**The electron Fokker-Planck equation for collisions with ions in a magnetized plasma **

A.A. Ware

**Abstract**

For those electrons whose Larmor radius is less then the Debye length, by averaging over binary collisions with ions and using a guiding-center approximation, a simple collision operator has been obtained which reproduces accurately the moments obtained from the existing more complex operators.

**Ionospheric accelerator **

T. Tajima, W. Horton, S. Nishikawa, T. Nishikawa

**Abstract**

Ionospheric acceleration of high energy particles by a short wavelength microwave pulse is discussed. The intense electromagnetic waves in an ionospheric (F2) or magnetospheric plasma can be self-trapped above a threshold power. The self-binding property and the consequent self-induced transparency of the triple soliton structure of two electromagnetic waves and a plasma wave allow the propagation of an intense electromagnetic pulse without the severe and wasteful distortion that accompanies low power pulse propagation. The effects of magnetospheric fluctuations on the particle beam transport are considered. The fluctuation-induced transport seems to be within the margin of tolerance for useful beam transport. Orbits of negatively charged particles are stable. While synchrotron radiation loss for electrons is prohibitive, that of muons and antiprotons is negligible. A corresponding terrestrial application is also suggested.

**Effect of scalar nonlinearity on the dipole vortex solution **

X. Su, W. Horton, P.J. Morrison, V.P. Pavlenko

**Abstract**

The dipole vortex solutions of the Hasegawa-Mima drift wave or equivalently, the quasi-geostrophic Rossby wave equation are shown to be split up into long-lived monopole vortices (cyclones and anticyclones) in the presence of a small scalar, i.e. KdV type, nonlinearity. The lifetime of the dipole vortex varies inversely with the strength of the scalar nonlinearity.

**Nonlinear evolution of a force-free arcade field driven by shear flow **

N. Bekki, T. Tajima, J.W. Van Dam, Z. Mikic, D.C. Barnes, D.D. Schnack

**Abstract**

The nonlinear evolution of a force-free magnetic field driven by slow shear flow on a boundary is investigated analytically and numerically. The Woltjer-Taylor theory is generalized to obtain the time-dependent magnetohydrodynamic (MHD) solution for the nonlinear force-free fields driven by shear flow. This solution agrees well with our resistive MHD simulations, which show that the force-free arcade field in a solar coronal plasma evolves quasi-statically through a series of force-free states before reconnection occurs.

**The effect of energetic trapped particles on the "ideal" internal kink mode **

Y.Z. Zhang, H.L. Berk, S.M. Mahajan

**Abstract**

The internal kink stability of a tokamak in the presence of energetic particles is studied. It is found that there exists a stable window when a finite population of energetic particles are present, and the relation between the predictions of the fishbone theory of Chen-White-Rosenbluth and the fishbone theory of Coppi-Porcelli is explained. The theory indicates why some experiments, like PDX and TFTR, are likely to see fishbone oscillations in conjunction with sawtooth modes, while other experiments can observe sawtooth suppression in presence of hot particles.

**Topics in tokamak stability and transport: Dynamic transition to second stability with auxiliary heating; Stability of global Alfven waves in an ignited plasma **

G. Fu

**Abstract**

The problem of access to the high-beta ballooning second-stability regime by means of auxiliary heating and the problem of the stability of global-shear Alfven waves in an ignited tokamak plasma are theoretically investigated. These two problems are related to the confinement of both the bulk plasma as well as the fusion-product alpha particles an dare fundamentally important to the operation of ignited tokamak plasma. First, a model that incorporates both transport and ballooning mode stability was developed in order to estimate the auxiliary heating power required for tokamak plasma to evolve in time self-consistently into a high-beta, globally self-stabilized equilibrium. The critical heating power needed for access to second stability is found to be proportional to the square root of the anomalous diffusivity induced by the ballooning instability. Next, the full effects of toroidicity are retained in a theoretical description of global-type-shear Alfven modes whose stability can be modified by the fusion-product alpha particles that will present in an ignited tokamak plasma. Toroidicity is found to induce mode coupling and to stabilize the so-called Global Alfven Eigenmodes (GAE).

**Exact and almost exact solutions to Vlasov-Maxwell system **

S.M. Mahajan

**Abstract**

Exact and almost exact solutions to the Vlasov-Maxwell systems describing a variety of plasma configurations with density, temperature and current gradients, are presented. Possible consequences of these solutions are also discussed.

**Self-similar evolution of nonlinear magnetic buoyancy instability **

K. Shibata, T. Tajima, R. Matsumoto

**Abstract**

A new type of self-similar solution of ideal magnetohydrodynamics (MHD) in the nonlinear stage of the undular model (k||B) of the magnetic buoyancy instability (the ballooning instability in fusion plasma physics or the Parker instability in astrophysics) is found through MHD simulation and theory. The linear theory developed agrees well with the authors' simulation in the early (linear) stage. The nonlinear stages of the instability in the simulation show the self-similar evolution.

**Nonlinear twist-kink instability of a coronal loop **

E. Zaidman, T. Tajima

**Abstract**

Three-dimensional magnetoinductive particle simulations are used to demonstrate that the mechanical twisting motion applied to a magnetized plasma column induces a current aligned to the external magnetic field direction, pinches the plasma and magnetic fields, and stores the energy in poloidal magnetic fields. As the twist motion continues, the field lines locally begin to wrap around the plasma more than one revolution. A strong MHD instability sets in that is a mixture of kink and ballooning modes, releasing the magnetic energy and causing destruction of coherent column structure and flows of turbulent plasma. A similar episode ensues, exhibiting relaxation oscillations. The buildup of poloidal fields and structure and its sudden release driven by the twist motion may be a model for the solar coronal loop dynamics which exhibits a slow energy buildup with some photospheric motion and a sudden energy release by flares.

**A new concept for muon catalyzed fusion reactor **

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

**Abstract**

A new concept for a muon catalyzed pure fusion reactor is considered. To our best knowledge this constitutes a first plausible configuration to make energy gain without resorting to fissile matter breeding by fusion neutrons, although a number of crucial physical and engineering questions as well as details have yet to be resolved. A bundle of DT ice ribbons (with a filling factor f) is immersed in the magnetic field. The overall magnetic field in the mirror configuration confines pions created by the injected high energy deuterium (or tritium) beam. The DT materials is long enough to be inertially confined along the axis of mirror. The muon catalyzed mesomolecule formation and nuclear fusion take place in the DT target, leaving α^{++} and occasionally (αμ)^{+} (muon sticking). The stuck muons are stripped fast enough in the target, while they are accelerated by ion cyclotron resonance heating when they circulate in the vaccum (or dilute plasma). The ribbon is (eventually) surrounded and pressure-confined by this coronal plasma, whereas the corona is magnetically confined. The overall bundle of ribbons (a pellet) is inertially confined. This configuration may also be of use for stripping stuck muons via the plasma mechanism of Menshikov and Ponomarev.

**Initial evolution of nonlinear magnetic islands in high temperature plasmas **

M. Kotschenreuther

**Abstract**

The evolution of nonlinear magnetic islands is computed in the kinetic collisionality regime called the semicollisional regime, which is appropriate to present fusion confinement devices. Realistic effects are included, such as the presence of small external field errors, radial electric fields, and omega. When present simultaneously, these effects can greatly change the stability of small amplitude nonlinear islands. Islands with Δ' > O can sometimes be prevented from growing to macroscopic size; it is also possible to produce moderate mode-number nonlinear instabilities in the plasma edge. Furthermore, island growth can be prevented by application of external fields with suitably chosen amplitude and frequency.

**US-Japan workshop of plasma & fluids turbulence **

W. Horton

**Exact area devil's staircase for the sawtooth map **

Q. Chen, J.D. Meiss

**Abstract**

The sawtooth mapping is a family of uniformly hyperbolic, piecewise linear, area-preserving maps on the cylinder. We construct the resonances, cantori, and turnstiles of this family and derive exact formulas for the resonance areas and the escaping fluxes. These are of prime interst for an understanding of the deterministic transport which occurs the stochastic regime. The resonances are shown to fill the full measure of phase space.

**A study of runaway electron confinement in the Asdex tokamak **

O.J. Kwon, P.H. Diamond

**Abstract**

he results of runaway electron confinement experiments from ASDEX are analyzed to elucidate the structure of electromagnetic turbulence that causes anomalous electron heat transport in the L-mode confinement regime. From a simple model, the radial correlation length (W) of the magnetic turbulence is determined to be about 1 mm. Using this value and that of the experimentally deduced electron thermal diffusivity, we determine the radial magnetic fluctuation level at the plasma edge in the L-mode to be (B_{r}/B_{0}) ≈ ← 2 x 10^{-4}. Scalings of W and B_{r}/B_{0} are deduced from parameter scans. From a comparison of these results with the predictions of various theoretical models, it is concluded that skin-depth turbulence, electromagnetic drift wave turbulence, rippling modes, and microtearing modes are inferior candidates and that resistive-ballooning modes offer the best possibility for a consistent interpretation of the data.

**Drift wave simulation in toroidal geometry **

M.J. Lebrun

**Abstract**

The drift wave, a general category of plasma behavior arising from a plasma inhomogeneity, is studied using the particle simulation method. In slab geometry, the drift wave (or universal mode) is stabilized by any finite amount of magnetic shear. In toroidal geometry, however, the coupling of the poloidal harmonics gives rise to a new branch of drift wave eigenmodes called the toroidicity -induced mode, which is predicted to be unstable in some regimes. The drift wave in a toroidal system is intrinsically three-dimensional, and is sensitive to the handling of the parallel electron dynamics, the (nearly) perpendicular wave dynamics, and the radial variation of magnetic field vector (shear). A simulation study must therefore be kinetic in nature, motivating the extension of particle simulation techniques to complex geometries. From this effort a three dimensional particle code in a toroidal coordinate system has been developed and applied to the toroidal drift wave problem. The code uses an (r, θ, φ) -type coordinate system, and a nonuniform radial grid that increases resolution near the mode-rational surfaces. Full ion dynamics and electron guiding center dynamics are employed. Further, the algorithm incorporates a straightforward limiting process to cylindrical geometry and slab geometry, enabling comparison to the theoretical results in these regimes. Simulations of the density-driven modes in toroidal geometry retain a single toroidal mode number (n = 9). In this regime, the poloidal harmonics are expected to be strongly coupled, giving rise to the marginally unstable toroidicity-induced drift mode. Analysis of the simulation data reveals a strong, low-frequency response that peaks near each mode rational surface. Further, the characteristic oscillation frequencies persist from one mode rational surface to the next, which identifies them as multiple harmonics of the toroidicity-induced mode. The lowest harmonic occurs at a frequency of ω/ω^{*} ~ 0.26, which is reasonably close to the prediction of linear theory. Interferogram analysis of these modes indicates a "ballooning" structure toward the outside of the torus. The amplitude of the potential is observed to grow exponentially for the m = 8 through m = 10 poloidal mode numbers, with a growth rate of approximately γ/ω ^{*} ~ 0.075. Saturation occurs at time t ~ 1000 Ω_{i}^{-1}, and may be caused by quasilinear flattening of the density profile.

**Fusion ignition experiment **

R. Carrera, E. Montalvo, M. Rosenbluth

**Abstract**

A fusion ignition experiment (IGNITEX) is described, the original concept for which was proposed by Rosenbluth, Weldon, and Woodson. In this concept, a single-turn-coil tokamak device produces a self-sustained fusion reaction. The basic idea is to employ a very high magnetic field and a very high plasma current to heat the plasma ohmically to thermonuclear temperatures and then to produce a stable ignited plasma. The experiment will permit the scientific study of a new regime of physics: alpha-heated plasmas.

**Free energy expressions for Vlasov equilibria **

P. Morrison, D. Pfirsch

**Abstract**

Hamiltonian or variational formulations of the Maxwell-Vlasov equation naturally yield expressions for the free energy available upon perturbation of an equilibrium. The noncanonical Hamiltonian, Hamilton-Jacobi, and Lagrangian formulations are used to obtain such expressions. It is concluded that all interesting equilibria are either linearly unstable or possess negative-energy modes.

**Drift waves vortices and anomalous transport **

W. Horton

**Abstract**

Theory and computer simulations are used to describe the inelastic vortex–vortex and vortex–wave interactions that lead to the quasicoherent transport of plasma across a constant magnetic field in 2-D systems. Monopole and dipole drift wave vortices with radii r_{0} large compared with the ion inertial scale length ρ_{s} are shown to produce transport at the rate un ∫ dσ(b) ≤ n_{v}v_{de}r_{0}, where n_{v} is the vortex line density and dσ(b) is the inelastic collision cross section for the impact parameter b. The transport during collisions and mergings is evaluated from the evolution of a passively convected scalar concentration of test particles.

**Some aspects of plasma kinetic theory: Energetic particle physics; Renormalization theory; Anomalous electron transport in tokamaks **

Y.Z. Zhang

**Abstract**

This dissertation discusses three special topics of collisionless plasma kinetic theory and their applications pertinent to high-temperature magnetic confinement experiments oriented to nuclear fusion research. The first special topic is the energetic particle physics. Appropriate equations are derived from first principles and applied to mirror configurations. A significant contribution is the prediction of a diamagnetic limit due to negative energy precessional waves. Discussion is given on whether experiments exist that violate this stability limit. The major theme of the second special topic is the construction of a set of rigorous formal systematology for nonlinear Vlasov equation with electromagnetic interactions in order to produce a renormalized perturbation theory to arbitrary higher order. This systematology makes it possible to clarify some ambiguities of previous theories as well as to improve a few aspects of calculational techniques. The last topic of the dissertation is the theoretical study of anomalous electron energy transport in tokamak. In combination with phenomenological analysis, a formula for local electron energy transport coefficient is proposed on the theoretical basis for strong microturbulence on the scale of the electromagnetic skin depth. The transport consequences of this formula are found in good agreement with a variety of tokamak experiments. Attempts have also been made to derive such a formula from first principle to clarify the underlying mechanism. Thus far these approaches have not proved to be successful.

**Wave energy flow conservation for propagation in inhomogeneous Vlasov-Maxwell equilibria **

H.J. Kull, H.L. Berk, P.J. Morrison

**Abstract**

Wave energy flow conservation is demonstrated for Hermitian differential operators that arise in the Vlasov-Maxwell theory for propagation perpendicular to a magnetic field. The energy flow can be related to the bilinear concomitant, for a solution and its complex conjugate, by using the Lagrange identity of the operator. This bilinear form obeys a conservation law and is shown to describe the usual WKB energy flow for asymptotically homogeneous regions. The additivity and uniqueness of the energy flow expression is discussed for a general superposition of waves with real and complex wave numbers. Furthermore, a global energy conservation theorem is demonstrated for an inhomogeneity in one-dimension and generalized reflection and transmission coefficients are thereby obtained.

**Radial fluctuation scale of ion temperature gradient driven turbulence **

P.W. Terry, J.N. Leboeuf, P.H. Diamond, D.R. Thayer, J.E. Sedlak, G.S. Lee

**Abstract**

It is shown that ion temperature gradient driven turbulence supports fluctuations with broader radial scales than those inferred previously on the basis of mixing length arguments derived from the eigenfunction with lowest radial eigenmode number. These fluctuations of greater radial extent are more strongly excited and are shown to result in fluctuation levels and transport coefficients that can be considerably larger than those previously predicted. It is expected that transport of this nature would maintain ion temperature and density profiles near marginal stability in neutral beam heated and ion cyclotron resonance heated plasmas.

**WKB theory of wave tunneling for Hermitian vector systems of integral equations **

H.J. Kull, R.J. Kashuba, H.L. Berk

**Abstract**

A general theory of wave tunneling in one dimension for Hermitian and nearly Hermitian vector systems of integral equations is presented. It describes mode conversion in terms of the general dielectric tensor of the medium and properly accounts for the forward and backward nature of the waves without regard to specific models. Energy conservation in the WKB approximation can be obtained for general Hermitian systems by the use of modified Furry rules that are similar to those used by Heading for second-order differential equations. Wave energy absorption can then be calculated perturbatively using the conservation properties of the dominant Hermitian operator. Operational graphical rules are developed to construct global wave solutions and to determine the direction of energy flow for spatially disconnected roots. In principle, these rules could be applied to systems with arbitrary mode complexity. Coupling coefficients for wave tunneling problems with up to four interacting modes are calculated explicitly.

**Interaction of a strong vortex with decaying turbulence **

P.W. Terry

**Abstract**

The evolution of a localized, axially symmetric vortex under the action of shear stresses associated with decaying two-dimensional turbulent vorticity which is inhomogeneous in the presence of the vortex is studied analytically. For a vortex which is sufficiently strong relative to the coefficient of turbulent eddy viscosity, it is shown that turbulent fluctuations in the vortex interior and diffusion of coherent vorticity by the turbulence localize to the vortex periphery. It is also found that the coefficient of diffusion is small compared to the coefficient of eddy viscosity.

**Toroidal electron temperature gradient driven drift modes **

W. Horton, B.G. Hong, W.M. Tang

**Abstract**

The electron temperature gradient in tokamak geometry is shown to drive a short wavelength lower hybrid drift wave turbulence resulting from the unfavorable magnetic curvature on the outside of the torus. Ballooning mode theory is used to determine the stability regimes and the complex eigenfrequencies. At wavelengths of the order of the electron gyroradius, the polarization is electrostatic and the growth rate is greater than the electron transit time around the torus. At longer wavelengths of the order of the collisionless skin depth, the polarization is electromagnetic with electromagnetic vortices producing the dominant transport. The small scale electrostatic component of the turbulence produces a small, of order (m_{e}/m_{i})^{1/2}, drift wave anomalous transport of both the trapped and passing electrons while the c/ω_{pe} scale turbulence produces a neo-Alcator (Nucl. Fusion 25, 1127 (1985)) type transport from the stochastic diffusion of the trapped electrons.