**Quasilinear evolution of tearing modes during magnetic reconnection **

W. Horton, T. Tajima, R.M. Galvao

**Abstract**

Particle simulations of magnetic reconnection show that the presence of a magnetic field parallel to the current sheet changes the character of reconnection from a laminar process to a weakly turbulent one. An analysis of the quasi-linear spectrum of tearing modes is presented which may account for some features of the 2-1/2D simulations. The turbulent incompressible plasma flow carrying reversed magnetic flux to the tearing layer is shown to produce a positive definite anomalous resistivity. The relationship to the negative anomalous resistivity of Biskamp and Welter (1983) is discussed.

**Theory of dissipative density gradient driven turbulence in the tokamak edge **

P.H. Diamond, P.W. Terry

**Abstract**

A theory of resistive, density-gradient-driven turbulence is presented and compared with tokamak edge fluctuation measurements. In addition to linear driving, the theory accounts for relaxation of the density gradient through a nonlinear process associated with emission from localized density fluctuation elements. From a fluid model for isothermal electrons in toroidal geometry, equations are obtained and solved analytically, retaining both coherent and incoherent contributions. The effect of collisions on the density blobs is treated. A Reynolds number parameterizes the magnitude of the turbulent scattering relative to the collisional viscous diffusion. The analytic results indicate that the spectrum is characterized by linewidths which increase as a function of the Reynolds number and may reach Δω/ω ≥ 1. Energy lies predominantly in the small wavenumbers (k_{⊥}ρ_{s}~0.1). For larger wavenumbers and frequency, the spectrum decays as k−^{17/6} and ω^{−2}. The fluctuation level scales as 1/k_{⊥}L_{n} and may reach −30% for parameters typical of the pretext edge. Particle diffusion is Bohm-like in magnitude but does not follow Bohm scaling, going instead as n^{2/3}T_{e}^{1/6}. The density fluctuations exhibit nonadiabatic character caused by the incoherent mode coupling.

**Kinetic theory of resistive ballooning modes **

P.H. Diamond, P.L. Similon, T. Hender, B.A. Carreras

**Abstract**

A linear and nonlinear kinetic theory of resistive ballooning modes that includes diamagnetic drifts and finite Larmor radius effects is presented. The linear stability of resistive ballooning modes is examined analytically and numerically. A renormalized resistive ballooning equation is derived, and the saturation level of the instabilities is analytically calculated. Finally, a calculation of the electron thermal conductivity for the large omega regime is presented.

**Computer modeling of fast collisionless reconnection **

J.N. Leboeuf, T. Tajima, F. Brunel, J. Sakai, C. Wu, J.M. Dawson

**Abstract**

Particle simulations of collisionless tearing, reconnection and coalescence of magnetic fields for a sheet pinch configuration show that reconnection is Sweet-Parker like in the tearing and island formation phase. It is much faster to explosive in the island coalescence state. Island coalescence is the most energetic process and leads to large ion temperature increase and oscillations in the merged state. Similar phenomena have been observed in equivalent MHD simulations. Coalescence and its effects, as observed in simulations, may explain many of the features of solar flares and coronal X-ray brightening.

**Drift waves in rotating plasmas **

J. Liu, W. Horton

**Abstract**

The stability of the electron drift wave is investigated in the presence of E×B plasma rotation typical of the central cell plasma in tandem mirrors. It is shown that a rotationally driven drift wave may occur at low azimuthal mode numbers. Conditions for rotational instabilities are derived. Quasilinear formulas are given for the anomalous transport associated with the unstable fluctuations.

**Beat-wave current drive in a magnetized plasma **

R.M. Galvao, T. Tajima

**Abstract**

Large currents are efficiency driven in magnetically confined plasmas by the ponderomotive force created by the heating of two electromagnetic waves. The heating waves are cyclotron waves propagating parallel to the magnetic field or light waves propagating obliquely to the magnetic field.

**Transport in Hamiltonian systems **

R. Mackay, J.D. Meiss, I. Percival

**Abstract**

We develop a theory of transport in Hamiltonian systems in the context of iteration of area preserving maps. Invariant closed curves present complete barriers to transport, but in regions without such curves there are still invariant Canton sets named cantori, which appear to form partial barriers. The flux through the gaps of the cantori is given by Mather’s differences in action. This gives useful bounds on transport between regions, and for one parameter families of maps it provides a universal scaling law when a curve has just broken. The bounds and scaling law both agree well with numerical experiments of Chirikov and help to explain an apparent disagreement with results of Greene. By dividing the largest irregular component of phase space into regions separated by the strongest partial barriers, and assuming that the motion is mixing within these regions, we present a global picture of transport, and indicate how it can be used, for example, to predict confinement times and to explain longtime tails in the decay of correlations.

**The effect of compressibility on the resistive ballooning mode **

T. Hender, B.A. Carreras, W. Cooper, J. Holmes, P.H. Diamond

**Abstract**

The linear stability of the resistive ballooning mode, as described by the resistive magnetohydrodynamic (MHD) model, is investigated both analytically and numerically. When the pressure evolution is approximated by fluid convection (reduced MHD model), an instability driven by geodesic curvature, with a growth rate γ~η^{1/3}β_{p}^{2/3}, is found. For conditions relevant to the Impurity Study Experiment (ISX-B), it is shown that for modest poloidal bets (β_{p}≈1), high current, and relatively high temperatures, compressibility has a significant stabilizing influence, relative to the pressure convection model. However, at high β_{p} (≥2), low current, and lower temperatures, compressibility has much less effect.

**Elimination of stochasticity in stellarators **

J.D. Hanson, J.R. Cary

**Abstract**

A method for optimizing stellarator vacuum magnetic fields is discussed. Application of this method shows that the stochasticity of vacuum magnetic fields can be made negligible by proper choice of the coil configuration. It is shown that the optimization increases the equilibrium Α-limit by factors of two or more over that of the simple, straight coil winding law. The method is general and is applicable to other systems in which stochasticity: (1) is a problem; yet (2) is affected by the design parameters.

**Stochasticity and transport in Hamiltonian systems **

R. Mackay, J.D. Meiss

**Abstract**

The theory of transport in nonlinear dynamics is developed in terms of "leaky" barriers which remain when invariant tori are destroyed. A critical exponent for transport times across destroyed tori is obtained which explains numerical results of Chirikov. The combined effects of many destroyed tori lead to power-law decay of correlations observed in many computations.

**Nonlinear interaction of toroidicity induced drift waves **

P.L. Similon, P.H. Diamond

**Abstract**

Drift modes in toroidal geometry are destabilized by trapped electron inverse dissipation and evolve to a nonlinearly saturated state. Using renormalized one-point turbulence theory for the nonlinear gyrokinetic equation in the ballooning representation, it is shown that ion Compton scattering is an effective saturation mechanism. Ion Compton scattering transfers wave energy from short to long perpendicular wavelength, where it is absorbed by ion resonance with extended, linearly stable, long-wavelength modes. The fluctuation spectrum and fluctuation levels are calculated using the condition of nonlinear saturation. Transport coefficients and energy confinement time scalings are determined for several regimes. Specifically, the predicted confinement time density scaling for an Ohmically heated discharge increases from n^{3/8} in the collisionless regime to n^{9/8} in the dissipative trapped electron regime.

**Linear relativistic gyrokinetic equation in general magnetically confined plasmas **

S. Tsai, J.W. Van Dam, L. Chen

**Abstract**

he gyrokinetic formalism for linear electromagnetic waves of arbitrary frequency in general magnetic field configurations is extended to include full relativistic effects. The derivation employs the small adiabaticity parameter ρ/L_{0} where ρ is the Larmor radius and L_{0} the equilibrium scale length. The effects of the plasma and magnetic field inhomogeneities and finite Larmor radii effects are also contained.

**Nonlinear kinetic theory of a single helicity tearing instability **

K. Swartz, R.D. Hazeltine

**Abstract**

The evolution of a single helicity tearing mode is analyzed by kinetic theory, including modification of the electron orbits by the islands themselves. Using flux coordinates appropriate to island geometry, analytic solutions for the time-dependent island width are obtained. These describe evolution from linear growth to nonlinear saturation in the collisionless limit, and transition from such saturation to slow Rutherford-type growth for increasing finite collisionality. Previous work is thus unified and generalized.

**Sustained self-reversal in the reversed field pinch **

A.Y. Aydemir, D.C. Barnes

**Abstract**

Spontaneous reversal of the toroidal field in a reversed-field pinch as a result of low-β (small J_{⊥} / J_{||}) resisitive kink mode activity is investigated with use of a three-dimensional magnetohydrodynamics code. Helical and three-dimensional steady reversed states are obtained. In three dimensions quasisteady fluctuating states are observed above a critical value of the pinch parameter θ.

**Influence of finite wavelengths on the quantum kicked rotator in the semi-classical regime **

J.D. Hanson, E. Ott, T.M. Antonsen

**Abstract**

The quantum kicked rotator, the classical limit of which is described by the standard map, is considered. Particular attention is devoted to a study of the effect of finite quantum wavelength on the detailed structure of phase space which appears in the classical limit. In the classical case, for large values of the nonlinearity parameter, most of the trajectories are ergodic. However, in addition to these ergodic trajectories, there can be small integrable regions of phase space, known as accelerator modes, which dominate the long-time evolution of the expected value of the particle energy. In this paper it is shown that this behavior is modified in the quantum case for small but finite values of the wavelength (i.e., Planck's constant). A simple model is presented to explain this modification. Based on our results, it is speculated that certain problems in the application of statistical concepts to intrinsically stochastic problems of classical mechanics may, in some cases, be mitigated by quantum effects.

**Institute for fusion studies progress report for the period 1 September 1982 to 31 August 1983 **

R.D. Hazeltine

**Collective ion acceleration by a reflexing electron beam: Model and scaling **

F. Mako, T. Tajima

**Abstract**

Analytical and numerical calculations are presented for a reflexing electron beam type of collective ion accelerator. These results are then compared to those obtained through experiment. By constraining one free parameter to experimental conditions, the self-similar solution of the ion energy distribution agrees closely with the experimental distribution. Hence the reflexing beam model appears to be a valid model for explaining the experimental data. Simulation shows in addition to the agreement with the experimental ion distribution that synchronization between accelerated ions and electric field is phase unstable. This instability seems to further restrict the maximum ion energy to several times the electron energy.

**Energetic particle stability of ballooning modes in tokamaks **

M.N. Rosenbluth, S. Tsai, J.W. Van Dam, M.G. Engquist

**Abstract**

Introduction of an anisctropic, highly energetic trapped-particle species into a tokamak may allow direct stable access to the high-beta regime of second stability. Under certain conditions, the mode at marginal stability acquires a real frequency close to the precessional drift frequency of the energetic particles, perhaps correlating with recent "fishbone" observations on PDX.

**Hamiltonian description of reduced MHD **

R.D. Hazeltine, P.J. Morrison

**Abstract**

Reduced magnetohydrodynamics (RMHD) has become a principal tool for understanding nonlinear processes, including disruptions, in tokamak plasmas. Although analytical studies of RMHD turbulence have been useful, the model’s impressive ability to simulate tokamak fluid behavior has been revealed primarily by numerical solution. The present work describes a new analytical approach, not restricted to turbulent regimes, based on Hamiltonian field theory. It is shown that the nonlinear (ideal) RMHD system, in both its high-beta and low-beta versions, can be expressed in Hamiltonian form. Thus a Poisson bracket, { , }, is constructed such that each RMHD field quantity, ξ_{i} , evolves according to ξ_{i} = {ξ_{i}, H} , where H is the total field energy. The new formulation makes RMHD accessible to the methodology of Hamiltonian mechanics; it has led, in particular, to the recognition of new RMHD invariants and even exact, nonlinear RMHD solutions. A canonical version of the Poisson bracket, which requires the introduction of additional fields, leads to a nonlinear variational principle for time-dependent RMHD.

**The heat conduction process in tokamak hot ion plasmas **

A.A. Ware

**Abstract**

The two-component ion distribution observed with active charge-exchange measurements on PDX are explained using the Fokker-Planck drift-kinetic equation and assuming ion self collisions are dominant for energy scattering. The energetic tail of the distribution, which is diffusing outwards in radius and down in energy, must retain an approximately constant effective temperature T_{H} = (-δlnf_{i}/m δε)^{ -1}. The discontinuity in the slope of ln f_{i} is shown to be the boundary between the inward and outward diffusion parts of f_{i} and is a form of contact discontinuity. Energy-scattering collisions with electrons or circulating-beam ions, when important, modify the constancy of T_{H}.

**Construction of three-dimensional vacuum magnetic fields with dense nested flux surfaces **

J.R. Cary

**Abstract**

Toroidal vacuum magnetic fields are analyzed by the surface of section technique and noncanonical perturbation theory. The surface of section analysis shows that it is easy to find magnetic fields with an outermost flux surface of relatively small aspect ratio. However, as the aspect ratio is decreased, so is the rotational transform of the outermost flux surface. A new method of averaging combined with noncanonical perturbation theory shows that island overlap can account roughly for this loss of rotational transforms. Finally, it is shown that one can significantly decrease the stochasticity and island structure by making small modifications to the magnetic field.

**Loop coalescence in flares and coronal x-ray brightening **

T. Tajima, F. Brunel, J. Sakai

**Abstract**

In connection with recent direct observations of interconnecting coronal loops, loop coalescence is being considered as a possibly important process for solar flares and coronal X-ray brightening phenomena. In the present investigation, it is proposed that the nonlinear development of the coalescence instability of current loops provides a coherent explanation of such observations and a phenomenon reported by Forrest et al. (1982), who observed amplitude oscillations in gamma-ray emission from the impulsive phase of a solar flare. It is pointed out that the present theory also offers a quantitative and natural explanation of such known characteristics as the impulsive nature of flares, the time scale of the impulsive phase, intense heating by flares, and formation of the high-energy tail of particle distribution. A plasma configuration which is unstable against the tearing and subsequent coalescence instabilities is studied.

**Magnetic reconnection driven by the coalescence instability **

A. Bhattacharjee, F. Brunel, T. Tajima

**Abstract**

A detailed study of magnetic reconnection at an x-point driven by the coalescence instability is presented. In particular, the effects of plasma compressibility and the magnitude of the toroidal field on the rate of reconnection were explored. For large toroidal fields, the plasma is almost incompressible and the destruction of magnetic flux proceeds linearly with time. However, when the attractive force between two neighboring islands is strong, and when the poloidal and the toroidal fields are comparable in magnitude, the reconnection rate is found to be faster. Implications for the steady-state models of magnetic reconnection are discussed.

**Statistical description of drift wave turbulence **

W. Horton

**Abstract**

Dissipative drift wave fluctuations are studied with the Terry-Horton nonlinear drift wave model. The kω spectral characteristics of the fluctuations are parameterized in terms of the nonlinear frequency ω_{k} and line-width v_{k} from computer simulations and the renormalized wave-kinetic equation. The probability distributions of the fluctuations are analyzed to assess the validity of the quasi-normal approximation made in the closure of the hierarchy of correlations in statistical turbulence theory.

**Impact of clumps on plasma stability and the nature of turbulence in a saturated state **

P.W. Terry, P.H. Diamond

**Abstract**

Phase-space density granulations (clumps) are studied using the theory of two point phase space density correlation. A mechanism of extraction of expansion free energy is described. This mechanism affects questions pertaining to nonlinear stability. Theories for two point correlation for the universal mode in a slab geometry with shear and trapped electrons in toroidal geometry are discussed. Results are presented which show destabilization of the universal mode and enhancement of the trapped electron growth rate. An analytic formula for the width of the frequency spectrum is obtained. By specifying the collective resonance damping mechanism, the wavenumber spectrum is also calculated. A formula or energy flux illustrates the impact of clumps on transport and energy confinement.

**Non-Maxwellian ion distributions caused by neoclassical heat conduction **

A.A. Ware

**Abstract**

It is shown that in tokamaks with neoclassical banana-regime heat conduction and ion self-collisions dominant for energy scattering, the distribution tail diffuses outwards radially and downwards in energy maintaining a constant effective temperature (-∂lnf_{i} / ∂ε)^{-1}, as observed in some experiments.

**Searching for integrable systems **

J.R. Cary

**Reduced magnetohydrodynamics and the Hasegawa-Mima equation**

R.D. Hazeltine

**Abstract**

Reduced magnetohydrodynamics consists of a set of simplified fluid equations which has become a principal tool in the interpretation of plasma fluid motions in tokamak experiments. The Hasegawa--Mima equation is applied to the study of electrostatic fluctuations in turbulent plasmas. The relations between these two nonlinear models is elucidated. It is shown that both models can be obtained from appropriate limits of a third, inclusive, nonlinear system. The inclusive system is remarkably simple.

**Ballooning mode calculations in stellarators **

H.L. Berk, M.N. Rosenbluth, J.L. Shohet

**Abstract**

Ballooning equations that are readily integrated by a field line following code are written. The asymptotic properties of the ballooning mode equation are analyzed to rederive the Mercier condition in a form particularly well suited for our computations. The low beta stability properties for three typical configurations, Proto-Cleo, Wistor U and a Heliac where the magnetic fields are due to actual vacuum plus the currents arising from a localized pressure gradient are evaluated.

**Three-dimensional structure of ICRF waves in tokamak plasmas **

S.I. Itoh, K. Itoh

**Abstract**

The three-dimensional structure of the ICRF (Ion Cyclotron Range of Frequencies) waves in a tokamak plasma is studied. The toroidal field gradient, radial density inhomogeneity and the poloidal-toroidal localizations of the antenna current are incorporated. The ICRF wave propagation and absorption in the two-ion species plasma (majority deuterion and minority hydrogen) are obtained by numerically solving the propagation equations in collisional cold plasmas. Two cases of the heating mechanisms, the ion-ion hybrid resonance and the cavity resonance (i.e., the forced excitation of damped eigenmode) are found in a three-dimensional configuration. Owing to the lack of homogeneity in the up-down direction, the global wave form, energy deposition profile and total energy absorption are affected, and considerable differences from the two-dimensional calculations are found. In the case of the ion-ion hybrid resonance, the energy deposition profile is localized near the hybrid resonance surface; the energy absorption integrated over the plasma column is independent of the damping rate. On the contrary, the cavity resonance realizes a strong heating over the whole plasma column by a coherent wave, which satisfies the cavity resonance condition. This heating occurs in the absence of the cyclotron and the hybrid resonances. When we calculate the loading impedance by the poloidal current antenna with Faraday shield, we find that the cavity resonance can contribute to the wave absorption as much as (or more than) the ion-ion hybrid resonance for wide range of plasma parameters.

**Variational methods for the three-dimensional inverse equilibrium problem in toroids **

A. Bhattacharjee

**Abstract**

A variational method is developed for three-dimensional magnetostatic equilibria in toroids. We represent equilibria in cylindrical inverse variables R(v, θ, ζ), φ (v, θ, ζ), and Z (v, θ, ζ), where v is a radial flux surface label, θ, a poloidal angle, and ζ, a toroidal angle. We Fourier-expand in θ and ζ and derive, from the variational principle, a set of ordinary differential equations for the amplitudes in v. Truncation of the infinite Fourier series leads to a reduced set of equations which we solve numerically by collocation to obtain two- and three-dimensional toroidal equilibria.

**Kinetic theory of Alfven waves **

S.M. Mahajan

**Abstract**

The addition of electron parallel dynamics in the description of an inhomogeneous current-carrying cylindrical plasma is shown to replace the magnetohydrodynamic continuum, associated with Alfven waves, by a discrete spectrum. The spectrum of the resulting discrete modes is determined analytically and numerically.

**Non-intrinsic ambipolar diffusion in turbulence theory **

H.L. Berk, K. Molvig

**Abstract**

Ambipolar flow in a turbulent plasma is investigated by combining a WKB treatment of the waves with a turbulent collision operator resulting from either quasi-linear theory or certain renormalized turbulence theories. If the wave momentum has a flow from outgoing waves, then particle diffusion is not intrinsically ambipolar, and the time variation of the electric potential profile is determined by the turbulent spectrum. However, in most cases of practical interest, as in the drift wave problem, this effect is small; and, in steady state, equal rates of stochastic diffusion are predicted for electrons and ions.

**Solitons in turbulent flow **

J.D. Meiss

**Scattering of regularized-long-wave solitary waves **

P.J. Morrison, J.D. Meiss, J.R. Cary

**Abstract**

The Lagrangian density for the regularized-long-wave equation (also known as the BBM equation) is presented. Using the trial function technique, ordinary differential equations that describe the time dependence of the position of the peaks, amplitudes, and widths for the collision of two solitary waves are obtained. These equations are analyzed in the Born and "equal-width'' approximations and compared with numerical results obtained by direct integration utilizing the split-step fast Fourier-transform method. The computations show that collisions are inelastic and that production of solitary waves may occur.