**Observations on energy transport in tokamaks: A new formula for turbulent transport **

Y.Z. Zhang, S.M. Mahajan

**Abstract**

Arguments for a new, modified Ohkawa type formula of anomalous electron energy transport due to electromagnetic microturbulence are presented and its predictions are compared with tokamak experiments.

**Toroidal study of sawtooth oscillations **

A. Aydemir, J. Wiley, D. Ross

**Abstract**

Tokamak sawtooth oscillations are studied with a nonreduced, fully toroidal, resistive MHD (magnetohydrodynamic) model that includes ohmic heating, and parallel and perpendicular thermal conduction. Effects of perpendicular transport in producing different types of sawteeth, varying from simple, period oscillations to giant sawteeth with temperature modulations of order unity, and compound sawteeth with multiple relaxations, are demonstrated. Some of the recent experimental observations from large tokamaks, such as the fast crash times and a presumed topological anomaly in the X-ray tomography pictures, thought to be inconsistent with the Kadomtsev reconnection model, are examined and possible explanations are offered.

**Relation of wave energy and momentum with the plasma dispersion relation in an inhomogeneous plasma **

H.L. Berk, D. Pfirsch

**Abstract**

The expressions for wave energy and angular momentum commonly used in homogeneous and near-homogeneous media is generalized to inhomogeneous media governed by a nonlocal conductivity tensor. The expression for wave energy applies to linear excitations in an arbitrary three-dimensional equilibrium, while the expression for angular momentum applies to linear excitations of azimuthally symmetric equilibria. The wave energy *E*_{wave} is interpreted as the energy transferred from linear external sources to the plasma if there is no dissipation. With dissipation, such a simple interpretation is lacking as energy is also thermally absorbed. However, for azimuthally symmetric equilibria, the expression for the wave energy in a frame rotating with a frequency ω can be unambiguously separated from thermal energy. This expression is given by *E*_{ wave} - ωL_{wave}l, where L_{wave} is the wave angular momentum defined in the text and l the azimuthal wavenumber and it is closely related to the real part of a dispersion relation for marginal stability. The imaginary part of the dispersion is closely related to the energy input into a system. Another useful quantity discussed is the impedance form, which can be used for three-dimensional equilibrium without an ignorable coordinate and the expression is closely related to the wave impedance used in antenna theory. Applications to stability theory are also discussed.

**Theory of trapped-ion-temperature-gradient-driven turbulence and transport in low-collisionality plasmas **

H. Biglari, P. Diamond, P. Terry

**Abstract**

A novel theory for the nonlinear evolution of the trapped-ion-temperature-gradient-driven mode, based on the turbulent trapping of resonant ions in the electrostatic potential of the waves, is proposed. A statistical description is adopted whereby the self-consistent evolution of the two-point correlation function of the trapped-particle distribution function is followed in phase space. Threshold-dependent, non-steady-state turbulence (nonlinear instability) is shown to develop when the decay of the correlation function is overcome by a source term that derives its free energy from the relaxation of the average distribution function. This nonlinear instability leads to anomalous thermal and particle transport that in turn reconfigure the equilibrium temperature and density profiles in such a way as to return the system toward its marginal point. Expressions for the nonlinear dispersion relation and threshold, as well as estimates of the thermal and particle transport level, are derived. The estimated flux levels are sufficiently high as to make any significant departure away from marginality unlikely. The scenario outlined serves to underscore the desirability for pellet injection in experimental devices such as the Compact Ignition Tokamak (Bull. Am. Phys. Soc. 32, 1921 (1987)) that operate in the very low ion collisionality regime where this mode would be expected to become relevant. As with a number of recent theories, the present work further reinforces the notion that unfavorably drifting trapped particles pose a serious menace to confinement and suggests inboard, off-axis radio-frequency heating as one means of reducing the size of this population, at least for the case of those energetic trapped particles created during auxiliary heating.

**Renormalized perturbation theory: Vlasov Poisson system, weak turbulence limit and gyrokinetics **

Y.Z. Zhang, S.M. Mahajan

**Abstract**

The self-consistency of the renormalized perturbation theory of Zhang and Mahajan (1985) is demonstrated by applying it to the Vlasov-Poisson system and showing that the theory has the correct weak turbulence limit. Energy conservation is proved to arbitrary high order for the electrostatic drift waves. The theory is applied to derive renormalized equations for a low-beta gyrokinetic system. Comparison of this theory with other current theories is presented.

**pTSC: Data file editing for the tokamak simulation code **

J.D. Meiss

**Abstract**

The code pTSC is an editor for the data files needed to run the Princeton Tokamak Simulation Code (TSC). pTSC utilizes the Macintosh interface to create a graphical environment for entering the data. As most of the data to run TSC consists of conductor positions, the graphical interface is especially appropriate.

**Stability of low-shear tokamaks **

F.L. Waelbroeck, R.D. Hazeltine

**Abstract**

It has been suggested that the recently observed fast sawtooth crashes are caused by a low-shear, pressure-driven ideal instability. This hypothesis is investigated, using asymptotic methods to solve the toroidal mode equations for a class of equilibria characterized by a low-shear central region in which q^{−1} is small, separated from the wall by a region with finite shear. A dispersion relation that differs significantly from previous results is obtained. An explicit expression for the growth rate is given for a model q profile.

**On tokamak equilibrium **

R.D. Hazeltine, M.H. Montgomery

**Abstract**

Large aspect-ratio tokamak equilibrium is studied, including effects, such as ellipticity of the flux surfaces, of second order in the inverse aspect ratio. To facilitate the interpretation of experimental data, the analysis uses simple coordinates that are readily evaluated, rather than coordinates based on the exact flux surfaces. An explicit formula for the safety factor, including second order terms, is presented, along with a brief discussion of rotational effects.

**Saturation of Kelvin Helmholtz fluctuations in a sheared magnetic field **

B. Scott, P. Terry, P. Diamond

**Abstract**

Kelvin--Helmholtz unstable flows are numerically investigated in the context of a sheared E x B flow profile and a sheared magnetic field in the collisional, electrostatic limit. In the extreme form of this limit, density fluctuations are small and the system is described by the nonlinear E x B vorticity dynamics. In order to focus on the role of magnetic shear localization, the computations are confined to two dimensions. For weak magnetic shear the fluctuations become turbulent and saturate by nonlinear cascade to small (dissipative) scales. In a strong magnetic shear regime near the linear stability boundary, nonlinear spatial broadening allows direct access to resistive shear dissipation, leading to saturation at small amplitude with nearly all the fluctuation energy in the longest-wavelength mode. This is in accordance with previous investigation using a statistical closure analysis (Phys. Fluids 29, 231 (1986)). The amount of broadening is proportional to the linear growth rate. The fluctuation amplitude scaling with magnetic shear is found to agree closely with the theory.

**Theoretical microtearing turbulence and related computational studies of constrained turbulent relaxation **

G. Craddock

**Abstract**

This thesis is devoted to two studies of magnetic turbulence. The model employed is that of drift semicollisional microtearing. The first study is the development of the analytic theory of electron temperature gradient driven microtearing turbulence near the tokamak edge. Edge turbulence is believed to play an important role in anomalous transport and edge confinement. Renormalized one point and two point equations are derived (using a quasi-gaussian closure theory) and solved in order to obtain the magnetic spectra of stationary microtearing turbulence. Spectral transfers are discussed in terms of the renormalized two point equations and equilibrium statistical mechanics. The resulting anomalous electron thermal diffusivity is applied to electron energy confinement in both ohmic and neutral beam heated discharges, in light of profile consistency. In the second study, the undriven (negligible electron temperature gradient) equations in 2-D are allowed to viscously relax (negligible resistivity) computationally from a random initial condition. The constraint of mean square flux conservation allows long-lived isolated, nongaussian, current filaments to appear out of the random flow, indicating a form of spatial intermittency. Implications of these results are discussed in terms of quasi-gaussian closure theories.

**Drift-Alfven kinetic stability in the ballooning mode approximation **

B. Hong, W. Horton, D. Choi

**Abstract**

The coupled drift-shear Alfven mode including the complete Bessel function gyroradius effect and the ∇_{⊥}B -curvature guiding center drift resonance of kinetic theory is solved for the toroidal ballooning mode eigenvalues and eigenfunctions. Comparisons between nonlocal (ballooning) and local kinetic theory and between nonlocal fluid and kinetic theory are made. The critical plasma pressure for kinetic ballooning mode instability is only the same as the magnetohydrodynamic (MHD) theory critical pressure β_{MHD} for η_{i}=0. The critical kinetic theory plasma pressure β_{K}(η_{i}) is well below β_{MHD} and the kinetic theory growth rate is unstable for all k. The MHD second stability region is also unstable in the kinetic theory. The kinetic theory growth rate is a maximum around k ≤ 0.3--0.5 for finite aspect ratio ε_{n}=r_{n}/R. The effects of trapped electrons are found to be weakly stabilizing both analytically and numerically, and the instability is still significant outside the ideal MHD stable window from the ion magnetic drift resonances when η_{i} ≥ 1. The kinetic growth rate is a function of the six dimensionless parameters k, q^{2}β, ε_{n}, s, η_{i}, and τ=T_{e}/T_{i}.

**New directions in plasma turbulence and anomalous transport **

P.W. Terry, P.H. Diamond, T.S. Hahm

**Abstract**

The nature and role of nonwavelike incoherent fluctuations in Vlasov plasma turbulence and transport are considered. In particular, electrostatic drift holes, which are localized, self-binding incoherent fluctuations giving large negative skewness, are described in detail. Both the three dimensional structure and dynamics of drift holes are discussed. The important role of incoherent fluctuations in plasma transport is underscored with an example, that of transport in drift-Alfven turbulence.

**Thermally driven convective cells and tokamak edge turbulence **

D.R. Thayer, P.H. Diamond

**Abstract**

A unified theory for the dynamics of thermally driven convective cell turbulence is presented. The cells are excited by the combined effects of radiative cooling and resistivity gradient drive. The model also includes impurity dynamics. Parallel thermal and impurity flows enhanced by turbulent radial diffusion regulate and saturate overlapping cells, even in regimes dominated by thermal instability. Transport coefficients and fluctuation levels characteristic of the saturated turbulence are calculated. It is found that the impurity radiation increases transport coefficients for high density plasmas, while the parallel conduction damping, elevated by radial diffusion, in turn quenches the thermal instability. The enhancement due to radiative cooling provides a resolution to the dilemma of explaining the experimental observation that potential fluctuations exceed density fluctuations in the edge plasma (eΦ/T_{e} > n/n_{0}).

**The relation between quantum and classical thresholds for multiphoton ionization of excited atoms **

R.S. Mackay, J.D. Meiss

**Abstract**

The ionization of hydrogen can be treated by classical theory when the initial quantum number is large and the photon energy is small. Classically, the electron motion is stochastic for high intensities and the resulting diffusion can lead to ionization. However, Casati et al. [Phys. Rev. Lett. 57, 823 (1986)] have found that the ionization threshold is often higher than the threshold for classical stochasticity. We present here a heuristic explanation: classical stochasticity will be suppressed when the phase-space area escaping through classical cantori each period of the electric field is small compared to Planck’s constant. We obtain a scaling law which agrees remarkably well with the numerical results of Casati et al.

**Neoclassical electron heat conduction in tokamaks performed by the ions **

A.A. Ware

**Abstract**

The increment to neoclassical ion heat conduction caused by electron collisions is shown to act like electron heat conduction since the energy is taken from and given back to the electrons at each diffusion step length. It can exceed electron neoclassical heat conduction by an order of magnitude.

**Crystal X-ray accelerator **

T. Tajima, M. Cavenago

**Abstract**

An ultimate linac structure is realized by an appropriate crystal lattice (superlattice) that serves as a ‘‘soft’’ irised waveguide for x rays. High-energy (∼40 keV) X rays are injected into the crystal at the Bragg angle to cause Bormann anomalous transmission, yielding slow-wave accelerating fields. Particles (e.g., muons) are channeled along the crystal axis.

**Superdense muonic matter **

T. Tajima

**Abstract**

A possible method of creation of superdense matter with approximate atomic density 4 x 10^{29}cm^{-3} is suggested. A pulsed beam of 10^{8} muons, with duration 3 x 10^{ -6}sec is shone on liquid hydrogen of volume approx. (300A)^{3}. A muon charge-exchanges with an electron in a hydrogen atom: with enough muonic hydrogen atoms, the compressibility tends to diverge and condensation into a much higher density state begins. The muon beam should be cooled by the ionization process and channeled through crystal axes before irradiation on the hydrogen specimen. When magnetic fields are present upon irradiation, the fields may be enhanced up to 10^{9} Gauss. A possible state of this matter is speculated.

**Momentum and thermal transport in neutral-beam-heated tokamaks **

N. Mattor, P.H. Diamond

**Abstract**

The report explores the relation between momentum and thermal transport in neutral-beam-heated tokamaks with subsonic toroidal rotation velocity. A theory of diffusive momentum transport driven by ion-temperature-gradient-driven turbulence (η_{i}-turbulence) is presented. In addition, the level of η_{i}-turbulence is enhanced by radially sheared toroidal rotation. The resulting ion shear viscosity χ_{φ} is found. The associated ion thermal diffusivity, χ_{i}, is identical to χ_{φ}. Thus, a scenario based on velocity-shear-enhanced η_{i}-turbulence is consistent with the experimentally observed relationship between thermal and momentum confinement.

**Tearing mode growth in a regime of weak magnetic shear **

S. Riyopoulos, R.D. Hazeltine

**Abstract**

The nonlinear growth for the m/ngreater than or equal to resistive tearing mode is studied in the case when the rational surface q(r_{0}) = m/n falls in a regime of weak magnetic shear, q(r_{0}) ≈ =0. The island width is determined self-consistently from the nonlinear, zero-helicity component of the perturbed magnetic flux that provides the local shear. It is found that the magnetic perturbation keeps growing exponentially in the nonlinear regime on a hybrid resistive-Alfvenic time scale, while the island width and the vorticity grow on a much slower time scale. Accordingly, much faster release of magnetic energy results for modes growing near minima of hollow q profiles.

**Nonlinear fluid equations for a low collisionality toroidal plasma **

M. Kotschenreuther

**Abstract**

Nonlinear fluid equations are rigorously derived to describe resonant resistive perturbations localized near a rational surface in a torus, including neoclassical dissipative effects such as rotation damping and bootstrap current production which arise when the collisional mean free path exceeds the equilibrium scale length. A systematically ordered analysis with two parallel scales is used, for a low aspect ratio, starting directly from the drift-kinetic equation. The radial mode width is formally taken of order the poloidal gyroradius, and frequencies are taken to be sufficiently rapid that the perturbed poloidal and toroidal ion flows do not necessarily relax to satisfy the neoclassical equilibrium relation. This is realistic for many modes, and leads to some departures from standard neoclassical results. Neoclassical effects for electrons are relatively smaller than for ions so to include both consistently next order ion corrections are also included. For simplicity temperature perturbations are neglected. The new dissipative terms are shown to satisfy a field theoretic generalization of the Onsager symmetries, and the equations satisfy an H theorem.

**Theory of trapped-particle-induced resistive fluid turbulence **

H. Biglari, P.H. Diamond

**Abstract**

A theory of anomalous electron heat transport, evolving from trapped-particle-induced resistive interchange modes, is proposed. These latter are a new branch of the resistive interchange-ballooning family of instabilities, destabilized when the pressure carried by the unfavorably-drifting trapped particles is sufficiently large to overcome stabilizing contributions coming from favorable average curvature. Expressions for the turbulent heat diffusivity and anomalous electron thermal conductivity at saturation are derived for two regimes of trapped particle energy: (1) a moderately-energetic regime, which is ''fluid-like'' in the sense that the unstable mode grows faster than the time that it takes for particles in this energy range to precess once around the torus; and (2) a highly-energetic regime, where the trapped species has sufficiently high energy as to be able to resonantly interact with the mode. Unlike previous theories of anomalous transport, the estimates of diffusion and transport obtained here are self-consistent, since the trapped particles do not ''see'' the magnetic flutter due to their rapid bounce motion. The theory is valid for moderate electron-temperature, high ion-temperature (auxiliary-heated) plasmas, and as such, is relevant for present and future-generation experimental fusion devices.

**Variational principle and stability of non-monotonic Vlasov-Poisson Equilibria **

P. Morrison

**Abstract**

The stability of nonmonotonic equilibria of the Vlasov-Poisson equation is assessed by using nonlinear constants of motion. The constants of motion make up the free energy of the system, which upon variation yields nonmonotonic equilibria. Such equilibria have not previously been obtainable from a variational principle, but here this is accomplished by the inclusion of a passively advected tracer field. Defniteness of the second variation of the free energy gives a sufficient condition for stability in agreement with Gardner’s theorem. Previously, we have argued that indefiniteness implies either spectral instability or negative energy modes, which are generically unstable when one adds dissipation or nonlinearity. Such is the case for the nonmonotonic equilibria considered.

**Area as a devil's staircase in twist maps **

Q. Chen

**Abstract**

In area-preserving maps, the area under an invariant set as a function of frequency is a devil's staircase. We show that this staircase is the derivative of the average action of the invariant set with respect to frequency. This implies that resonances fill the phase space completely when there are no invariant curves.

**The free energy principle **

P. Morrison, M. Kotschenreuther

**Abstract**

This paper is concerned with instability of equilibria of Hamiltonian, fluid and plasma dynamical systems. Usually the dynamical equilibrium of interest is not the state of thermodynamic equilibrium, and does not correspond to a free energy minimum. The relaxation of this type of equilibrium is conventionally considered to be initiated by linear instability. However, there are many cases where linear instability is not present, but the equilibrium is nonlinearly unstable to arbitrarily small perturbations. This paper is about general free energy expressions for determining the presence of linear or nonlinear instabilities. These expressions are simple and practical, and can be obtained for all equilibria of all ideal fluid and plasma models. By free energy, we mean the energy change upon perturbations of the equilibrium that respect dynamical phase space constraints. This quantity is measured by a self-adjoint quadratic form, called δ^{2}F. The free energy can result in instability when δ^{2}F is indefinite; i.e., there exist accessible perturbations that lower the free energy of the system.

**Low beta rigid mode stability criterion for an arbitrary larmor radius plasma **

H.L. Berk, H.V. Wong

**Abstract**

The low beta flute interchange dispersion relation for rigid displacement perturbation of axisymmetric plasma equilibria with arbitrary Larmor radius particles and field line curvature, large compared to the plasma radius, is derived. The equilibrium particle orbits are characterized by two constants of motion, energy and angular momentum, and a third adiabatic invariant derived from the rapid radial motion. The Vlasov equation is integrated, assuming that the mode frequency, axial ''bounce'' frequency, and particle drift frequency are small compared to the cyclotron frequency, and it is demonstrated that the plasma response to a rigid perturbation has a universal character independent of Larmor radius. As a result the interchange instability is the same as that predicted from conventional MHD theory. However, a new prediction, more optimistic than earlier work, is found for the low density threshold of systems like Migma, which are disc-shaped, that is, the axial extent δz is less than the radial extent r_{0}. For δz_{r}_{0} much << 1, the stability criterion is determined by the total particle number. Whereas the older theory (δz_{r0} >> 1) predicted instability at about the densities achieved in actual Migma experiments, the present theory (δz_{r}_{0} >> 1) indicates that the experimental results were for plasmas with particle number below the interchange threshold.

**Dynamic global transition to second stability in an auxiliary heated tokamak **

G.Y. Fu, J.W. Van Dam, M.N. Rosenbluth

**Abstract**

A simple transport model is developed to study the dynamic evolution of an auxiliary heated tokamak plasma during the transition to a high-beta, ballooning mode second-stable equilibrium. The effect of the ballooning mode stability on the transport is incorporated by prescribing an enhanced thermal diffusion in the unstable region. The resultant highly nonlinear transport equation is solved numerically as an initial value problem, and also analytically by means of boundary layer theory. In particular, the auxiliary heating power P required for global transition of a flux-conserving tokamak plasma to the second stability regime is found to scale P τ_{E}∝√ξ_{max}, where ξ_{max} is the instability-induced thermal conductivity enhancement factor and τ_{E} is the confinement time in the ballooning stable regime.

**Orbit extension method for finding unstable orbits **

Q. Chen, J.D. Meiss, I.C. Percival

**Abstract**

The orbit extension method is a technique for building long orbits from shorter ones. Combined with Newton's method, it gives a powerful and efficient tool for finding various orbits of area-preserving twist maps on a cylinder. It is particularly valuable for unstable orbits. We develop this method and use it to find ordered periodic orbits, homoclinic orbits, and heteroclinic orbits between two resonances. Flux from one resonance to another is obtained as a by-product of the heteroclinic orbits.

**Nonlinear dynamics of tearing modes in the reversed field pinch **

J.A. Holmes, B.A. Carreras, P.H. Diamond, V.E. Lynch

**Abstract**

The results of investigations of nonlinear tearing-mode dynamics in reversed field pinch plasmas are described. The linear instabilities have poloidal mode number m=1 and toroidal mode numbers 10 ≤ n ≤ 20, and the resonant surfaces are therefore in the plasma core. The nonlinear dynamics result in dual cascade processes. The first process is a rapid m=1 spectral broadening toward high n, with a simultaneous spreading of magnetic turbulence radially outward toward the field-reversal surface. Global m=0 perturbations, which are driven to large amplitudes by the m=1 instabilities, in turn trigger the m=1 spectral broadening by back coupling to the higher n. The second process is a cascade toward large m and is mediated by m=2 modes. The m=2 perturbations have the structure of localized, driven current sheets and nonlinearly stabilize the m=1 modes by transferring m=1 energy to small-scale dissipation. The calculated spectrum has many of the qualitative features observed in experiments.

**Plasma current sustained by fusion charged particles in a field reversed configuration **

H.L. Berk, H. Momota, T. Tajima

**Abstract**

The distribution of energetic charged particles generated by thermonuclear fusion reactions in a field reversed configuration (FRC) are studied analytically and numerically. A fraction of the charged fusion products escapes directly while the others are trapped to form a directed particle flow parallel to the plasma current. It is shown that the resultant current density produced by these fusion charged particles can be comparable to background plasma current density that produces the original field reversed configuration in a D-^{3}He reactor. Self-consistent equilibria arising from the currents of the background plasma and proton fusion products are constructed where the Larmor radius of the fusion product is of arbitrary size. Reactor relevant parameters are examined, such as how the fusion reactivity rate varies as a result of supporting the pressure associated with the fusion products. We also model the synchrotron emission from various pressure profiles and quantitatively show how synchrotron losses vary with different pressure profiles in an FRC configuration.

**A mechanism for rapid sawtooth crashes in tokamaks - II **

A.Y. Aydemir

**Abstract**

The standard picture of Kadomtsev reconnection process predicts sawtooth crash times that are longer than those observed in the present day large tokamaks. Ideal kink modes are investigated as a possible mechanism for these fast crashes, by use of fully toroidal, compressible, full magnetohydrodynamic equations. In systems with low shear, parallel-current and pressure-driven modes are identified well below the previously accepted poloidal beta limits. Linear and nonlinear calculations show good agreement with experiments and indicate that such modes may explain fast collapse times reported in the recent literature.

**Bremsstrahlung from channeled charged particles: application to a crystal x-ray accelerator **

M. Cavenago, T. Tajima

**Abstract**

The ultimate acceleration structure for a very high energy accelerator may be provided by the solid state crystal lattice. We evaluate the interaction between high energy particles and lattice particles. The mean free path of electrons due to bremsstrahlung is less severe when they are channeled. Even when particles are channeled, however, the bremsstrahlung losses are proportional to the energy of the particle. For a muon or heavier particles this may be tolerable in practical application.

**Incompressible description of Rayleigh-Taylor instabilities in laser ablated plasmas **

H.J. Kull

**Abstract**

An incompressible fluid model of the ablative Rayleigh--Taylor instability (Phys. Fluids 29, 2067 (1986)) is generalized to include self-consistent diffuse boundaries. With diffuse boundaries the incompressible model is found to be in excellent agreement with a number of previous stability studies of laser ablation. The present theory can predict the scaling of the instability cutoff over an extended parameter range and its dependence on the heat conduction law. It is found that more favorable stability behavior can be obtained both for weak and strong thermal diffusion. Furthermore a strong dependence of the stabilization mechanism on the functional form of the heat conductivity is indicated. Representative conditions for laser ablation are identified and discussed in detail.

**Reduced fluid descriptions of toroidally confined plasma with finite ion temperature effects **

C.T. Hsu

**Abstract**

Fluid descriptions of toroidally confined plasma with FLR effects are studied, based on a generalized, energy-conserving, self-consistent, nonlinear reduced-fluid model (HHM). The model, derived via a fluid approach starting from moment equations, differs from Braginskii`s fluid system in retaining O(ρ_{i}^{2}) terms (where ρ_{i} is the ion gyroradius) and most of the non-ideal effects. Hence, many of the well-known reduced-fluid models can be reproduced from HHM by simply specifying scales of some parameters such as ρ_{i} and β. On the other hand, a Pade approximation of the full FLR system, obtained from the simplified version of HHM, is also presented. This simplified model is not only energy-conserving and much easier to access, but also can be shown to retain FLR effects quite accurately. It is therefore remarked that this version should deserve further analytical and numerical studies. The possible applications of HHM are discussed in a general way so that further detailed studies can readily follow. In particular, linear toroidal drift-tearing modes with finite ion-temperature effects are studied. In addition, the non-canonical Hamiltonian theory and it`s application to the reduced system are discussed. This fast developing theory has been useful for studying the equilibria and nonlinear instability of the fluid system.

**Two-dimensional shear flow turbulence: A statistical theory of vortex filaments **

T. Chiueh, P.H. Diamond, P. Terry

**Abstract**

A theory of two-dimensional free shear flow turbulence is proposed. In particular, the role of interactions of fluctuations with the mean flow is distinguished from that of interactions among fluctuations. The former are responsible for the extraction of free energy necessary for maintaining the turbulence, while the latter lead to the enstrophy cascade which generates small-scale eddies. On the length scales of greatest interest, turbulence is anisotropic because of the influence of mean flow shear. At smaller scales, self-similar enstrophy transfer occurs and yields the familiar power law spectrum. Free energy extraction is accompanied by mean flow relaxation. Two distinct mechanisms are involved. In addition to the usual quasilinear diffusion, there is a nonlinear drift which acts to enhance localized fluctuations and flatten the mean flow profile. The saturated state is characterized by collective resonances which must be nonlinearly damped in order to balance their generation by seed-eddy induction. This balance gives rise to a condition for the determination of the nonlinear damping rate, and thus the turbulence level. The spatial structure of the saturated fluctuations is also predicted.

**Dissipative trapped particle modes in tandem mirrors **

H.L. Berk, C.Y. Chen

**Abstract**

The theory for the dissipative trapped particle modes is developed for a tandem mirror taking into account equilibrium rotational effects, electron temperature gradients, and an axial ambipolar potential, for a broad range of collisional parameters. The electrons are treated as a Maxwellian plasma that occupies the central cell and anchor cell regions. It is assumed that the eigenfunction is piecewise constant with abrupt transitions between the anchor and central cell regions. It is found that when ω/ν_{p}≥1, with ω the mode frequency and ν_{p} the Pastukhov loss rate, that the energy conservation structure of the collision operator produces important changes to previously developed theories. A solution to the problem is achieved by using the solution for the lifetime of an electron in an ambipolar trap, taking into account the global energy conservation. The energy conservation structure also allows a self-consistent description of dissipative instabilities when thermal gradient and electric fields are present. At very high collision frequency, a new dispersion relation is obtained, which exhibits an axially rotational shear drive coupled to radial temperature gradients producing instability. Numerical studies are presented for some parameters, with the deviation from previous theory highlighted.

**Laser self-trapping for plasma fiber accelerator**

D.C. Barnes, T. Kurki-Suonio, T. Tajima

**Abstract**

A short-pulsed intense laser is injected into an underdense plasma to sustain a self-trapped photon channel. With either high-enough intensity or strong-enough focusing the optical beam causes total electron evacuation on the beam axis. Under appropriate conditions this laser and plasma fiber system can provide a slow wave structure of the electromagnetic wave that is suitable for high-energy acceleration.

**Simulation of ion cyclotron resonance heating through resonant absorption in two-ion species plasma **

T. Tajima, S. Riyopoulos, V. Demchenko

**Abstract**

Particle simulation of two-ion hybrid cyclotron resonance heating (ICRH) of a magnetized hydrogen plasma with deuteron minority by magnetosonic waves launched from the low magnetic field side is reported. Depending on the minority concentration, partial transmission and partial reflection of the incoming waves off the two-ion hybrid resonance layer occur, in contrast to the mode conversion mainly taking place during incidence from the high field side. Preferential minority heating is observed, as the minority cyclotron resonance is close to the two-ion hybrid resonance layer.

**Hamiltonian four-field model for nonlinear tokamak dynamics **

R.D. Hazeltine, C.T. Hsu, P.J. Morrison

**Abstract**

The Hamiltonian four-field model is a simplified description of nonlinear tokamak dynamics that allows for finite ion Larmor radius physics, as well as other effects related to compressibility and electron adiabaticity. Much simpler than some previous descriptions of the same physics, it still preserves essential features of the underlying exact dynamics. In particular, because it is a Hamiltonian dynamical system it conserves the appropriate Casimir invariants, as well as avoiding implicit, unphysical dissipation. Here the model is derived and interpreted, its Hamiltonian nature is demonstrated, and its constants of motion are extracted.