**Enhancement of the ion neoclassical heat conduction due to electron and subthermal ion energy-scattering collisions **

A.A. Ware

**Abstract**

It is shown that rippling instabilities can tap the density gradient expansion free energy source through the density dependence of the neoclassical resistivity. Linear analyses show that the region where neoclassical rippling modes are significantly excited extends from the edge of the plasma to the region where ν_{*e} ≤ 1. Since these modes are non-dispersive, diamagnetic effects are negligible in comparison to the nonlinear decorrelation rate at saturation. Thus, the relevant regime is the ‘strong turbulence’ regime. The turbulent radial diffusivities of the temperature and the density are obtained as eigenvalues of the renormalized eigenmode equations at steady state. The density gradient acts to enhance the level of turbulence, compared to that driven by the temperature gradient alone. The saturated turbulent state is characterized by: current decoupling, the breakdown of Boltzmann relation, a radical mode scale of density fluctuations exceeding tht of temperature fluctuations, implying that density diffusivity exceeds temperature diffusivity, and that density fluctuation levels exceed temperature fluctuation levels. Magnetic fluctuation levels are negligible.

**Self-consistent radial sheath **

R.D. Hazeltine

**Abstract**

The boundary layer arising in the radial vicinity of a tokamak limiter is examined, with special reference to the TEXT tokamak. It is shown that sheath structure depends upon the self-consistent effects of ion guiding-center orbit modification, as well as the radial variation of E x B-induced toroidal rotation. Reasonable agreement with experiment is obtained from an idealized model which, however simplified, preserves such self-consistent effects. It is argued that the radial sheath, which occurs whenever confining magnetic field-lines lie in the plasma boundary surface, is an object of some intrinsic interest. It differs from the more familiar axial sheath because magnetized charges respond very differently to parallel and perpendicular electric fields.

**Theory of weak ion temperature gradient driven turbulence near the threshold of instability **

N. Mattor, P.H. Diamond

**Abstract**

In this paper a theory of weak ion temperature gradient-driven turbulence near the threshold of instability is presented. The model considers kinetic ions and adiabatic electrons in a sheared slab geometry. Linear theory shows that for η_{th < ηi < ηth+(1+Ti/Te)Ln/Ls (where ηth=0.95 is the instability threshold and L n/Ls < 1) then γ < ω and a weak turbulence theory applies. The nonlinear wave kinetic equation indicates that ion Compton scattering is the dominant nonlinear saturation process, and it is shown that the energy scatters to the linearly stable low ky modes. The resulting fluctuation spectrum (peaked about k⊥ρ i = 1) is much lower than that suggested by naive extrapolation from the strong turbulence regime. The resulting ion thermal conductivity is also extremely low, so that strong ion heating can be expected to drive the ion temperature gradient to a level where this weakly turbulent threshold regime is surpassed. Thus the critical ηi relevant to magnetic confinement experiments is not the linear instability threshold but the point where the diminutions of the weak turbulence regime vanish, and the transport increases to the strong fluid turbulence level.}

**Statistical mechanics of a two-field model of drift wave turbulence **

F.Y. Gang, B.D. Scott, P.H. Diamond

**Abstract**

The absolute equilibrium statistical mechanics of a two-field model of drift wave turbulence is investigated with emphasis on predicting the direction of spectral transfer, and on exploration of the role of density-potential correlation in constraining the nonlinear transfer process. The results indicate that departure from the adiabatic relation ˜n/n_{0} = e˜φ/T_{e} allows transfer of total energy to small scales in fully developed turbulence. This prediction is in distinct contrast to intuition based on the one-field Hasegawa-Mima model. The results are verified by direct numerical simulation of the basic mode coupling equations.

**Numerical investigation of a plasma beam entering transverse magnetic fields **

J. Koga, J.L. Geary, T. Tajima

**Abstract**

We study plasma beam injection into transverse magnetic fields using both electrostatic and electromagnetic particle-in-cell (PIC) codes. In the case of small beam momentum or energy (low drift kinetic β) we study both large and small ion gyroradius beams. Large ion gyroradius beams with a large dielectric constant ε >> (M/m)^{1/2} are found to propagate across the magnetic field via E x B drifts at nearly the initial injection velocity, where ε = 1 + (ω_{pi}^{2})/(Ω_{i}^{ 2}) and (M/m) is the ion to electron mass ratio. Beam degradation and undulations are observed in agreement with previous experimental and analytical results. When ε is on the order of (M/m)^{1/2}, the plasma beam propagates across field lines at only half its initial velocity and loses its coherent structure. When ε << (M/m)^{1/2}, the beam particles decouple at the magnetic field boundary, scattering the electrons and slightly deflecting the ions. For small ion gyroradius beam injection a flute type instability is observed at the beam magnetic fields interface. In the case of large beam momentum or energy (high drift kinetic β) we observe good penetration of a plasma beam which shields the magnetic field from the interior of the beam (diagmagnetism).

**Theory of neoclassical pressure-gradient-driven turbulence **

O.J. Kwon, P.H. Diamond, H. Biglari

**Abstract**

The nonlinear evolution and saturation of neoclassical pressure- gradient-driven turbulence (NPGDT), evolving from linearly unstable bootstrap current modes, are investigated. The theoretical model is based on "neoclassical MHD equations" which are valid in the banana-plateau regimes of collisionality. Modes with poloidal wavelengths shorter than radial wavelengths are shown to be suppressed. From nonlinear saturation conditions, the turbulent pressure diffusivity is determined as an eigenvalue of the renormalized equations. Levels and radial scales of turbulence are determined from the pressure diffusivity and are shown to exceed mixing length estimates by powers of a nonlinear enhancement factor. The problem of the electron heat transport due to stochastic magnetic fields driven by NPGDT is revisited. The reconsideration of the radial structure of magnetic flutter leads to estimates of the electron heat transport and magnetic fluctuation levels which differ qualitatively and quantitatively from previous calculations.

**Pressure gradient-driven modes in finite beta toroidal plasmas **

B.G. Hong, W. Horton, D. Choi

**Abstract**

When the ion temperature gradient is finite, the ideal kinetic theory plasma destabilizes the MHD mode below the critical beta of MHD theory. For a temperature gradient η_{i}>2/3 and inverse aspect ratio ε (T_{i})<0.35 the electrostatic toroidal η_{i}-mode is unstable. The coupling between the two modes is investigated by solving the fourth-order system describing the shear Alfven-drift wave and the ion acoustic wave in the finite β Tokamak. The ion kinetic velocity integral including the gyroradius effects and the ion magnetic drift resonances are used to obtain gamma k(k, q, β, η_{i}, s, ε_{n}, q, τ) and γ_{k}/(k_{x}^{2}) for the modes. The study emphasizes the β- and q-dependence of the transport associated with γ_{k}/(k_{x}^{2}).

**Drift wave monopoles with magnetic shear **

X.N. Su, P.J. Morrison, W. Horton

**Abstract**

A model that incorporates both the effects of temperature gradients and magnetic shear on the drift wave monopole solutions is analyzed. In the case where the former effect is treated improperly and the latter is neglected, it was shown in Ref. 1 that there exist exact monopole solutions, which can further be shown [Ref. 4] to be equivalent to the existence of a point spectrum for a nonlinear eigenvalue problem. When both the effects are included, this spectrum becomes a banded continuous spectrum. An eigenvalue of this spectrum is associated with a localized vortex structure that undulates in space about a fixed level, eventually matching to a radiative ion acoustic tail. A novel separatrix crossing technique is used to investigate this problem.

**Computer simulation of Alfven wave heating **

J.L. Geary, J.N. Leboeuf, T. Tajima

**Abstract**

The first particle simulation study of shear Alfven wave resonance heating is presented. Particle simulation codes self-consistently follow the time evolution of the individual and collective aspects of particle dynamics as well as wave dynamics in a fully nonlinear fashion. Alfven wave heating is a possible means of increasing the temperature of magnetized plasmas. A new particle simulation model has been developed for this application that incorporates Darwin's formulation of the electromag- netic fields with a guiding center approximation for electron motion perpendicular to the ambient magnetic field. The implementation of this model and the examination of its theoretical and computational properties are presented. With this model, we examine several cases of Alfven wave heating in both uniform and nonuniform simulation systems in a two-dimensional slab. For the inhomogeneous case studies, the kinetic Alfven wave develops in the vicinity of the shear Alfven resonance region. The electrons kinetically react with the wave through a collisionless process. The electron velocity distribution function flattens about the parallel phase velocity of the wave. This modifi- cation of the plasma properties is related to changes in the spatial structure of the wave. The electron heating rate is in good agreement with the Landau damping model. The ions gain energy by oscillating in the wave electric fields.

**Particle simulation of toroidicity-induced drift modes **

M.J. Lebrun, T. Tajima

**Abstract**

The linear and nonlinear behavior of the toroidicity induced drift mode is studied using a three dimensional, natural coordinate electrostatic particle code. In this simulation an instability involving radially extended toroidal drift modes, showing a banded frequency structure, is observed for the first time. Nonlinear phenomena such as the enhancement of instability by toroidal mode multiplicity, saturation, and particle transport are also reported.

**MHD Alfven stability in ignited toroidal plasmas **

G.Y. Fu, J.W. Van Dam, M.N. Rosenbluth, D.W. Ross, Y.Z. Zhang, H.L. Berk, S.M. Mahajan

**Abstract**

The presence of fusion-product alpha particles in an ignition tokamak can significantly modify the stability behavior of the burning plasma. In this paper, the effects of toroidicity are retained in a theoretical description of two global-type shear Alfven modes that can be destabilized by the alpha particles. Toroidicity can lead to stabilization of the global shear Alfven eigenmode, but it induces a shear Alfven gap eigenmode whose growth rate is even larger, although these modes may saturate nonlinearly. The stabilizing influence of highly energetic particles on low-frequency MHD ballooning and internal kink modes is also explored; in particular, ion diamagnetic frequency effects may stabilize fishbone oscillations, as has been observed with monster sawteeth in JET with ICRF heating.

**Two dimensional magnetohydrodynamic model of emerging magnetic flux in the solar atmosphere **

K. Shibata, T. Tajima, R.S. Steinolfson, R. Matsumoto

**Abstract**

The nonlinear undular mode of the magnetic buoyancy instability in an isolated horizontal magnetic flux embedded in a two-temperature layered atmosphere (solar corona-chromosphere/photosphere) is investigated using a two-dimensional magnetohydrodynamic code. The results show that the flux sheet with β of about 1 is initially located at the bottom of the photosphere, and that the gas slides down the expanding loop as the instability develops, with the evacuated loop rising as a result of enhanced magnetic buoyancy. The expansion of the magnetic loop in the nonlinear regime displays self-similar behavior. The rise velocity of the magnetic loop in the high chromosphere (10-15 km/s) and the velocity of downflow noted along the loop (30-50 km/s) are consistent with observed values for arch filament systems.

**The quasi-interchange mode as a mechanism for fast sawtooth crashes **

F. Waelbroeck

**Abstract**

t 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 which differs significantly from previous results is obtained. The new instability displays no threshold with regard to the poloidal β. An explicit expression for the growth rate is given for a model q profile. The linear growth rates are found to rise steeply near marginal stability but not sufficiently so as to account for the rapid onset of growth observed in the experiments. The nonlinear corrections to the mode growth are evaluated to lowest order in the amplitude, using a low-β expansion of the reduced Magnetohydrodynamic equations. These equations are shown to be identical to the full Magnetohydrodynamic equations in the linear regime except for the neglect of parallel kinetic energy and the effects of compressibility. The calculation differs from previous bifurcation analyses by the inclusion of toroidal curvature effects including coupling to non-resonant harmonics. The nonlinear forces are found to be stabilizing for the profiles investigated. These forces lead to the appearance of stable bifurcated equilibria above marginal stability. In this respect our conclusions resemble those for the zero-β, free-boundary kink. The amplitude of the bifurcated equilibria, however, is found to be much larger than that for the fixed-boundary, finite -shear kink.

**Diffusive processes in the cross-field flow of intense plasma beams **

B. Newberger, N. Rostoker

**Abstract**

We consider magnetic field diffusion in the presence of strongly magnetized electrons (ω_{c}_{ e}τ_{c}_{0/>1}) as a mechanism for the rapid penetration observed in cross-field flows of high-β plasma beams. The diffusion has been investigated in several cases which are amenable to analytic solution. The flux penetration times are found to be insensitive to the particular configuration. Comparison with two experiments is made. Agreement within the limits of the experiments is found. Both require an anomalous collision rate which is consistent with observed fluctuations in one case but apparently not the other.

**Enhancement of the ion neoclassical heat conduction due to electron and subthermal ion energy-scattering collisions **

A.A. Ware

**Abstract**

This paper considers the extra components of neoclassical ion heat conduction driven by energy scattering collisions with electrons and superthermal ions. These components are important in auxiliary heated discharges and can be the most important if *the* plasma ion distribution has an enhanced non-Maxwellian tail.

**Two case studies of stochastic transport: Anomalous transport in two drift waves, and collisionless reconnection **

I. Doxas

**Abstract**

One-wave E x B motion in a slab geometry is described by a hamiltonian with a phase space composed of an infinite square lattice of counterrotating rolls. When a small amount of a second wave is added, the hamiltonian becomes time dependent, the energy of a single particle is no longer conserved, and all energy values are accessible to the particle. Collisionless magnetic reconnection is studied in the context of a reversed field with a small normal component b modeling the geomagnetic tail. The magnetic moment is adiabatically conserved far from the reversal layer, but it changes in small increments Δμ as the particle crosses the layer. The magnitude of Δμ is calculated and the analytic expression is found to agree well with numerical calculations for ε < 1, where ε is the small parameter in the adiabatic expansion. A test particle code is used to study the time evolution of ensembles of particles placed in the model magnetic field. The code gives ion temperatures of a few keV and earthward drift velocities of 400-900 km/s in the Plasma Sheet Boundary Layer, in good quantitative agreement with observed values.

**Overfocusing in a migma and exyder configurations **

H.L. Berk, H.V. Wong

**Abstract**

The theory of overfocusing is developed for a self-colliding storage ring (Exyder) and for a classical migma configuration. Forces due to external and self-generated magnetic fields are considered. The band of energy containment is calculated for a model external magnetic field configuration. The forces due to self-generated magnetic fields are in migma and Exyder can also cause overfocusing, which sets a beta limit on a migma disc and a luminosity limit on Exyder. The luminosity can increase substantially if the self-colliding storage ring is partially unneutralized.

**Nonlinear Parker instability of isolated magnetic flux in a plasma **

K. Shibata, T. Tajima, R. Matsumoto, T. Horiuchi, T. Hanawa, R. Rosner, Y. Uchida

**Abstract**

The nonlinear evolution of the Parker instability in an isolated horizontal magnetic-flux sheet embedded in a two-temperature layer atmosphere is studied by using a two-dimensional MHD code. In the solar case, this two-layer model is regarded as a simplified abstraction of the sun's photosphere/chromosphere and its overlying much hotter (coronal) envelope. The horizontal flux sheet is initially located in the lower temperature atmosphere so as to satisfy magnetostatic equilibrium under a constant gravitational acceleration. Ideal MHD is assumed, and only perturbations with k parallel to the magnetic-field lines are investigated. As the instability develops, the gas slides down the expanding loop, and the evacuated loop rises as a result of enhanced magnetic buoyancy. In the nonlinear regime of the instability, both the rise velocity of a magnetic loop and the local Alfven velocity at the top of the loop increase linearly with height and show self-similar behavior with height as long as the wavelength of the initial perturbation is much smaller than the horizontal size of the computing domain.

**Stochastic and collisional diffusion in two-dimensional periodic flows **

I. Doxas, W. Horton, H.L. Berk

**Abstract**

he global effective diffusion coefficient D* for a two-dimensional system of convective rolls with a time dependent perturbation added, is calculated. The perturbation produces a background diffusion coefficient D, which is calculated analytically using the Menlikov-Arnold integral. This intrinsic diffusion coefficient is then enhanced by the unperturbed flow, to produce the global effective diffusion coefficient D*, which we can calculate theoretically for a certain range of parameters. The theoretical value agrees well with numerical simulations.

**A fluid-ion particle-electron model for low-frequency plasma instabilities **

P.M. Lyster, J.N. Leboeuf

**Abstract**

We have developed a hybrid (particle/fluid) computer code for the study of quasi-neutral micro-instabilities for inhomogeneous plasmas that are immersed in a magnetic field. The ions are treated in the fluid approximation, retaining perpendicular E×B and polarization drifts as well as the parallel momentum and ion temperature equations. The electrons are represented as particles with perpendicular E×B drifts and parallel kinetics, thus exactly describing the effects of trapped electrons and electron-wave resonances.

**Periodic orbits for reversible, symplectic mappings **

H. Kook, J. Meiss

**Abstract**

A 2N dimensional symplectic mapping which satisfies the twist condition is obtained from a Lagrangian generating function F(q,q'). The q's are assumed to be angle variables. Reversible maps of this form have 2^{N+1} invariant symmetry sets topologically equivalent to N dimensional planes. We conjecture that such maps have at least 2^{N} symmetric periodic orbits for each frequency ω = (m,n). Furthermore, we conjecture there is an orbit which minimizes the periodic action for each ω, and 2^{N} - 1 other "minimax" orbits. There is "dominant" symmetry plane on which the minimizing orbit is never observed to occur. As a parameter varies, the symmetric orbits are observed to undergo symmetry breaking bifurcations, creating a pair nonsymmetric orbits. A pair of coupled standard maps provides a four dimensional example. For this case the two dimensional symmetry planes give a cross section of the resonances and a visualization of the Arnold web.

**Relation between beam driven seed-current and rotation in steady state FRC **

M. Okamoto, H.L. Berk, J.H. Hammer

**Abstract**

We consider an field steady state reversed configuration whose current is maintained by a steady state beam. Without quadrupole fields, back current can be inhibited by the Ohkawa effect if Z_{b} < Z_{eff}, where Z_{b} and Z_{eff} are the beam charge number and effective charge number of background ions. However, the resulting rotation of the plasma often leads to instability. For systems, with a large bootstrap effect, the rotation can be moderate, but it is then difficult to contain fusion products. An additional problem is that the Ohkawa effect due to alpha particles tends to dissemble the equilibrium. It has previously been shown that the presence of a quadrupole field inhibit back current. Here we show that a steady state flux can be maintained with moderate input power in both reactors and present day experiments with the resulting rotation slow enough to fulfill stability conditions. However, experimental means must be devised to supply a continual source of particles and additional energy.

**Trapped particle instability and flute instability in tandem mirror; scale separation closure in Alfven wave turbulence **

C. Chen

**Abstract**

Two different topics are discussed: instability in Tandem mirrors and closure schemes in turbulence theories. In the tandem mirror study, an appropriate quadratic functional form is constructed taking into account equilibrium rotational effects, electron temperature gradients, ion Landau damping, and electron-collisional effects. The collisional effects, which are described by the Fokker-Planck equation, are evaluated by using the solution for the particle and energy lifetime of an electron in an ambipolar trap and a magnetic trap. With the quadratic form, a dispersion relation that covers both the trapped-particle modes and the flute modes is derived and discussed. The numerical studies utilizing parameters suitable to the TMX-U mirror (Livermore) are conducted. The second topic is to study the renormalized turbulence theory. After analyzing the process of the wave-wave incoherent interaction, it is argued that the incoherent source term for most turbulence systems lacks fine structure in the k-w space. Based on the concept of scale separation between the coherent response function and the incoherent source, a closure scheme for the renormalized turbulence equation is proposed. A model dealing with the shear-Alfven wave turbulence is numerically solved.

**Stability thresholds of a disk-shaped migma **

H. Wong, M.N. Rosenbluth, H.L. Berk

**Abstract**

The stability of a Migma disk is reexamined to determine the threshold to the interchange instability. It is shown that a previous calculation (Z. Naturforsch. Teil A 42, 1208 (1987)), which assumes a rigid mode eigenfunction, is inaccurate at the predicted particle number for marginal stability. As a result the integral equation for the system must be solved. A variational method of solution is developed and is shown to give good agreement with a direct numerical solution. The threshold for instability is found to be sensitive to the details of the distribution function. For highly focused systems, where all ions pass close to the axis, the threshold particle number (N_{u1}) for instability is substantially below that predicted by rigid mode theory (N_{rigid}) (8ε^{2}, where ε = r_{1}r_{L}, r_{1} is the spread in the distance of closest approach to the axis, and r_{L} the ion Larmor radius). At a higher density, a second band of stability appears that again destabilizes at yet a higher particle number (N_{u2}).

**Transport model of optical beams in plasma **

T. Kurki-Suonio, T. Tajima

**Abstract**

A new kind of a particle simulation algorithm sustainable for following long time scale evolution of electromagnetic beams in plasma is presented. The algorithm is based on particle and field equations averaged over the rapid laser oscillations and the model constitutes the electromagnetic counterpart of the Zakharov’s model for electrostatics. Computational remarks upon implementation of the algorithm are given. Test results of the code applied to the Rayleigh spread and self-focusing of an intense laser beam in a plasma are discussed.

**Nonlinear growth of the quasi-interchange instability **

F.L. Waelbroeck

**Abstract**

In this paper nonlinear effects on the growth of a pressure-driven, interchange-like mode are investigated. This mode is thought to be responsible for the sawtooth crashes observed in the Joint European Torus (JET) and successfully accounts for most of their features. The analysis presented here differs from previous bifurcation calculations by the inclusion of toroidal coupling effects. Toroidal curvature, which is important for pressure-driven modes, destroys the helical symmetry which is typical of kink-like instabilities.