of the moment equations. All dynamically relevant parameters of the model can be computed analytically. The system is both Kirchhoffian and Hamiltonian, ensuring that the power input from the solar wind is divided into physically realizable energy subcomponents, a property not shared by data-based filters. The model provides a consistent mathematical formalism in which different models of the solar wind driver, ionospheric dissipation, global field configuration, and substorm trigger mechanism can be inserted, and the coupling between the different parts of the system investigated.

**Particle Canonical Variables and Guiding Center Hamiltonian
Up to Second Order in the Larmor Radius**

H. Vernon Wong

**Abstract**

A generating function, expressed as a power series in the particle Larmor radius, is used to relate an arbitrary set of magnetic field line coordinates to particle canonical variables. A systematic procedure is described for successively choosing the generating function at each order in the Larmor radius so that the transformed particle Hamiltonian is independent of the Larmor phase angle. The particle guiding center Hamiltonian up to second order in the Larmor radius is thereby derived. The analysis includes finite equilibrium electrostatic fields and time dependent electromagnetic field perturbations. The transformations which relate magnetic flux coordinates to particle canonical variables are also discussed.

**A Topological Knot in a Dissipative Fifth-Order System**

N. Bekki

**Abstract**

In order to show that some quasiperiodic orbits of a fifth-order system are embedded in a three-dimensional subspace, we investigate numerically main projections onto a three-dimensional subspace from the five-dimensional space. We find that the quasiperiodic orbits are topologically equivalent to a (p,q)-torus knot, which has q strands traveling p times meridionally about two-torus in a three-manifold. In terms of a braid word for the torus knot, we finally obtain a (2,7)-torus knot in the fifth-order system through the complicated bifurcations under the parameter variation. This suggests that topological invariants embedded in a three-manifold can be extracted from realistic dissipative higher-dimensional dynamical systems.

**Primary Flows, the Solar Wind and the Corona**

S.M. Mahajan, R. Miklaszewski, K.I. Nikol'skaya, and N.L. Shatashvili

**Abstract**

Based on the conjectured existence of primary solar emanations (plasma flows from the solar surface), a model for the origin of the Fast Solar Wind, and the creation and heating of the coronal structures is developed. Preliminary results reproduce many of the salient observational features.

**Simultaneous Beltrami Conditions in Coupled Vortex
Dynamics**

Z. Yoshida and S.M. Mahajan

**Abstract**

The two-fluid model of a plasma describes the strong coupling between the magnetic and the fluid aspects of the plasma. The Beltrami condition which demands alignment of vortices and flows becomes a system of simultaneous equations in the magnetic field and the flow velocity. Combining these equations yields the double curl Beltrami equation. General solvability of the equation has been proved using the spectral theory of the curl operator. The set of solutions contains field configurations which can be qualitatively different from the conventional constant-a-Beltrami fields (which are naturally included in the set). The larger new set may help us understand a variety of structures generated in plasmas.

**Scale Separation in Two-Fluid Plasmas and its Implications
for Dynamo Theory**

S.M. Mahajan and Z. Yoshida

**Abstract**

By relating the velocity and the magnetic fields, the Hall term in the two-fluid model of a plasma leads to a singular perturbation that couples physical quantities varying at vastly different length scales. In a Beltrami model of the steady states, then, the dynamo mechanism emerges naturally. The scale separation also suggests a dissipative mechanism for heating the solar coronal structures embedded in relatively smooth magnetic fields.

**Shear-Driven Wave Oscillations in Astrophysical Flux
Tubes**

Andria D. Rogava, Stefaan Poedts, and S.M. Mahajan

**Abstract**

Velocity shear induced wave transformations in a cylindrical flux tube with a parallel plasma flow are studied. All the MHD modes sustained by the system - the Alfvén (AW), the slow magnetosonic (SMW), and the fast magnetosonic (FMW) waves - are found to be coupled through the agency of the velocity shear. The coupling leads to reciprocal transformations of the waves with corresponding energy exchange between them and between the waves and the background flow. The individual wave transformation events happen perpetually and irregularly in the whole space occupied by the flow, establishing the regime of shear-driven wave oscillations throughout the flow. The importance of this phenomenon for the generation of solar hydromagnetic waves, for the transmission of the waves through the transition region, for coronal heating and the acceleration of the solar wind is discussed. The possible appearance of the phenomenon in other classes of astrophysical shear flows with cylindrical or quasi-cylindrical geometry (e.g., jets in AGN's and accretion columns in X-ray pulsars) is anticipated.

**Substorm Trigger Conditions**

W. Horton, H. Vernon Wong, and J.W. Van Dam

**Abstract**

Critical conditions for the onset of fast interchange dynamics in the stressed geotail during the growth phase of the substorm are derived. We compare the ideal MHD interchange stability conditions (Hurricane, 1997) with kinetically modified interchange-kink motions. It is shown that fast interchange growth is possible only in the near-Earth boundary of the plasma sheet where the local plasma pressure is near unity since compressibility stabilizes the high beta geotail. The growth rate is proportional to the local current density and exceeds the ion bounce frequency in the local region of b< 1. Only after sufficient thinning of the current sheet will the kinetic theory growth rate exceed the bounce frequency of the ions which is the effective condition for the substorm MHD-space-time scale unloading of the plasma energy stored in the geotail.

**Spontaneous Hole Clump Pair Creation**

H.L. Berk, B.N. Breizman, J. Candy, M. Pekker, and N.V. Petviashvili

**Abstract**

A nonlinear theory is presented for the spontaneous formation of a hole-clump pair in the phase space of a system whose equilibrium is just above the linear threshold for instability. The first case studied takes the damping to be a purely linear response with the nonlinear instability drive due to a single wave-particle resonance with particles that have an inverted distribution function. Analytic results shows that the hole and clump can each support a Bernstein-Greene-Kruskal nonlinear wave, with the trapping frequency of particles comparable to the linear growth rate without dissipation. The power that is dissipated to the background plasma is balanced by the energy extracted from the inverted equilibrium distribution by the moving phase space structures. This motion produces frequency sweeping of the fields. The second case studied has no extrinsic dissipation. Instead, a second resonance is taken, which affects a different species of particles whose distribution function decreases with energy. For an electrostatic interaction, we consider cases for which the mass ratio of the destabilizing to stabilizing species is: (i) much less than unity; (ii) equal to unity; (iii) much greater than unity. Case (i) gives results that are similar to the linear dissipation model, while cases (ii) and (iii) saturate without any frequency sweeping. However, in case (ii), the saturated level is proportional to the total linear growth rate, while the saturation level in (iii) is nearly the same as the saturation level in a system where the stabilizing species is not present. In the third case we show that frequency sweeping can reappear in the problem of a collisionless destabilizing heavy species, with collisions affecting the stabilizing light species. When the collision frequency is relatively large, so that the light species in effect have a linear response, the problem reverts to the first case. More subtle and speculative explanations are given to explain why, at lower collisionality, holes and clumps also emerge.

**Nonlinear Modeling of Kinetic Plasma Instabilities**

J. Candy, H.L. Berk, B.N. Breizman and F. Porcelli

**Abstract**

Many kinetic plasma instabilities, in quite dierent physical systems, share a genuinely similar mathematical structure near isolated phase-space islands. For this reason, dynamical features such as faster-than-exponential growth of the instability, as well as nonlinear frequency sweeping, are found to be uni- versal. Numerical delta f methods, which follow the evolution of the (nonlinear) perturbed distribution function along single-particle orbits, have been applied to analytic models which include a continuous particle source, resonant par- ticle collisions, and wave damping. The result is a series of codes which can reliably model the nonlinear evolution of kinetic instabilities, including some specic to tokamak plasmas, over experimentally relevant timescales. New re- sults include: (i) nonlinear simulations of two-species, one-degree-of-freedom plasmas; (ii) simulations of shbone bursts in tokamak plasmas; (iii) nonlinear modeling of beam-driven toroidal Alfvén eigenmode activity in tokamaks.

**Incomplete devil's staircase in a model of magnetoconvection**

N. Bekki and T. Karakisawa

**Abstract**

A fifth-order autonomous system of magnetoconvection is numerically investigated in the context of a transition from two-torus to three-torus through the third Hopf bifurcation. By use of a return-map constructed from the Poincare section for the orbits of two-torus, it is first shown that a phase-locking series of winding numbers creates an incomplete devil's staircase as the magnetic Prandtl number is varied in a certain parameter region. A relation between hierarchies of a set of winding numbers is leading to some scaling-laws and to a final stable two-torus; an accumulated rational winding number, which suggests strongly the existence of some topological invariants through the complicated bifurcations under the parameter variation.

**Forecasting Auroral Electrojet Activity from Solar
Wind Input with Neural Networks**

R.S. Weigel, W. Horton, T. Tajima, and T. Detman

**Abstract**

Neural networks for reconstructing the chaotic attractor in the nonlinear dynamics of the solar wind driven, coupled magnetosphere-ionosphere (MI) system are developed. Two new methods which improve predictive ability are considered: a gating method which accounts for different levels of activity, and a preconditioning algorithm which allows the network to ignore very short time fluctuations during training. The two networks are constructed using the Bargatze et al. [1985] substorm database that contains solar wind speed and interplanetary magnetic field (IMF) along with ionospheric electrojet indices, AE and AL. Both networks are found to produce improvements in predictability, and the significance of the performance increase of the gated network is demonstrated using the bootstrap model testing method.

**The Radial Electric Field in Tokamak with Reversed
Magnetic Shear**

P. Zhu, W. Horton, H. Sugama

**Abstract**

Neoclassical theory with the impurity rotational velocity is used to evaluate the radial electric field, E_r in tokamaks. The result of using the complete matrix method for the deuterium-carbon plasma is compared with a reduced analytic formula for determining E_r [Ernst et al. (1998)]. The analytic formula is shown to overestimate the E_r magnitude and its gradient. Two transport measures of the effect of the E_r shear are compared for the reverse shear and enhanced reversed shear discharges in TFTR [Mazzucato et al. (1996)]. We show that the combined E_r and magnetic shear measure, Upsilon_s, from linear stability theory gives a higher correlation with the observed transition between the two discharges than the vorticity measure w_s from E_r shear alone.

**Classification, Casimir Invariants, and Stability of
lie-Poisson Systems**

Jean-luc Thiffeault

**Abstract**

We classify Lie-Poisson brackets that are formed from Lie algebra ex- tensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, independent under coordinate transforma- tions, and is achieved with the techniques of Lie algebra cohomology. For extensions of order less than five, we find that the number of normal forms is small and they involve no free parameters. A special extension, known as the Leibniz extension, is shown to be the unique "maximal" extension.

We derive a general method of finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir invariants of all brackets of order less than five are explicitly computed, using the concept of coextension. We obtain the Casimir invariants of Leibniz extensions of arbitrary order. We also offer some physical insight into the nature of the Casimir invariants of compressible re- duced magnetohydrodynamics.

We make use of the methods developed to study the stability of exten- sions for given classes of Hamiltonians. This helps to elucidate the distinction between semidirect extensions and those involving cocycles. For compressible reduced magnetohydrodynamics, we find the cocycle has a destabilizing effect on the steady-state solutions.

**Destabilization of TAE Modes by Particle Anisotropy**

H. Vernon Wong and H.L. Berk

**Abstract**

Plasmas heated by ICRF produce energetic particle distribution functions which are sharply peaked in pitch-angle, and we show that at moderate toroidal mode numbers, this anisotropy is a competitive and even dominant instability drive when compared with the universal instability drive due to spatial gradient. The universal drive, acting alone, destabilizes only co-propagating waves (i.e. waves propagating in the same toroidal direction as the diamagnetic flow of the energetic particles), but stabilizes counter-propagating waves (i.e. waves propagating in the opposite toroidal direction as the diamagnetic flow of the energetic particles). Nonetheless, we show that in a tokamak, it is possible that particle anisotropy can produce a larger linear growth rate for counter-propagating waves, and provide a mechanism for preferred destabilization of the counter-propagating TAE modes that are sometimes experimentally observed.

**Nonlinear Splitting of Fast Particle Driven Alfvén
Eigenmodes: Observation and Theory**

A. Fasoli, B.N. Breizman, D. Borba, R.F. Heeter, M.S. Pekker, and S.E. Sharapov

**Abstract**

The measured spectra of fast particle driven Alfvén eigenmodes in the JET tokamak plasma are interpreted on the basis of a first principle nonlinear model of near-threshold kinetic instabilities. The observed splitting of the Alfvén modes is shown to be due to the combined effect of resonant wave-particle interaction and collision-like relaxation of the resonant particles. The instability growth rate and the effective collision frequency of the resonant particles are determined.

**Alfvén Wave Particle Interaction in Finite-Dimensional
Self-Consistent Field Model**

Nikhil Padhye and W. Horton

**Abstract**

A low-dimensional Hamiltonian model is derived for the acceleration of ions in finiteamplitude Alfvén waves in a finite pressure plasma sheet. The reduced low-dimensional wave-particle Hamiltonian is useful for describing the reaction of the accelerated ions on the wave amplitudes and phases through the self-consistent fields within the envelope approximation. As an example, we show for a single Alfvén wave in the central plasma sheet of the Earth's geotail, modeled by the linear pinch geometry called the Harris sheet, the time variation of the wave amplitude during the acceleration of fast protons.

**Global Gyrokinetic Simulations of Tokamak Transport**

G. Furnish, W. Horton, Y. Kishimoto, M.J. LeBrun, and T. Tajima

**Abstract**

A kinetic simulation code based on the gyrokinetic ion dynamics in global general metric (including a tokamak with circular or noncircular cross-section) has been developed. This gyrokinetic simulation is capable of examining the global and semi-global driftwave structures and their associated transport in a tokamak plasma. We investigate the property of the ion temperature gradient (ITG) or etai driven drift waves in a tokamak plasma. The emergent semi-global drift wave modes give rise to thermal transport characterized by the Bohm scaling.

**Particle Dynamics and its Consequences in Wakefield
Acceleration in a High Energy Collider**

S. Cheshkov, T. Tajima, W. Horton, and K. Yokoya

**Abstract**

The performance of a wake field accelerator in a high energy collider application is analyzed by use of a nonlinear dynamics map built on a simple theoretical model of the wake field generated by the laser pulse (or whatever other method) and a code based on this map [1].The crucial figures of merit for such a system other than the final energy include the emittance (that determines the luminosity). The more complex the system is, the more "opportunities" the system has to degrade the emittance (or entropy of the beam). Thus our map guides us to identify where the crucial elements lie that affect the emittance. If the focusing force of the wake field is strong when there is a jitter in the position (or laser aiming) of each stage coupled with the spread in the individual particle betatron frequencies, particles experience a phase space mixing. This effect sensitively controls the emittance degradation. We investigate these effects both in a uniform plasma and in a plasma channel. We also study the effect of beam loading. Further, we briefly consider collision point physics issues for a collider expected or characteristic of such a construction based on a scenario for the multi-staged wake field accelerators.

**Overview of Nonlinear Theory of Kinetically Driven
Instabilities**

H.L. Berk and B.N. Breizman

**Abstract**

An overview is presented of the theory for the nonlinear behavior of instabilities driven by the resonant wave particle interaction. The approach should be applicable to a wide variety of kinetic systems in magnetic fusion devices and accelerators. Here we emphasize application to Alfvén wave driven instability, and the principles of the theory are used to interpret experimental data.

**Stability of Short Wavelength Tearing and Twisting
Modes**

F. L. Waelbroeck

**Abstract**

The stability and mutual interaction of tearing and twisting modes in a torus is governed by matrices that generalize the well-known D' stability index. The diagonal elements of these matrices determine the intrinsic stability of modes that reconnect the magnetic field at a single resonant surface. The off-diagonal elements indicate the strength of the coupling between the different modes. We show how the elements of these matrices can be evaluated, in the limit of short wavelength, from the free energy driving radially extended ballooning modes. We apply our results by calculating the tearing and twisting D' for a model high-beta equilibrium with circular flux surfaces.

**Interchange Trigger for Substorms in a Nonlinear Dynamics
Model**

W. Horton, H. Vernon Wong, R. Weigel, and I. Doxas

**Abstract**

Here we describe the generalization of the Lorenz model of Rayleigh--Benard
convection to the finite pressure plasma interchange dynamics. We include
the usual three ODE's of the Lorenz model and two new modes describing
the coupling of the plasma convection to the shear Alfvén waves
in the system. Thus, we have a *d=5* phase space with an attractor
or chaotic attractor depending on the system parameters. The system describes
the creation of field line currents driven by the onset of the convective
interchange which we interpret as the symmetry breaking in the ambient
region~1 current system to form the initial phase of the substorm current
wedge.

**Negative Energy Modes and Gravitational Instability
of Interpenetrating Fluids**

A. R. R. Casti, P. J. Morrison, and E. A. Spiegel

**Abstract**

We study the longitudinal instabilities of two interpenetrating fluids interacting only through gravity. When one of the constituents is of relatively low density, it is possible to have a band of unstable wave numbers well separated from those involved in the usual Jeans instability. If the initial streaming is large enough, and there is no linear instability, the indefinite sign of the free energy has the possible consequence of explosive interactions between positive and negative energy modes in the nonlinear regime. The effect of dissipation on the negative energy modes is also examined.

**Classification and Casimir Invariants of Lie-Poisson
Brackets**

Jean-Luc Thiffeault and¬ P.J. Morrison

**Abstract**

We classify Lie-Poisson brackets that are formed from Lie algebra extensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, and is achieved partially through the use of Lie algebra cohomology. For extensions of order less than five, the number of normal forms is small and they involve no free parameters. We derive a general method of finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir invariants of all low-order brackets are explicitly computed. We treat in detail a four field model of compressible reduced magnetohydrodynamics.

**Laboratory Laser Acceleration and High Energy Astrophysics:
Gammaüray Bursts and Cosmic Rays**

T. Tajima and Y. Takahashi

**Abstract**

Recent experimental progress in laser acceleration of charged particles (electrons) and its associated processes has shown that intense electromagnetic pulses can promptly accelerate charged particles to high energies and that their energy spectrum is quite hard. On the other hand some of the high energy astrophysical phenomena such as extremely high energy cosmic rays and energetic components of gamma-ray bursts cry for new physical mechanisms for promptly accelerating particles to high energies. We suggest that the basic physics involved in laser acceleration experiments sheds light on some of the underlying mechanisms and their energy spectral characteristics of the promptly accelerated particles in these high energy astrophysical phenomena.

**Resonant Instabilities in Synchrotron Accelerators
from Space Charge Effects**

M.B. Ottinger, T. Tajima, and K. Hiramoto

**Abstract**

The space-charge effects of low-energy Gaussian beams in synchrotron accelerators can significantly affect the beam particle trajectories, including altering the regions of beam instability. We show that the tuneshift of a Gaussian beam is not uniform throughout the beam, but decreases as a function of particle amplitude. From the amplitude dependence of the tuneshift, we derive equations for the region of beam instability due to integer resonances and coupled resonances. We demonstrate the validity of these equations through particle simulation.

**A Nonlinear Particle Dynamics Map of Wake Field Acceleration
in a Linear Collider**

T. Tajima, S. Cheshkov, W. Horton and K. Yokoya

**Abstract**

The performance of a wake field accelerator in a high energy collider application is analyzed. In order to carry out this task, it is necessary to construct a strawman design system (no matter how preliminary) and build a code of the systems approach (a typical systems code approach was used, for instance, in SSC studies [1]). A nonlinear dynamics map built on a simple theoretical model of the wake field generated by the laser pulse (or whatever other method) is obtained and we employ this as a base for building a system with multi-stages (and components) as a high energy collider. The crucial figures of merit for such a system other than the final energy include the emittance (that determines the luminosity). The more complex the system is, the more "opportunities" the system has to degrade the emittance (or entropy of the beam). Thus our map guides us to identify where the crucial elements lie that affect the emittance. We find that a strong focusing force of the wake field coupled with a possible jitter of the axis (or laser aiming) of each stage and a spread in the betatron frequencies arising from different phase space positions for individual particles leads to a phase space mixing. This sensitively controls the emittance degradation. We show that in the case of a uniform plasma the effect of emittance growth is large and may cause serious problems. We discuss possibilities to avoid it and control the situation.

**FluxüForce Relation and Plasma Flows in Toroidal
Plasmas**

K. C. Shaing

**Abstract**

A flux-force relation for toroidal plasmas is derived from the kinetic definition of the flux. It is shown that in Hamada coordinates fluxes are driven by plasma viscosity in the long-mean-free-path regime. The results are identical to those derived from the fluid equation. The significance of this derivation on the determination of plasma flows is discussed.

**Structure of Parallel-Velocity-Shear Driven Modes in
Toroidal Plasmas**

J-Q. Dong, W.B. Xu, Y.Z. Zhang, and W. Horton

**Abstract**

It is shown that the Fourier-ballooning representation is appropriate for the study of short wavelength drift-like perturbation in toroidal plasmas with a parallel velocity shear (PVS). The radial structure of the mode driven by a PVS is investigated in a torus. The Reynolds stress created by PVS turbulence and proposed as one of the sources for a sheared poloidal plasma rotation is analyzed. It is demonstrated that a finite ion temperature may strongly enhance the Reynolds stress creation ability from PVS driven turbulence. The correlation of this observation with the requirement that ion heating power be higher than a threshold value for the formation of an internal transport barrier is discussed.

**Predictive Tests of ITG-Based Models of Tokamak Heat
Transport on ITER-Database Discharges**

M. Erba, W. Horton, and M. Ottaviani

**Abstract**

Three gyro-Bohm transport models are applied to the L-mode discharge of the ITER profile database for predictive tests of plasma confinement. All three models are based on the ion temperature gradient (ITG) driven turbulence but emphasize different aspects of the resulting turbulent transport. It is found that the model using the long wave-length component of the turbulence for the radial correlation length yields predictions with the lowest overall residual error for the given database. Extrapolations for the performance of ITER are given for the models.

**Collisionless Transport Parallel to the Magnetic Field
in a Toroidal Plasma**

R. D. Hazeltine

**Abstract**

Transport parallel to the magnetic field of a toroidal plasma confinement system is investigated through kinetic theory, with emphasis on the long mean-free path limit. The crucial differences between transport on rational and irrational (ergodic) magnetic surfaces is discussed in detail. A collisionless transport law, involving a non-local operator that accounts for toroidal topology, is derived for parallel heat conduction on irrational magnetic surfaces. In the rational surface case, perpendicular diffusion is included in the kinetic equation to avoid singularity; this allows a calculation of the width and amplitude of resonant temperature perturbations that will be excited by heat sources with sufficiently broad Fourier spectra.

**Semiconductor Optics with Ultra Short Laser Pulses**

S.M. Mahajan and Ivane Murusidze

**Abstract**

In order to describe the propagation of the optical laser pulses in
semiconductors, we begin with the wave equation in its most general form.
Taking into account both the laser field-induced polarization **P**(**r**,t),
and the laser-driven current **j**(**r**,t), the Maxwell's macroscopic
equations can be combined in the form \begin{equation} \nabla\times\nabla\times**E**+{1\over
c^2}\,{\partial^2**E** \over\partial t^2} =-{4\pi\over c^2}\,{\partial^2**P**\over\partial
t^2}-{4\pi\over c^2}{\partial**j** \over\partial t}, \end{equation}
where **E** is the transverse electric field (\nabla**.** **E**(**r**,t)=0).

**Frequency Up-Conversion and Trapping of Ultrashort
Laser Pulses in Semiconductor Plasmas**

V.I. Berezhiani, S.M. Mahajan, and R. Miklaszewski

**Abstract**

It is shown that the interaction of ultrashort laser pulses with nonstationary semiconductor plasmas can, under appropriate conditions, lead to a variety of interesting phenomena including controlled upshifting of the laser frequency leading to the possibility of tunable lasers in a wide range of frequencies, and trapping (nonpropagation) of a substantial part of the incident pulse.

**Plasma Turbulence**

W. Horton and G. Hu

**Abstract**

The origin of plasma turbulence from currents and spatial gradients in plasmas is described and shown to lead to the dominant transport mechanism in many plasma regimes. A wide variety of turbulent transport mechanism exists in plasmas. In this survey we summarize some of the universally observed plasma transport rates.