**Resistive tearing instability with equilibrium shear flow **

X.L. Chen, P.J. Morrison

**Abstract**

The effect of equilibrium velocity shear on the resistive tearing instability has been systematically studied, using the boundary layer approach. Both the constant-ψ tearing mode, which has a growth rate that scales as S^{−3/5}, and the nonconstant-ψ tearing mode (Δ'(αS)^{−1/3} > 1), which has a growth rate that scales as S^{−1/3}, are analyzed in the presence of flow. Here S is the usual ratio of the resistive diffusion and Alfven times. It is found that the shear flow has a significant influence on both the external ideal region and the internal resistive region. In the external ideal region, the shear flow can dramatically change the value of the matching quantity Δ'. In the internal resistive region, the tearing mode is sensitive to the flow shear at the magnetic null plane: G'(0). When G'(0) is comparable to the magnetic field shear, F'(0), the scalings of the constant-ψ tearing mode are changed and the Δ' > 0 instability criterion is removed, provided G'(0)G''(0) − F'(0) F''(0) ≠ 0. The scalings of the nonconstant-ψ tearing mode remain unchanged. When the flow shear is larger than the magnetic field shear at the magnetic null plane, both tearing modes are stabilized. Finally, the transition to ideal instability is discussed.

**High energy beam transport in crystal channels **

B. Newberger, T. Tajima

**Abstract**

The transport of high-energy accelerating charged particles in channeling conditions in a crystalline solid is considered by means of a Fokker-Planck model. Multiple scattering on electrons, appropriate to the channeling of positive particles, and radiation damping due to the emission of channeling radiation are also included. Analytic solutions have been obtained for the case of a harmonic channeling potential. Without acceleration, diffusion due to multiple scattering occurs. With acceleration, the adiabatic damping retards this, although the reduction in critical channeling angle with increasing energy eventually competes. For light particles (positrons), the emission of channeling radiation can lead to a steady state. Implications for crystal accelerator schemes are discussed.

**Statistical-mechanical methods of cold nuclear fusion in metal hybrides **

S. Icimaru, A. Nakano, S. Ogata, S. Tanaka, H. Iyetomi, T. Tajima

**Abstract**

Screening action of the s-d hybridized electrons in PdH_{z} and TiH_{z} is analyzed in the Fermi-Thomas approximation. The resulting interaction between hydrogen exhibits an attractive part arising from interference between the H-induced s-electrons and the valence electrons. The screening potentials due to many-body effects between the electron-screened protons are examined through solution to the hypernetted-chain equations. The nuclear reaction rates between hydrogen isotopes are calculated at various temperatures by taking account of statistical-mechanical enhancement arising from the increment in the Coulombic chemical potential of a reacting pair before and after nuclear reaction. Remarkable isotopic and temperature-dependent effects are predicted.

**General-relativistic plasma physics in the early universe **

K.A. Holcomb, T. Tajima

**Abstract**

We apply the ‘‘3+1’’ formalism of Thorne and Macdonald to construct the linearized theory of a general-relativistic electron-positron plasma in the early Universe. Close formal correspondence between the theory of such plasmas and that of their special-relativistic counterparts is demonstrated. The time variation of the plasma modes due to the expansion of the background is determined for the case of a radiation-dominated Universe; it is found that the frequencies of the basic modes redshift like the frequency of a free photon. A simple kinetic argument is used to justify the neglect of creation and annihilation (collisional) effects. The formulation is sufficiently straightforward to be readily amenable to numerical implementation. Our results can be applied to the study of the origin of primordial intergalactic magnetic fields, as well as to the problem of matter fluctuations in the early Universe.

**Ion temperature gradient driven turbulence in tokamaks with flat density profiles **

N. Mattor, P. Diamond

**Abstract**

The theory of ion temperature gradient-driven turbulence in tokamaks is extended to the flat density regime. The values of ion and electron thermal conductivities, χ_{ i} and χ_{e}, momentum diffusivity χ_{φ}, and particle flux Γ_{ r} are also calculated. These formulas extend previous calculations, which were restricted to the regime *L*_{n} < (* L*_{s}L_{T})^{1/2} (Phys. Fluids 29, 3291 (1986); Phys. Fluids 31, 1180 (1988)). This allows an assessment of the role of ion temperature gradient turbulence in H modes, where the density gradient is often observed to be flattened in the plasma core.

**Screening of the hybridized 4D-1S electrons in PdDx and nuclear reaction rates between hydrogen isotopes **

S. Ichimaru, A. Nakano, S. Ogata, H. Iyetomi, T. Tajima

**Abstract**

Screening action of the hybridized 4d-1s electrons in PdDx is analyzed in the Fermi-Thomas approximation; charge-form factors for Pd and D are derived. The resulting D-D interaction is a sensitive function of both density x of the deuterons and energy levels E_{1s} of D-induced s-electron states; it exhibits an attractive part arising from interference between the 1s-screening electrons and strongly coupled (r_{s}≅2) valence electrons. Nuclear reaction rates of hydrogen isotopes in Pd are calculated at various combinations of x and E_{1s} by including D-D many-body effects through the ion-sphere potential; effects of fluctuations in x and/or E_{1s} are discussed.

**Nonlinear self-focusing of optical beams in plasmas **

T. Kurki-Suonio

**Abstract**

Motivated by the need for a novel, ultra-high energy particle accelerator, the nonlinear interaction between a short optical pulse and plasma is studied. Assuming a steady state in which the ponderomotive force on the electrons is balanced by the electrostatic force, it is found that, for appropriate parameters, the inherent diffraction of a Gaussian beam can be balanced or overcome by plasma lensing brought about by the nonlinear effects. The critical quantity is determined to be the total power of the beam so that for values greater than P_{cr} ≈ 10^{10}(ω_{0} / ω_{p})^{2}W the optical beam launched into a plasma will self-focus. A steady asymptotic beam profile solution of a solitary nature is also obtained. A phase -averaged particle simulation code appropriate for studying transport of optical beams in plasmas was developed and is described. The theoretical predictions obtained are tested using this simulation code, and a good qualitative agreement between theory and simulation is obtained.

**Particle simulation in curvilinear coordinate systems **

M.J. Lebrun, T. Tajima

**Abstract**

Methods are presented for particle simulation of plasmas in a nearly arbitrary coordinate metric and describe a toroidal electrostatic simulation code that evolved from this effort. A Mercier-type coordinate system is used, with a nonuniform radial grid for improved cross-field resolution. A fast iterative method for solving the Poisson equation is employed, and the interpolation and/or filtering technique shown to be momentum and energy conserving in the continuum limit. Lorentz ion and drift electron species are used. The code was thoroughly tested for its reproduction of linear and nonlinear physics, and was applied to the toroidal drift wave problem and its impact on anomalous transport in tokamaks.

**Nonlinear behavior of magnetohydrodynamic modes near marginally stable states I: General formulation & application to the non-resonant kink modes in a reversed field pinch & to the quasi-interchange modes in a tokamak **

N. Nakajima

**Abstract**

Two types of nonlinear equations describing the time development of modes near marginally stable states in an inhomogeneous medium are obtained through a general formulation that employs a perturbation expansion around the marginally stable state under the assumption of a single helicity. One type of nonlinear equation has a Hamiltonian form that may be interpreted as the equation of motion for a particle in the potential field of a central force; the other type leads to the Landau equation, which is well known in fluid dynamics. The former equation is obtained when the linear operator is degenerate at the marginally stable state, which corresponds to the case when the linear dispersion relation has a double root for the frequency at the marginally stable state, whereas the latter is obtained when the linear operator is nondegenerate, i.e., the linear dispersion relation has a single root. In the framework of magnetohydrodynamics, the former corresponds to the nonresonant ideal modes, and the latter to the resistive modes. The nonlinear behavior of the nonresonant kink modes in a reversed field pinch and of the quasi-interchange modes in a tokamak are examined with the application of the general formulation. It is shown that new stable helical equilibria bifurcate near the initial axisymmetric equilibrium, so that the plasma nonlinearly oscillates around the new bifurcated equilibrium, which leads to nonlinear saturation of the nonresonant kink modes in a reversed field pinch and of the quasi-interchange mode in a tokamak. The nonlinear stabilizing effects causing bifurcation of the equilibrium are interpreted as quasilinear effects. Compressibility reduces the nonlinear stabilizing effects through changing the quasilinear components and is important even when the modes are near marginally stable states.

**Equilibrium of a plasma in the fluid- and Vlasov-Maxwell systems **

S.M. Mahajan, W. Li

**Abstract**

It is shown that a recently constructed exact solution of the Vlasov equation describing a plasma with density and temperature gradients can be expressed in terms of the constants of motion. The distribution function is then used to illustrate the differences between a Vlasov and a one fluid description. In fluid theory, only the pressure profile is determined (unless one postulates an equation of state), while the Vlasov description leads to a separate determination of density (g), and temperature (ψ^{2}) profiles; the equation of state, g = ψ^{3/2}^{β}, comes out naturally in the latter case.

**Enhanced radiation driven by a DC electric field **

T. Tajima, A.O. Benz, M. Thaker, J.N. Leboeuf

**Abstract**

Direct radiation by runaway electrons under a constant (dc) electric field is investigated. In a one-and-two-halves-dimensional relativistic EM code, an electron beam propagates along the external magnetic field parallel to the dc field and quickly decays into a runaway tail sustained by the dc field. Electrostatic and transverse waves are observed at various (fixed for each particular run) angles of wave propagation. Both plasma waves and EM radiation are strongly enhanced by the runaway tail. In the linear and early nonlinear beam stages, the EM wave energy is slightly enhanced as the associated electrostatic component of the waves (together with the dc field) traps and detraps electrons. In the late nonlinear (runaway) stage and with sufficiently large observing angle, bursts of EM wave energy occur, accompanied by fast perpendicular spreading of the distribution function, and they coincide with clamping of runaway electron momenta. A possible application is to msec radio spikes associated with solar flares. In this situation, the possibility of the present mechanism yielding radiation temperatures in excess of 10 to the 15th K is not out of the question.

**Stability of the global Alfven eigenmode in the presence of fusion alpha particles in an ignited tokamak plasma **

G.Y. Fu, J.W. Van Dam

**Abstract**

The stability of global Alfven eigenmodes is investigated in the presence of super-Alfvenic energetic particles, such as fusion-product alpha particles in an ignited deuterium--tritium tokamak plasma. Alpha particles tend to destabilize these modes when ω_{*α} > ω_{A}, where ω_{A} is the shear-Alfven modal frequency and ω_{*α} is the alpha particle diamagnetic drift frequency. This destabilization due to alpha particles is found to be significantly enhanced when the alpha particles are modeled with a slowing-down distribution function rather than with a Maxwellian distribution. However, previously neglected electron damping due to the magnetic curvature drift is found to be comparable in magnitude to the destabilizing alpha particle term. Furthermore, the effects of toroidicity are also found to be stabilizing, since the intrinsic toroidicity induces poloidal mode coupling, which enhances the parallel electron damping from the sideband shear-Alfven Landau resonance. In particular, for typical ignition tokamak parameters, global Alfven eigenmodes are found to be completely stabilized by either the electron damping that enters through the magnetic curvature drift or the damping introduced by finite toroidicity.

**Influence of attractive interaction between deuterons in Pd on nuclear fusion **

T. Tajima, H. Iyetomi, S. Ichimaru

**Abstract**

It is shown that in a heavily deuterated palladium metal a pair of deuterons exhibits attractive interaction at short distances (≈0.1–0.7 Å) due to strong Coulomb correlations in the ion-sphere model and due to the screening action of localized 4d electrons. This mechanism can lead to enhanced thermonuclear reactions at room temperatures some 50 orders of magnitude faster than that in a D_{2} molecule. Characteristic signatures of predicted nuclear reactions are described.

**Excitation of toroidicity-induced shear Alfven eigenmode by fusion alpha particles in an ignited tokamak **

G.Y. Fu, J.W. Van Dam

**Abstract**

The toroidicity-induced shear Alfvén eigenmode is found to be destabilized by fusion alpha particles in an ignited tokamak plasma.

**Transition from neoclassical to turbulent electron diffusion **

D.E. Kim, D. Choi, C.W. Horton, P.N. Yushmanov, V.V. Parail

**Abstract**

Electron diffusion in tokamak geometry is calculated including both the Coulomb collisional pitch angle scattering and the electromagnetic drift wave fluctuation spectrum. In the weak fluctuation limit the neoclassical banana-plateau diffusion coefficient is recovered. At higher fluctuation levels typical of the mixing length amplitude the anomalous transport formulas of drift wave turbulence are recovered using test particle simulations.

**Anomalous electron thermal conduction from magnetic turbulence **

B.G. Hong, W. Horton

**Abstract**

The electron thermal balance equation from the Braginskii equations with the finite Larmor radius heat flux is analyzed for the space-time-average power balance in the presence of electromagnetic fluctuations. Formulas for the anomalous thermal flux associated with the E×B motion and the magnetic δB_{⊥} fluctuations are derived and evaluated for the c/ω_{pe} scale electromagnetic turbulence typical of the ∇_{Te}-driven short wavelength drift modes. The result is compared with several Ohkawa-type formulas for the anomalous electron thermal energy transport.

**Profiles of a self-focused optical beam in a plasma **

T. Kurki-Suonio, P.J. Morrison, T. Tajima

**Abstract**

Self-focusing of an intense optical beam in a plasma is studied, including the nonlinear effects of both the relativistic electron mass and the ponderomotive potential due to the electromagnetic wave. An exact steady asymptotic solution of beam propagation in a localized solitary wave form is obtained in slab geometry. Amplitude-width scaling relations are obtained, which imply that the width is limited to be less than square root of three of the collisionless skin depth. In the nonrelativistic limit, keeping only the relativistic mass effects, our solution reduces to the solution obtained by Schmidt and Horton. Solutions where the beam profile is of oscillatory nature, which correspond to the presence of the steady solution of a multi-beamlet, are also presented. Finally, the asymptotic nature of the solitary wave is tested using a recently developed numerical particle simulation code.

**Short wavelength electron temperature gradient driven drift wave turbulence **

N. Bekki, W. Horton, B.G. Hong, T. Tajima

**Abstract**

A local, hydrodynamic model of the short wavelength electrostatic drift waves driven unstable by the electron temperature gradient in toroidal geometry is used to find the saturation level and the mode coupling to the longer wavelength collisionless skin depth c/ω_{pe} magnetic turbulence. We show that the k-spectrum is peaked in the c/ω_{pe} wavelength region at the mixing length amplitude. The level of magnetic turbulence is sufficient to produce the empirical neo-Alcator type of confinement formula.

**Diffusion in symplectic maps **

H. Kook, J. Meiss

**Abstract**

The characteristic function method is used to obtain the diffusion tensor for symplectic maps. At lowest order the quasilinear result is obtained, and a series in higher-order correlations is developed. Comparison of the theory to numerical experiments is given using a four-dimensional example of Froeschlé. The experiments agree well with the theory for moderately large parameters. Arnol’d diffusion for the ‘‘thick-layer’’ case is discussed. It is shown that the short-time correlations in one canonical plane affect the diffusion in the other plane even in the limit of zero coupling. Accelerator modes exist for the Froeschlé example and cause divergences in the diffusion, but these only appear when the accelerating region is included in the ensemble.

**The dynamics of ion rings in highly conductive plasmas **

P.M. Lyster, R.N. Sudan

**Abstract**

A numerical study employing a hybrid computer code (particle ions/fluid electrons) is made of the dynamics of translating ion rings in a highly conductive plasma. The translational kinetic energy of such rings is lost through the generation of Alfven waves. When the initial axial velocity of the ring is greater than the local Alfven speed in the plasma, a typical slowing down length is *z*_{A} =(*a*_{r}/4ζ)*v*^{4}_{z}0/(*v*^{2}_{θ}*v*^{2}_{ A}), where *a*_{r} is the ring radial width, ζ=|δ*B*/*B*_{0}|, δ*B* is the ring self-magnetic-field strength on axis, *B*_{0} is the strength of the axial magnetic field, *v*_{z}0 is the initial translational velocity of the ring, *v*_{θ} is the average toroidal ring particle velocity, *v*_{A}=*B*_{0}/(4πρ)^{1/2} is the Alfven velocity, and ρ is the plasma mass density. Also discussed is the merging of an ion ring with a spheromak.

**Suppression of sawtooth oscillations due to hot electrons and hot ions **

Y.Z. Zhang, H.L. Berk

**Abstract**

The theory of m = 1 kink mode stabilization is discussed in the presence of either magnetically trapped hot electrons or hot ions. For instability hot ion requires particles peaked inside the q = 1 surface, while hot electrons require that its pressure profile be increasing at the q = 1 surface. Experimentally observed sawtooth stabilization usually occurs with off-axis heating with ECRH and near axis heating with ICRH. Such heating may produce the magnetically trapped hot particle pressure profiles that are consistent with theory.

**New developments in the theory of ion-temperature-gradient-driven turbulence in tokamaks **

N. Mattor

**Abstract**

This work considers three aspects of ion-temperature-gradient driven turbulence (η_{i}-turbulence) in tokamaks. Chapter 1 is a primer for those not familiar with the basics of this instability. Chapter 2 presents a theory of weak η_{i}-turbulence near the threshold of instability. The model considers kinetics ions and adiabatic electrons in a sheared slab geometry. The nonlinear wave kinetic equation indicates that ion Compton scattering is the dominant nonlinear saturation process. The wave kinetic equation is reduced to a differential equation for the spectrum, from which it is shown that the energy scatters to the linearly stable low k_{y} modes. The spectrum of fluctuation levels (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 similarly low, so that strong ion heating can be expected to drive the ion temperature gradient past this threshold regime. Chapter 3 presents a theory of diffusive momentum transport driven by η_{ i}-turbulence, in order to investigate the relation between momentum and thermal transport in neutral-beam-heated tokamaks with subsonic toroidal rotation velocity. Chapter 4 is a study of η_{i}-turbulence in the presence of flat density profiles, relevant to the high confinement discharges ( H-modes) on the DIII-D tokamak. Chapter 5 is a summary of this work, and a list of suggestions for further investigation.

**Application of Newton's method to Lagrangian dynamical systems **

H. Kook, J. Meiss

**Abstract**

An algorithm for Newton's method is presented to find periodic orbits for N degree-of-freedom Lagrangian mappings. The computation time of the method is proportional to n, the period of the orbit, and the required storage space is reduced to omicron(n). The construction of the algorithm is based upon block-diagonalization of the Hessian matrix of the action function. The index of the action function, which is closely related to orbital stability, is obtained during execution of the algorithm.