**Magnetic fusion with high energy self-colliding ion beams **

N. Rostoker, F. Wessel, B. Maglich, A. Fisher

**Abstract**

Self-consistent equilibria are obtained for high beta plasma where almost all of the ions are energetic with a gyroradius of the order of the plasma scale length. Magnetohydrodynamics would not apply to such a plasma. Recent experiments with tokamaks suggest that it would be insensitive to microinstabilities. Several methods are described for creating the plasma with intense neutralized ion beams.

**Fast ion-driven Bernstein instabilities **

K.R. Chen

**Abstract**

We investigate a new mechanism, the two-energy-stream cyclotron instability, for fast ions (e.g., fusion products) to drive electrostatic waves and to slow down. The instability comes from a relativistic effect, which dominates conventional phase overtaking as the axial phase velocity exceeds the speed of light. Both a single particle model and a dispersion relation are developed in order to illuminate the physics insights and scaling laws. We present numerical results and discuss nonlinear processes. The mechanism is essential for the dynamics of the fast ions in both D-D and D-T devices.

**Explicit, analytical and numerical solution of the nonlinear Vlasov Poisson system **

S.M. Mahajan, V. Skarka, E. Fijalkow

**Abstract**

In order to describe the time evolution of an inhomogeneous collisionless plasma, the nonlinear Vlasov equation is solved perturbatively, using the subdynamics approach and the diagrammatic techniques. The solution is given in terms of a double perturbation series: one with respect to the nonlinearities and the other with respect to the interaction between particles. The infinite sum of interaction terms can be performed exactly due to the property of dynamical factorization. Following the methodology, the exact solution in each order with respect to nonlinearities is computed. For a choice of initial perturbation the first-order exact solution is numerically integrated in order to find the local density excess. The approximate analytical solution is found to be in excellent agreement with exact numerical integration as well as with ab initio numerical simulations. Analytical computation gives a better insight into the problem, and it has the advantage to be simpler, and also accessible, in some range of parameters where it is difficult to find numerical solutions.

**Amplification mechanism of ion ripple lasers and its possible application **

K.R. Chen, J. Dawson

**Abstract**

The ion-ripple laser (IRL) is an advanced scheme for generating coherent high-power radiation, in which a relativistic electron beam propagates obliquely through an ion ripple in a plasma. Its amplification mechanism is described by a low gain theory, while the linear growth rate is given by the dispersion relation. The efficiency of the lasing is determined by the nonlinear saturation mechanism discussed. By proper choice of device parameters, sources of microwaves, optical, and perhaps even X-rays can be made. The availability of tunable sources for wide wavelength regimes, coherence and high-power, as well as lower cost and simplicity of equipment, are emphasized.

**Anomalous ion thermal transport in hot ion plasmas by the ion temperature gradient mode **

J-Y. Kim, W. Horton, B. Coppi

**Abstract**

Experiments show that the observed radial profiles of the ion thermal conductivity χ_{i} have the opposite shapes with those obtained from the ion temperature gradient mode (η_{i} mode) turbulence model by the traditional mixing length estimate. In this work, this radial profile problem is reconsidered with an electromagnetic study of the linear stability of the toroidal η_{i} mode and a new rule for choosing the mixing length. It is first shown that the electromagnetic effect gives a significant stabilizing effect on the toroidal η_{i} mode, and that the observed reduction of χ_{i}(r) in the core region can be explained by this electromagnetic effect. Secondly, in view of earlier numerical simulations showing the transfer of fluctuation energy to larger scales that those for the fastest growth rate, as well as fluctuation measurements indicating longer radial correlation lengths, a new mixing length formula is proposed to explain the radial increase of the χ_{i}. It is shown the new formula fits well the observed χ_{i}(r) profiles in two TFTR supershot discharges and also gives the scaling law in the current and the magnetic field which agrees better with experiment than the conventional formula.

**Resistive MHD studies of tearing mode instability with equilibrium shear flow and magnetic reconnection **

L. Ofman

**Abstract**

Magnetic reconnection and tearing can play an important role in fusion experiments and in space plasmas. The magnetohydrodynamic (MHD) study of the linear and nonlinear evolution of the resistive tearing mode instability in the presence of equilibrium shear flow, and the reconnection of an x-point magnetic field configuration are discussed. Numerical solutions of the linearized time-dependent MHD equations and growth rate scaling are obtained. The results of the computations are compared to previous work, and the computed growth rate scalings agree with analytical predictions. The introduction of viscosity and small equilibrium shear flow alters the growth rate scaling considerably. When the shear flow is large, the growth rate behaves in a more complex way, and Kelvin-Helmholtz instability effects are present. The linear evolution of the double tearing mode with equilibrium shear flow and viscosity is investigated numerically. The dispersion relation for the growth rate of the double tearing instability is generalized to include flow. Relatively small shear flow at the resonant surfaces has a stabilizing effect on the double tearing mode. For Reynolds number comparable or larger than the magnetic Reynolds number a stabilizing effect is found. The nonlinear evolution of the tearing mode instability with equilibrium shear flow is investigated via numerical solutions of the resistive incompressible two dimensional MHD equations. The simulations are initiated with solutions of the linearized MHD equations. Magnetic energy release decreases, and the saturation time increases with shear flow. The validity of the numerical solutions is tested by verifying that the total energy and helicity are conserved. The reconnection of two-dimensional stressed x-type neutral point magnetic fields is studied via solution of the nonlinear incompressible and compressible resistive MHD equations. Solutions of the linear eigenvalue problem are obtained and compared to the MHD simulations.

**Stochastic diffusion and Kolmogorov entropy in regular and random Hamiltonians **

M. Isichenko, W. Horton, J-Y. Kim, Hen and Choi

**Abstract**

The scalings of the E×B turbulent diffusion coefficient D and the Kolmogorov entropy K with the potential amplitude φ˜ of the fluctuation are studied using the geometrical analysis of closed and extended particle orbits for several types of drift Hamiltonians. The high-amplitude scalings, D α φ˜ ^{2} or φ˜ ^{0} and K α log φ˜, are shown to arise from different forms of a periodic (four-wave) Hamiltonian φ˜(x,y,t), thereby explaining the controversy in earlier numerical results. For a quasirandom (six-wave) Hamiltonian numerical data for the diffusion D α φ˜ ^{0.92±0.04} and the Kolmogorov entropy K α φ˜ ^{0.56±0.17} are presented and compared with the percolation theory predictions D_{p}α φ˜ ^{0.7}, K_{p}α φ˜ ^{0.5}.

**Two dimensional aspects of toroidal drift waves in the ballooning representation **

Y.Z. Zhang, S.M. Mahajan, X.D. Zhang

**Abstract**

By systematically doing the higher order theory, the predictions of the conventional ballooning theory (CBT) are examined for non-ideal systems. For the complex solvability condition to be satisfied, radial variation of the lowest order mode amplitude needs to be invoked. It turns out, however, that even this procedure with its concomitant modifications of eigenvalues and eigenstructures, is not sufficient to justify the predictions of many CBT solutions; only a small set of CBT solutions could be put on a firm footing. To demonstrate our general conclusions, theoretical and numerical results are presented for system of fluid drift waves non-adiabatic electron response.

**Vortices associated with toroidal ion-temperature gradient driven fluctuations **

W. Horton, D. Jovanovic, J. Rasussen

**Abstract**

The three nonlinear hydrodynamic equations for potential, parallel ion velocity and ion pressure used in simulations of the toroidal ion-temperature-gradient-driven fluctuations and transport in a shear magnetic field are analyzed for coherent vortex structures. Two types of vortex structures are found: one type for weak shear that is a generalization of the usual modon vortex construction and the second type of solution for strong magnetic shear where the convective nonlinearity in the parallel velocity field generates a cubic trapping nonlinearity in the vorticity equation. These vortex structures show the possibility of explaining the saturated states observed in the numerical simulations as self-organized nonlinear states in contrast to wave turbulence.

**Quasi-three dimensional electron holes in magnetized plasmas **

D. Jovanovic, W. Horton

**Abstract**

Using the electron drift-kinetic equation and a hydrodynamic description for the ions, new nonlinear vortex equations are derived taking into account the parallel trapping of the electrons in the positive potential regions. It is shown that the usual integration procedure for finding the coherent vortex structures for the E x B flows in the fluid description can be generalized to include the parallel acceleration eE_{||} ∂ f / ∂ ν_{||} producing the electron holes in the phase space. An example is considered in some detail.

**Chaotic transport by rossby waves in shear flow **

D. del Castillo, P.J. Morrison

**Abstract**

Transport and mixing properties of Rossby waves in shear flow are studied using tools from Hamiltonian chaos theory. The destruction of barriers to transport is studied analytically, by using the resonance overlap criterion and the concept of separatrix reconnection, and numerically by using Poincaré sections. Attention is restricted to the case of symmetric velocity profiles with a single maximum; the Bickley jet with velocity profile sech^{2} is considered in detail. Motivated by linear stability analysis and experimental results, a simple Hamiltonian model is proposed to study transport by waves in these shear flows. Chaotic transport, both for the general case and for the sech^{2} profile, is investigated. The resonance overlap criterion and the concept of separatrix reconnection are used to obtain an estimate for the destruction of barriers to transport and the notion of banded chaos is introduced to characterize the transport that typically occurs in symmetric shear flows. Comparison between the analytical estimates for barrier destruction and the numerical results is given. The role of potential vorticity conservation in chaotic transport is discussed. An area preserving map, termed standard nontwist map, is obtained from the Hamiltonian model. It is shown that the map reproduces the transport properties and the separatrix reconnection observed in the Hamiltonian model. The conclusions reached are used to explain experimental results on transport and mixing by Rossby waves in rotating fluids.

**On neutral plasma oscillations **

B.A. Shadwick, P.J. Morrison

**Abstract**

We examine the conditions for the existence of spectrally stable neutral modes in a Vlasov-Poisson plasma and show that for stable equilibria of systems that have unbounded spatial domain, the only possible neutral modes are those with phase velocities that correspond to stationary inflection points of the equilibrium distribution function. It is seen that these neutral modes can possess positive or negative free energy.

**Dielectric energy vs. plasma energy and Hamiltonian action-angle variables for the Vlasov equation **

P.J. Morrison, D. Pfirsch

**Abstract**

Expressions for the energy content of one-dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well-known dielectric energy, E_{D}, is compared with the exact plasma free energy expression, δ^{2}F, that is conserved by the Vlasov–Poisson system [Phys. Rev. A 40, 3898 (1989) and Phys. Fluids B 2, 1105 (1990)]. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbation of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes δ^{2}F=E_{D}. In the case of unstable and corresponding damped modes it is seen that δ^{2}F ≠E_{D}; in fact δ^{2}F = 0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since E_{D} is nonzero in this limit, whereas δ^{2}F remains zero. In the case of general perturbations about stable equilibria, the two expressions are not equivalent; the exact energy density is given by an expression proportional to ω||E(k,ω)||^{2}||ε(k,ω)||^{2}/ε_{I}(k,ω), where E(k,ω) is the Fourier transform in space and time of the perturbed electric field (or equivalently the electric field associated with a single Van Kampen mode) and ε(k,ω) is the dielectric function with ω and k real and independent. The connection between the new exact energy expression and the at-best approximate E_{D} is described. The new expression motivates natural definitions of Hamiltonian action variables and signature. A general linear integral transform (or equivalently a coordinate transformation) is introduced that maps the linear version of the noncanonical Hamiltonian structure, which describes the Vlasov equation, to action-angle (diagonal) form.

**The Rayleigh-Taylor instability in an expanding plasma **

S. Cable, T. Tajima

**Abstract**

In a number of fluid systems, overall fluid expansion has a retarding effect on the growth of Rayleigh-Taylor (RT) instabilities: the growth of RT instabilities relative to the expansion of the fluid is slowed and is often sub-exponential. We give two new analytical examples of this phenomena of reduced growth or stabilization: one with incompressible fluids and one with an adiabatic fluid. Confirmation of this phenomenon is also obtained from a new MHD code constructed specifically for modeling fluids undergoing nearly homogenous (but not necessarily isotropic) expansion or contraction. In the code, expansion is included by making each point of the computational grid co-moving with a predetermined overall expansion, which is equivalent to using an expanding metric.

**Kinetic theory of toroidicity-induced Alfven eigenmodes **

R. Mett, S. Mahajan

**Abstract**

An analytic kinetic description of the toroidicity-induced Alfvén eigenmode (TAE) is presented. The theory includes electron parallel dynamics nonperturbatively, an effect that is found to strongly influence the character, and damping of the TAE−contrary to previous theoretical predictions. A parallel conductivity model that includes collisionless (Landau) damping on the passing electrons and collisional damping on both trapped and passing electrons is used. Together, these mechanisms damp the TAE more strongly than previously expected. This is because the TAE couples (or merges) with the kinetic Alfvén wave (KAW) within the gap region under conditions that depend on the gap size, the shear, the magnitude of the conductivity, and the mode numbers. The high damping could be relevant to recent experimental measurements of the TAE damping coefficient. In addition, the theory predicts a "kinetic'' TAE, whose eigenfreqeuency lies just above the gap, whose existence depends on finite conductivity, and that is formed by the coupling of two KAW's.

**Scenarios for the nonlinear evolution of alpha particle induced Alfven wave instability **

H.L. Berk, B.N. Breizman, H. Ye

**Abstract**

Various nonlinear scenarios are given for the evolution of energetic particles that are slowing down in a plasma and simultaneously excite the background plasma waves. Depending on the relationships between the source, the background damping, and the classical transport rate, either a steady state or pulsations arise. At the predicted saturation levels, anomalous particle transport is rather low. However, if the particle orbits are stochastic at the amplitude level needed to balance the growth rate with the wave trapping frequency, a phase space ‘‘explosion’’ occurs, giving enhanced transport.

**Low frequency fluctuations in plasma magnetic fields **

S. Cable, T. Tajima

**Abstract**

It is shown that even a nonmagnetized plasma with temperature T sustains zero-frequency magnetic fluctuations in thermal equilibrium. Fluctuations in electric and magnetic fields, as well as in densities, are computed. Four cases are studied: a cold, gaseous, isotropic, nonmagnetized plasma; a cold, gaseous plasma in a uniform magnetic field; a warm, gaseous plasma described by kinetic theory; and a degenerate electron plasma. For the simple gaseous plasma, the fluctuation strength of the magnetic field as a function of frequency and wave number is calculated with the aid of the fluctuation-dissipation theorem. This calculation is done for both collisional and collisionless plasmas. The magnetic-field fluctuation spectrum of each plasma has a large zero-frequency peak. The peak is a Dirac δ function in the collisionless plasma; it is broadened into a Lorentzian curve in the collisional plasma. The plasma causes a low-frequency cutoff in the typical blackbody radiation spectrum, and the energy under the discovered peak approximates the energy lost in this cutoff. When the imposed magnetic field is weak, the magnetic-field wave-vector fluctuation spectra of the two lowest modes are independent of the strength of the imposed field. Further, these modes contain finite energy even when the imposed field is zero. It is the energy of these modes that forms the zero-frequency peak of the nonmagnetized plasma. In deriving these results, a simple relationship between the dispersion relation and the fluctuation power spectrum of electromagnetic waves is found. The warm plasma is shown, by kinetic theory, to exhibit a zero-frequency peak in its magnetic-field fluctuation spectrum as well. For the degenerate plasma, we find that electric-field fluctuations and number-density fluctuations vanish at zero frequency; however, the magnetic-field power spectrum diverges at zero frequency.

**The effect of charge-exchange on plasma flows **

P.M. Valanju, M. Calvin, R.D. Hazeltine, E. Solano

**Abstract**

A simple drift-kinetic derivation of the expressions for the poloidal ion and impurity flows in the presence of charge-exchange drag and ion-impurity collisions is presented.

**Effects of orbit squeezing on ion transport in the banana regime in tokamaks **

K.C. Shaing, R.D. Hazeltine

**Abstract**

It is shown that ion transport in the banana regime in tokamaks is reduced in the presence of a strong shear in the radial electric field Er , as is often observed in the edge region. For simplicity, the ordering with ρ_{pi}||d ln E_{r} /dr|| >> 1 but c||E_{r}||/B_{pvti} < 1 is adopted. Here, ρ_{pi} is the ion poloidal gyroradius, B_{p} is the poloidal magnetic field strength, v_{ti} is the ion thermal speed, and c is the speed of light. A kinetic transport theory similar to those for bumpy tori and stellarators is developed to show that the ion thermal conductivity χ_{i} is reduced by a factor of roughly S^{−3/2}, where S = 1 − (ρ_{pi} d ln E_{r} /dr)(cE_{r} B_{pvti}). The result reflects more than simple orbit squeezing: The fraction of trapped ions is also modified by S.

**The PLplot plotting library programmer's reference manual version 4.0**

M. Lebrun, G. Furnish, T. Richardson

**Mode structure and continuum damping of high-n toroidal Alfven eigenmodes **

M.N. Rosenbluth, H.L. Berk, J.W. Van Dam, D.M. Lindberg

**Abstract**

An asymptotic theory is described for calculating the mode structure and continuum damping of short-wavelength toroidal Alfvén eigenmodes (TAE). The formalism somewhat resembles the treatment used for describing low-frequency toroidal modes with singular structure at a rational surface, where an inner solution, which for the TAE mode has toroidal coupling, is matched to an outer toroidally uncoupled solution. A three-term recursion relation among coupled poloidal harmonic amplitudes is obtained, whose solution gives the structure of the global wave function and the complex eigenfrequency, including continuum damping. Both analytic and numerical solutions are presented. The magnitude of the damping is essential for determining the thresholds for instability driven by the spatial gradients of energetic particles (e.g., neutral-beam-injected ions or fusion-product alpha particles) contained in a tokamak plasma.

**Excitation of solitons by an external resonant wave with a slowly varying phase velocity **

I. Aranson, B. Meerson, T. Tajima

**Abstract**

A mechanism is proposed for the excitation of solitons in nonlinear dispersive media. The mechanism employs an external pumping wave with a varying phase velocity, which provides a continuous resonant excitation of a nonlinear wave in the medium. Two different schemes of a continuous resonant growth (continuous phase locking) of the induced nonlinear wave are suggested. The first of them requires a definite time dependence of the pumping-wave phase velocity and is relatively sensitive to the initial wave phase. The second employs the dynamic autoresonance effect and is insensitive to the exact time dependence of the pumping-wave phase velocity. It is demonstrated analytically and numerically, for a particular example of a driven Korteweg–de Vries (KdV) equation with periodic boundary conditions, that as the nonlinear wave grows, it transforms into a soliton, which continues growing and accelerating adiabatically. A fully nonlinear perturbation theory is developed for the driven KdV equation to follow the growing wave into the strongly nonlinear regime and describe the soliton formation.

**Magnetic surfaces in a steady-state tokamak **

R. Kinney, T. Tajima, H. Irie

**Abstract**

The effects of self-consistently interacting internal currents are modeled in a tokamak plasma with given external toroidal and poloidal magnetic fields. The unperturbed external magnetic surfaces are described through the well-known nonlinear "standard map.'' When the magnetic field is allowed to carry an internal current, the self-interaction of these currents disturbs the integrity of the magnetic surfaces. A computational study of the effects of the interacting internal current filaments measures the diffusion of field lines from the unperturbed surfaces, and finds the self-interaction to be a significant effect that always serves to increase diffusion. Perfect surfaces are not maintained even when magnetic islands would not otherwise overlap. Diffusion from the current interaction dominates when current fluctuations reach a fraction of the applied field.

**Self-consistent theory for ion gyroresonance **

H. Ye, A.N. Kaufman

**Abstract**

For describing ion gyroresonance processes, a complete set of self-consistent Vlasov–Maxwell equations is derived by systematically transforming a self-consistent action principle from particle coordinates to guiding-center/oscillation-center coordinates. They include the oscillation-center Vlasov equation; the equations for wave–particle resonant interactions; and Maxwell equations for the background electromagnetic fields. These equations satisfy local conservation laws for energy, momentum, and angular momentum, constructed by using the Noether algorithm. A heuristic interpretation based on the theory of linear mode conversion in ray phase space is also presented for ion gyroresonance processes, which suggests a method for obtaining analytic solutions in general geometry.

**On zero frequency magnetic fluctuations in plasmas **

T. Tajima, S. Cable, R.M. Kulsrud

**Abstract**

A plasma sustains fluctuations of electromagnetic fields and particle density even in thermal equilibrium and such fluctuations have a large zero-frequency peak. The level of fluctuations in the plasma for a given wavelength and frequency of electromagnetic fields is calculated through the fluctuation–dissipation theorem. The frequency spectrum shows that the energy contained in this peak is complementary to the energy "lost" by the plasma cutoff effect. The level of the zero (or nearly zero) frequency magnetic fields is computed as <B^{2}>^{0}/8π = (1/2π^{3})T(ω_{p}/c)^{3}, where T and ω_{p} are the temperature and plasma frequency. The relation between the nonradiative and radiative fluctuations is elucidated. Both a simple collision model and a kinetic theoretic treatment are presented with essentially the same results. The size of the fluctuations is λ ~ (c/ω_{p})(η/ω)^{1/2}, where η and ω are the collision frequency and the (nearly zero) frequency of magnetic fields oscillations. Perhaps the most dramatic application of the present theory, however, is to the cosmological plasma of early epoch (say t = 10^{−2}–10^{0} sec after the Big Bang). Implications of these magnetic fields in the early Universe are discussed.