**Area Preserving Nontwist Maps: Periodic Orbits and Transition to Chaos **

D. del Castillo Negrete, J.M. Greene, P.J. Morrison

**Abstract**

Area preserving nontwist maps, i.e. maps that violate the twist condition, are considered. A representative example, the standard nontwist map that violates the twist condition along a curve called the shearless curve, is studied in detail. Using symmetry lines and involutions, periodic orbits are computed and two bifurcations analyzed: periodic orbit collisions and separatrix reconnection. The transition to chaos due to the destruction of the shearless curve is studied. This problem is outside the applicability of the standard KAM (Kolmogorov-Arnold-Moser) theory. Using the residue criterion we compute the critical parameter values for the destruction of the shearless curve with rotation number equal to the inverse golden mean. The results indicate that the destruction of this curve is fundamentally different from the destruction of the inverse golden mean curve in twist maps. It is shown that the residues converge to a six-cycle at criticality.

**On the Quasi-hydrostatic Flows of Radiatively Cooling Self-Gravitating Gas Clouds **

B. Meerson, E. Megged, T. Tajima

**Abstract**

Two model problems are considered, illustrating the dynamics of quasi-hydrostatic flows of radiatively cooling, optically thin self-gravitating gas clouds. In the first problem, spherically symmetric flows in an unmagnetized plasma are considered. For a power-law dependence of the radiative loss function on the temperature, a one-parameter family of similarity solutions is found. We concentrate on a constant-mass cloud, one of the cases when the similarity indices are uniquely selected. In this case, the problem can be formally reduced to the classical Lane-Emden equation and therefore solved analytically. The cloud is shown to undergo radiative condensation, if the gas specific heat ratio γ > 4/3. The condensation proceeds either gradually or in the form of (quasi-hydrostatic) collapse. For γ < 4/3, the cloud is shown to expand. The second problem addresses a magnetized plasma slab that undergoes quasi-hydrostatic radiative cooling and condensation. The problem is solved analytically, employing the Lagrangian mass coordinate.

**Shear Flow Effects on Ion Thermal Transport in Tokamaks**

T. Tajima, Y. Kishimoto, W. Horton, J.Q. Dong

**Abstract**

From various laboratory and numerical experiments, there is clear evidence that under certain conditions the presence of sheared flows in a tokamak plasma can significantly reduce the ion thermal transport. In the presence of plasma fluctuations driven by the ion temperature gradient, the flows of energy and momentum parallel and perpendicular to the magnetic field are coupled with each other. This coupling manifests itself as significant off-diagonal coupling coefficients that give rise to new terms for anomalous transport. The authors derive from the gyrokinetic equation a set of velocity moment equations that describe the interaction among plasma turbulent fluctuations, the temperature gradient, the toroidal velocity shear, and the poloidal flow in a tokamak plasma. Four coupled equations for the amplitudes of the state variables radially extended over the transport region by toroidicity induced coupling are derived. The equations show bifurcations from the low confinement mode without sheared flows to high confinement mode with substantially reduced transport due to strong shear flows. Also discussed is the reduced version with three state variables. In the presence of sheared flows, the radially extended coupled toroidal modes driven by the ion temperature gradient disintegrate into smaller, less elongated vortices. Such a transition to smaller spatial correlation lengths changes the transport from Bohm-like to gyrobohm-like. The properties of these equations are analyzed. The conditions for the improved confined regime are obtained as a function of the momentum-energy deposition rates and profiles. The appearance of a transport barrier is a consequence of the present theory.

**"Heavy Light Bullets" in Electron-Positron Plasma **

V.I. Berezhiana, S.M. Mahajan

**Abstract**

The nonlinear propagation of circularly polarized electromagnetic waves with relativistically strong amplitudes in an unmagnetized hot electron-positron plasma with a small fraction of ions is investigated. The possibility of finding localized solutions in such a plasma is explored. It is shown that these plasmas support the propagation of "heavy light bullets"; nondiffracting and nondispersive electromagnetic (EM) pulses with large density bunching.

**More on Core-Localized Toroidal Alfvén Eigenmodes**

H.L. Berk, J.W. Van Dam, D. Borba, J. Candy, G.T.A. Huysmans, S. Sharapov

**Abstract**

A novel type of ideal toroidal Alfvén eigenmode, localized in the low-shear core region of a tokamak plasma, is shown to exist, whose frequency is near the upper continuum of the toroidal Alfvén gap. This mode converts to a kinetic-type toroidal Alfvén eigenmode above a critical threshold that depends on aspect ratio, pressure gradient, and shear. Opposite to the usual ideal toroidal Alfvén eigenmode, this new mode is peaked in amplitude on the small-major-radius side of the plasma.

**Stabilization of the Resistive Shell Mode in Tokamaks**

R. Fitzpatrick, A.Y. Aydemir

**Abstract**

The stability of current-driven external-kink modes is investigated in a tokamak plasma surrounded by an external shell of finite electrical conductivity. According to conventional theory, the ideal mode can be stabilized by placing the shell sufficiently close to the plasma, but the non-rotating `resistive shell mode`, which grows as the characteristic L/R time of the shell, always persists. It is demonstrated, using both analytic and numerical techniques, that a combination of strong edge plasma rotation and dissipation somewhere inside the plasma is capable of stabilizing the resistive shell mode. This stabilization mechanism is similar to that found recently by Bondeson and Ward (1994), except that it does not necessarily depend on toroidicity, plasma compressibility or the presence of resonant surfaces inside the plasma. The general requirements for the stabilization of the resistive shell mode are elucidated.

**Quantitative Predictions of Tokamak Energy Confinement from First-Principles Simulations with Kinetic Effects**

M. Kotschenreuther, W. Dorland

**Abstract**

A first-principles model of anomalous thermal transport based on numerical simulations is presented, with stringent comparisons to experimental data from the Tokamak Fusion Test Reactor (TFTR) [Fusion Technol. 21, 1324 (1992)]. This model is based on nonlinear gyrofluid simulations, which predict the fluctuation and thermal transport characteristics of toroidal ion-temperature-gradient-driven (ITG) turbulence, and on comprehensive linear gyrokinetic ballooning calculations, which provide very accurate growth rates, critical temperature gradients, and a quasilinear estimate of χe/χi. The model is derived solely from the simulation results. More than 70 TFTR low confinement (L-mode) discharges have been simulated with quantitative success. Typically, the ion and electron temperature profiles are predicted within the error bars, and the global energy confinement time within ±10%. The measured temperatures at r/a ≅ 0.8 are used as a boundary condition to predict the temperature profiles in the main confinement zone. The dramatic transition to the improved confinement in the supershot regime is also qualitatively explained. Further work is needed to extend this model of core heat transport to include particle and momentum transport, the edge region, and other operating regimes besides the ITG-dominated L mode. Nevertheless, the present model is very successful in predicting thermal transport in the main plasma over a wide range of parameters.

**Singular Eigenfunctions for Shearing Fluids I**

N.J. Balmforth, P.J. Morrison

**Abstract**

The authors construct singular eigenfunctions corresponding to the continuous spectrum of eigenvalues for shear flow in a channel. These modes are irregular as a result of a singularity in the eigenvalue problem at the critical layer of each mode. They consider flows with monotonic shear, so there is only a single critical layer for each mode. They then solve the initial-value problem to establish that these continuum modes, together with any discrete, growing/decaying pairs of modes, comprise a complete basis. They also view the problem within the framework of Hamiltonian theory. In that context, the singular solutions can be viewed as the kernel of an integral, canonical transformation that allows us to write the fluid system, an infinite-dimensional Hamiltonian system, in action-angle form. This yields an expression for the energy in terms of the continuum modes and provides a means for attaching a characteristic signature (sign) to the energy associate with each eigenfunction. They follow on to consider shear-flow stability within the Hamiltonian framework. Next, the authors show the equivalence of integral superpositions of the singular eigenfunctions with the solution derived with Laplace transform techniques. In the long-time limit, such superpositions have decaying integral averages across the channel, revealing phase mixing or continuum damping. Under some conditions, this decay is exponential and is then the fluid analogue of Landau damping. Finally, the authors discuss the energetics of continuum damping.

**Reynolds Stress of Localized Toroidal Modes**

Y.Z. Zhang, S.M. Mahajan

**Abstract**

An investigation of the two-dimensional (2-D) toroidal eigenmode problem reveals the possibility of a new consistent 2-D structure, the dissipative ballooning mode of the second kind (BM-II). In contrast to the conventional ballooning mode, the new mode is poloidally localized at π/2 (or −π/2), and possesses significant radial asymmetry. The radial asymmetry, in turn, allows the dissipative BM-II to generate considerably larger Reynolds stress as compared to the standard slab drift-type modes. It is also shown that a wide class of localized dissipative toroidal modes are likely to be of the dissipative BM-II nature, suggesting that at the tokamak edge, the fluctuation generated Reynolds stress (a possible source of poloidal flow) can be significant.

**Numerical simulation of bump-on-tail instability with source and sink**

H.L. Berk, B.N. Breizman, M. Pekker

**Abstract **

A numerical procedure has been developed for the self-consistent simulation of the nonlinear interaction of energetic particles with discrete collective modes in the presence of a particle source and dissipation. A bump-on-tail instability model is chosen for these simulations. The model presents a kinetic nonlinear treatment of the wave–particle interaction within a Hamiltonian formalism. A mapping technique has been used in this model in order to assess the long time behavior of the system. Depending on the parameter range, the model shows either a steady-state mode saturation or quasiperiodic nonlinear bursts of the wave energy. It is demonstrated that the mode saturation level as well as the burst parameters scale with the drive in accordance with the analytical predictions. The threshold for the resonance overlap condition and particle global diffusion in the phase space are quantified. For the pulsating regime, it is shown that when γ_{L}0.16 ΔΩ, where γ_{L} is the linear growth rate for the unperturbed system and ΔΩ is the frequency separation of neighboring resonances, overlap occurs together with an amplification of the free energy release compared to what is expected with the saturation of nonoverlapping modes. The effect of particle losses on the wave excitation is included in the model, which illustrates in a qualitative way the bursting collective losses of fast ions/alpha particles due to Alfvén instabilities.

**Exactly conservative integrators **

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

**Abstract**

Traditional explicit numerical discretizations of conservative systems generically predict artificial secular drifts of any nonlinear invariants. In this work we present a general approach for developing explicit nontraditional algorithms that conserve such invariants exactly. We illustrate the method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator-prey model, and the Kepler problem. The ideas are discussed in the context of symplectic (phase--space-conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.

**Neoclassical and anomalous transport in axisymmetric toroidal plasmas with electrostatic turbulence **

H. Sugama, W. Horton

**Abstract**

Neoclassical and anomalous transport fluxes are determined for axisymmetric toroidal plasmas with weak electrostatic fluctuations. The neoclassical and anomalous fluxes are defined based on the ensemble-averaged kinetic equation with the statistically averaged nonlinear term. The anomalous forces derived from that quasilinear term induce the anomalous particle and heat fluxes. The neoclassical banana-plateau particle and heat fluxes and the bootstrap current are also affected by the fluctuations through the parallel anomalous forces and the modified parallel viscosities. The quasilinear term, the anomalous forces, and the anomalous particle and heat fluxes are evaluated from the fluctuating part of the drift kinetic equation. The averaged drift kinetic equation with the quasilinear term is solved for the plateau regime to derive the parallel viscosities modified by the fluctuations. The entropy production rate due to the anomalous transport processes is formulated and used to identify conjugate pairs of the anomalous fluxes and forces, which are connected by the matrix with the Onsager symmetry.

**Lie group analysis of plasma-fluid equations **

Raul Acevedo

**Abstract**

Lie group methods for nonlinear partial differential equations are implemented to study, analytically, a subset of the full solution space of a family of plasma-fluid models. The solutions obtained by this method are known as group invariant solutions. The basic set of equations considered comprise the three-field fluid model due to Hazeltine (HTFM), which was obtained to describe nonlinear large aspect ratio tokamak physics. This model contains as particular limits the physics of the Charney-Hasegawa-Mima equation (CHM) and reduced magnetohydrodynamics (RMHD), which are two other models known to describe some features of nonlinear behavior of tokamak plasmas. Lie's method requires a large number of systematic calculations to determine the Lie point symmetries of the system of differential equations. These symmetries form a Lie group and describe the geometrical invariance of the equations. The Lie symmetries have been calculated for the systems mentioned above by using a symbolic manipulation program. A detailed analysis of the physical meaning of these symmetries is given. Using the Lie algebraic properties of the generators of the symmetries, a reduction of the number of independent variables for the full nonlinear systems of equations is calculated, which in turn yields simplified equations that sometimes can be solved analytically. A discussion of some of the reductions and solutions generated by this technique is presented. The results show the feasibility of using Lie methods to obtain analytical results for complicated nonlinear systems of partial differential equations that describe physically interesting situations.

**Normal modes and continuous spectra **

N.J. Balmforth, P.J. Morrison

**Abstract**

In theory of fluids, plasmas, and stellar systems, we frequently encounter the question of the stability of equilibria. The answer is provided in part on determining the evolution of an infinitesimal disturbance away from equilibrium, an approach that usually goes by way of a normal mode expansion. This approach can at times be very powerful, and amounts to solving an eigenvalue problem. It can, however, run into difficulty in circumstances for which that eigenvalue problem is, in some sense, irregular. What we might call regular eigenvalue problems involve the solution of a set of ordinary differential equations with regular coefficients on a domain of finite size. Here we are concerned with situations for which the eigenvalue problem is irregular and the resulting spectrum is at least partly continuous. This kind of a spectrum can arise as a result of solving the problem on an infinite domain, in which case there is simply no quantization condition. Of more interest are problems in which the set of ordinary differential equations is not autonomous and contains coefficients that become singular at points within the domain. In physical situations, singularities in the equations governing the evolution of an infinitesimal disturbance can result from a variety of effects, and they do not always affect the form of the eigenspectrum. An important class of problems for which the singularity has direct repercussions on the eigenspectrum occurs in fluids, plasmas, and stellar systems. These are ideal problems in which there are wave-mean flow or wave-particle resonances that result in the creation of a continuous eigenvalue spectrum. In these circumstances, coefficients in the differential problem are formally singular at the point at which resonance occurs. Moreover, that point is determined by the speed of a wavelike perturbation or, equivalently, the eigenvalue. In this paper we follow the directions indicated by Van Kampen for more general problems than the relatively simple plasma and fluid equilibria considered by Van Kampen and Case. We first describe the general method. Then, in the general context, the problem of plasma oscillations is reviewed. The remaining sections on parallel shear flow, shear flow in shallow water theory, incompressible circular vortices, and differentially rotating disks, are the bulk of the paper. We conclude with a discussion of the uses of singular eigenfunctions.

**On inertial-range scaling laws **

J.C. Bowman

**Abstract**

Inertial-range scaling laws for two- and three-dimensional turbulence are re-examined within a unified framework. A new correction to Kolmogorov's k^{−5/3} scaling is derived for the energy inertial range. A related modification is found to Kraichnan's logarithmically corrected two-dimensional enstrophy-range law that removes its unexpected divergence at the injection wavenumber. The significance of these corrections is illustrated with steady-state energy spectra from recent high-resolution closure computations. Implications for conventional numerical simulations are discussed. These results underscore the asymptotic nature of inertial-range scaling laws.

**Magnetic viscosity by localized shear flow instability in magnetized accretion disks **

R. Matsumoto, T. Tajima

**Abstract**

Differentially rotating disks are subject to the axisymmetric instability for perfectly conducting plasma in the presence of poloidal magnetic fields (Balbus & Hawley 1991). For nonaxisymmetric perturbations, we find localized unstable eigenmodes whose eigenfunction is confined between two Alfven singularities at ω_{D} = +/- ω_{A}, where ω_{D} is the Doppler-shifted wave frequency and ω_{A} = k_{parallel} ν_{A} is the Alfven frequency. The radial width of the unstable eigenfunction is Δx is ≈ ω_{A}/(Ak_{y}), where A is Oort's constant and k_{y} is the azimuthal wavenumber. The growth rate of the fundamental mode is larger for smaller values of k_{y}/k_{z}. The maximum growth rate when k_{y}/k_{z} is ≈ 0.1 is ≈ 0.2 Ω for the Keplerian disk with local angular velocity Ω. It is found that the purely growing mode disappears when k_{y}/k_{z} is > 0.12. In a perfectly conducting disk, the instability grows even when the seed magnetic field is infitesimal. Inclusion of the resistivity, however, leads to the appearance of an instability threshold. When the resistivity η depends on the instability-induced turbulent magnetic fields δB as η (mean value of δB^{2}), the marginal stability condition self consistently determines the α-parameter of the angular momentum transport due to magnetic stress. For fully ionized disks, the magnetic viscosity parameter αB is between 0.001 and 1. Our three-dimensional MHD simulation confirms these unstable eigenmodes. It also shows that the α-parameter observed in simulation is between 0.01 and 1, in agreement with theory. The observationally required smaller α in the quiescent phase of accretion disks in dwarf novae may be explained by the decreased inoization due to the temperature drop.

**Envelope evolution of a laser pulse in an active medium **

D.L. Fisher, T. Tajima, M.C. Downer, C.W. Siders

**Abstract**

We show that the envelope velocity, *v*_{env}, of a short laser pulse can, via propagation in an active medium, be made less than, equal to, or even greater than c, the vacuum phase velocity of light. Simulation results, based on moving frame propagation equations coupling the laser pulse, active medium, and plasma, are presented, as well as equations that determine the design of superluminous and subluminous *v*_{env} values. In this simulation the laser pulse evolves in time in a moving frame as opposed to earlier work [D. L. Fisher and T. Tajima, Phys. Rev. Lett. 71, 4338 (1993)], where the profile was fixed. The elimination of phase slippage and pump depletion effects in a laser wake field accelerator is discussed as a particular application. Finally, we discuss media properties necessary for an experimental realization of this technique.

**Nonlinear instability and chaos in plasma wave-wave interactions. I. Introduction **

C.S. Kueny, P.J. Morrison

**Abstract**

Conventional linear stability analyses may fail for fluid systems with an indefinite free-energy functional. When such a system is linearly stable, it is said to possess negative energy modes. Instability may then occur either via dissipation of the negative energy modes, or nonlinearly via resonant wave–wave coupling, leading to explosive growth. In the dissipationless case, it is conjectured that intrinsic chaotic behavior may allow initially nonresonant systems to reach resonance by diffusion in phase space. In this and a companion paper (submitted to Phys. Plasmas), this phenomenon is demonstrated for a simple equilibrium involving cold counterstreaming ions. The system is described in the fluid approximation by a Hamiltonian functional and associated noncanonical Poisson bracket. By Fourier decomposition and appropriate coordinate transformations, the Hamiltonian for the perturbed energy is expressed in action-angle form. The normal modes correspond to Doppler-shifted ion-acoustic waves of positive and negative energy. Nonlinear coupling leads to decay instability via two-wave interactions, and to either decay or explosive instability via three-wave interactions. These instabilities are described for various integrable systems of waves interacting via single nonlinear terms. This discussion provides the foundation for the treatment of nonintegrable systems in the companion paper.

**Analytical and numerical studies of ion mobility near the tokamak plasma edge **

H. Xiao, R.D. Hazeltine, P.M. Valanju

**Abstract**

The effects of radial electric field on charged particle motion and transport in the toroidal magnetic system have been studied both analytically and numerically. The effects of radial electric field on particle orbits are examined, allowing for the relatively large and strongly sheared field observed in some experiments. It is found that ion radial mobility due to the combined effects of radial electric field and charge exchange collisions can dramatically affect the ion transport and orbit loss near the tokamak edge. These properties may help understand the formation of transport barrier near the tokamak plasma edge during high confinement mode (H-mode) discharge and explain the asymmetry between bias voltage and confinement in biased-electrode-induced H-mode.

**Dynamics and transport in rotating fluids and transition to chaos in area preserving nontwist maps **

D. del. Castillo Negrete

**Abstract**

Dynamics and transport in rotating fluids in general, and in a rotating annulus experiment in particular, are studied. A derivation of the quasigeostrophic equation for the rotating annulus experiment is presented, followed by a discussion of exact time-independent axisymmetric solutions. Linear quasigeostrophic theory is used to study both the instability of axisymmetric solutions and the propagation of Rossby waves in the experiment. Transport of passive scalars by waves in shear flow is studied using few degrees-of-freedom Hamiltonian models. Two transport problems are considered: the destruction of transport barriers and the statistical description of particle motion. The first problem is studied using the resonance overlap criterion and a criterion for separatrix reconnection. Both criteria are compared with numerical and experimental results. For the second problem, the statistical description of particle motion, the variance of the particle displacement is computed and evidence of anomalous diffusion is presented. The probability distribution functions of trapping and flight events are computed and shown to exhibit power law behavior. The transport results obtained in the models are compared with the experimental results. It is shown that the general Hamiltonian for traveling waves in symmetric shear flow is degenerate and, thus, exhibits topological changes in phase space due to separatrix reconnection. From this general Hamiltonian, an area preserving map called the standard nontwist map is constructed. This map violates the twist condition along a curve called the shearless curve. Using the method of approximating KAM curves by periodic orbits and the residue criterion, a numerical study of the destruction of the shearless curve is presented. This is a novel problem for which KAM theory and other important results can not be applied, due to the violation of the twist condition. The results obtained are interpreted in light of the renormalization group and it is concluded that the destruction of the shearless KAM curve with winding number equal to the inverse golden mean is described by a period-six fixed point of the renormalization group operator. This new fixed point defines a new universality class for the transition to chaos in Hamiltonian systems.

**Analytical studies of the effects of charge-exchange on a magnetized plasma **

M.D. Calvin

**Abstract**

We analytically calculate the neutral particle distribution and its effects on ion heat and momentum transport in three dimensional magnetized plasmas with arbitrary temperature and density profiles. A general variational principle taking advantage of the simplicity of the charge-exchange (CX) operator is derived to solve self-consistently the neutral-plasma interaction problem. To facilitate an extremal solution, we use the short CX mean-free-path λ_{x} ordering, Further, a non-variational, analytical solution providing a full set of transport coefficients is derived by making the realistic assumption that the product of the CX cross section with relative velocity is constant. The effects of neutrals on plasma energy loss and rotation appear in simple, sensible forms. The presence of ionized impurities in the plasma are also considered, and the effects of CX drag and ion-impurity collisions on plasma flows in the short λ_{x} regime are presented. Finally, the long λ_{x} regime is analyzed in slab geometry by finding an appropriate Green's function for the neutral kinetic equation and by solving recursively for the neutral distribution function. Our results are found to agree favorably with previous work.

**1/ f noise in two-dimensional fluids **

S.B. Cable, T. Tajima

**Abstract**

We derive an exact result on the velocity fluctuation power spectrum of an incompressible two-dimensional fluid. Employing the fluctuation-dissipation relationship and the enstrophy conversation, we obtain the frequency spectrum of a 1/*f* form.

**Comparisons of nonlinear toroidal turbulence simulations with experiment **

W. Dorland, M. Kotschenreuther, P.M. Valanju, W.H. Miner, Jr. J.Q. Dong, W. Horton, F.L. Waelbroeck, T. Tajima, M.J. Lebrun, Et al.

**Abstract**

The anomalously large thermal transport observed in tokamak experiments is the outstanding physics-based obstacle in the path to a commercially viable fusion reactor. Although decades of experimental and theoretical work indicate that anomalous transport and collective instabilities in the gyrokinetic regime are linked, no widely accepted description of this transport yet exists. Here, detailed comparisons of first-principles gyrofluid and gyrokinetic simulations of tokamak microinstabilities with experimental data are presented. With no adjustable parameters, more than 50 TFTR L-mode discharges have been simulated with encouraging success. Given the local plasma parameters and the temperatures at r/a ≅ 0.8, the simulations typically predict T_{i}(r) and T_{e}(r) within +/- 25% throughout the core and confinement zones. In these zones, the predicted thermal diffusivity increases with minor radius robustly. For parameters typical of r/a greater than 0.8, toroidal stability studies confirm the importance of impurity density gradients as a source of free energy potentially strong enough to explain the large edge thermal diffusivity, as first emphasized by Coppi, *et al*. Advanced confinement discharges have also been simulated. The dramatic increase of T_{i}(0) observed in Supershots is recovered by our model for dozens of simulated experiments. Finally, simulations of VH and PEP mode-like plasmas show that velocity-shear stabilization of toroidal microinstabilities is quantitatively significant for realistic experimental parameters.

**Two-dimensional ballooning transformation with applications to toroidal Alfven eigenmodes **

X.D. Zhang

**Abstract**

A general formulation for high-n (n is the toroidal mode number) modes in an axisymmetric toroidal plasma is presented, based on the two dimensional (2-D) ballooning transformation. It is shown that this formulation is more general than the conventional ballooning theory, and reduces to the conventional theory in a special case. Toroidal Alfven waves are studied using the 2 -D ballooning formulation. A perturbation theory is systematically developed for the continuum damping of the toroidal Alfven eigenmode (TAE). A formula, similar to the Fermi golden rule for decaying systems in quantum mechanics, is derived for the continuum damping rate of the TAE; the decay (damping) rate is expressed explicitly in terms of the coupling of the TAE to the continuum spectrum. Numerical results are obtained and compared to previous calculations. Kinetic effects on toroidal Alfven waves are studied. Multiple -gap coupling is included automatically by the 2-D ballooning formulation. A new branch of modes, the kinetic toroidal Alfven eigenmodes (KTAE), emerges as a result of kinetic effects. This mode resides just above the toroidal shear Alfven gap, and has a structure similar to the TAE. Numerical results for the kinetic damping rates for the TAE and the KTAE are obtained, and multiple-gap coupling effects are studied by comparing with the single gap theory of Mett and Mahajan (Phys. Fluids B 4 2885 (1992)).

**Potentials and bound states **

W.F. Buell, B.A. Shadwick

**Abstract**

We discuss several quantum mechanical potential problems, focusing on those which highlight commonly held misconceptions about the existence of bound states. We present a proof, based on the variational principle, that certain one dimensional potentials always support at least one bound state, regardless of the potential's strength. We examine arguments concerning the existence of bound states based on the uncertainty principle and demonstrate, by explicit calculations, that such arguments must be viewed with skepticism.

**Impurity effects on linear and nonlinear ion temperature gradient driven modes **

D. Jovanovic, W. Horton

**Abstract**

Linear and nonlinear stages in the development of the ion-temperature-gradient-driven drift-wave instability are studied analytically in the presence of shear flows, magnetic shear, inhomogeneity, and curvature. In the linear regime, it is shown that the toroidal η_{i} mode is destabilized by a small amount of impurities only if there exists an impurity buildup at the plasma edge. In the nonlinear regime two types of coherent structures are found: a generalized Hasegawa–Mima dipole vortex in the weak magnetic shear case, and a periodic, vortex-chain solution in the strong shear case, which corresponds to the saturated, large-amplitude drift-tearing mode.

**Generalized relaxation theory and vortices in plasmas **

S.R. Oliveira, T. Tajima

**Abstract**

We present a generalization of the relaxation theory based on the canonical momentum of each species fluid in a multicomponent plasma. The generalized helicity, as a topological quantity, has a lifetime larger than the lifetime of the energy. We then propose a simple variational principle that suggests the existence of vortex structures. We study localized solutions, assuming the existence of a separatrix. Two-dimensional and three-dimensional solutions are studied for an electron-positron-proton plasma. Ideal magnetohydrodynamic three-dimensional localized vortices are studied as well. Possible cosmological implications are discussed.

**Energetic particle drive for toroidicity-induced Alfven eigenmodes and kinetic toroidicity-induced Alfven eigenmodes in a low-shear tokamak**

B.N. Breizman, S.E. Sharapov

**Abstract**

The structure of toroidicity-induced Alfven eigenmodes (TAE) and kinetic TAE (KTAE) with large mode numbers is analysed and the linear power transfer from energetic particles to these modes is calculated in the low-shear limit when each mode is localized near a single gap within an interval whose total width Δ^{out} is much smaller than the radius *r*_{m} of the mode location. Near its peak where most of the mode energy is concentrated, the mode has an inner scale length Δ^{in}, which is much smaller than Δ^{out}. The scale Δ^{in} is determined by toroidicity and kinetic effects, which eliminate the singularity of the potential at the resonant surface. This work examines the case when the drift orbit width of energetic particles Δ_{b} is much larger than the inner scale length Δ^{in}, but arbitrary compared to the total width of the mode. It is shown that the particle-to-wave linear power transfer is comparable for the TAE and KTAE modes in this case. The ratio of the energetic particle contributions to the energetic particle drive for the TAE and KTAE modes is then roughly equal to the inverse ratio of the mode energies. It is found that in the low-shear limit the energetic particle drive for the KTAE modes can be larger than that for the TAE modes.

**A wavenumber-partitioning scheme for two-dimensional statistical closures **

J.C. Bowman

**Abstract**

A technique of wavenumber partitioning that conserves both energy and enstrophy is developed for two-dimensional statistical closures. This advance facilitates the computation of energy spectra over seven wavenumber decades, a task that will clearly remain outside the realm of conventional numerical simulations for the foreseeable future. Within the context of the test-field model, the method is used to demonstrate Kraichnan's logarithmically-corrected scaling for the enstrophy inertial range and to make a quantitative assessment of the effect of replacing the physical Laplacian viscosity with an enhanced hyperviscosity.

**Nonlinear symmetric stability of planetary atmospheres **

J.C. Bowman, T.G. Shepherd

**Abstract**

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.

**Wake fields in semiconductor plasmas **

V.I. Berezhiani, S.M. Mahajan

**Abstract**

It is shown that an intense short laser pulse propagating through a semiconductor plasma will generate longitudinal Langmuir waves in its wake. The measurable wake field can be used as a diagnostic to study nonlinear optical phenomena. For narrow gap semiconductors (for example, InSb) with Kane-type dispersion relation, the system can simulate, at currently available laser powers, the physics underlying wake-field accelerators.

**Hybrid model in general geometry **

M. Yagi, T. Tajima, M. Lebrun

**Abstract**

We propose hybrid model equations in toroidal (or more general) geometry for magnetically confined plasmas. This is suitable for low frequency toroidal modes, for example, the trapped electron and current diffusive ballooning instabilities. This model consists of fluid ions and drift kinetic electrons. We discuss the numerical algorithm of these model equations. The linear dispersion relation of this model equations that defines the requirements of the model for describing these modes is also discussed.