**Transport near the onset of stochasticity **

J.D. Meiss

**Abstract**

For two-degree-of-freedom Hamiltonians, (e.g., a particle in a 2-D potential or the flow of magnetic field lines) an invariant torus in phase space acts as an absolute barrier for trajectories. When an invariant torus is destroyed by a perturbation, a remnant remains with gaps. This ''cantorus'' forms a formidable barrier even well into the stochastic regime. We show that correlation functions decay algebraically invalidating the common assumptions of chaos. The decay rate is given by a universal exponent, obtained from self-similar scaling.

**Ambipolarons: Solitary wave solutions for the radial electric field in a plasma **

D.E. Hastings, R.D. Hazeltine, P.J. Morrison

**Abstract**

The ambipolar radial electric field in a nonaxisymmetric plasma can be described by a nonlinear diffusion equation. This equation is shown to possess solitary wave solutions. A model nonlinear diffusion equation with a cubic nonlinearity is studied. An explicit analytic step-like form for the solitary wave is found. It is shown that the solitary wave solutions are linearly stable against all but translational perturbations. Collisions of these solitary waves are studied and three possible final states are found: two diverging solitary waves, two stationary solitary waves, or two converging solitary waves leading to annihilation.

**Current interruption by density depression **

J.S. Wagner, T. Tajima, S.I. Akasofu

**Abstract**

Using a one-dimensional electrostatic particle code, we examine processes associated with current interruption in a collisionless plasma when a density depression is present along the current channel. Current interruption due to double layers was suggested by Alfvén and Carlqvist (1967) as a cause of solar flares. At a local density depression, plasma instabilities caused by an electron current flow are accentuated, leading to current disruption. Our simulation study encompasses a wide range of the parameters in such a way that under appropriate conditions, both the Alfvén and Carlqvist (1967) regime and the Smith and Priest (1972) regime take place. In the latter regime the density depression decays into a stationary structure (lsquoion-acoustic layerrsquo) which spawns a series of ion-acoustic lsquosolitonsrsquo and ion phase space holes travelling upstream. A large inductance of the current circuit tends to enhance the plasma instabilities.

**Fluid model of explosive coalescence **

T. Tajima, J.I. Sakai

**Abstract**

An explosive reconnection process associated with the nonlinear evolution of the coalescence instability is found through studies of the electromagnetic particle simulation and the magnetohydrodynamic particle simulation. The explosive coalescence is a process of magnetic collapse, in which we find the magnetic and electrostatic field energies and temperatures (ion temperature in the coalescing direction, in particular) explode toward the explosion time t_{0} as (t_{0} - t)^{-8/3}, (t_{0} - t)^{-4}, and (t_{0} - t)^{-8/3}, respectively for a canonical case. Single-peak, double-peak, and triple-peak structures of magnetic energy, temperature, and electrostatic energy, respectively, are observed on the simulation as overshoot amplitude oscillations and are theoretically explained. The heuristic model of Brunel and Tajima is extended to this explosive coalescence in order to extract the basic process. Since the explosive coalescence exhibits self-similarity, a temporal universality, we theoretically search for a self-similar solution to the two-fluid plasma equations.

**Alfven wave continuum with toroidicity **

S. Riyopoulous, S.M. Mahajan

**Abstract**

The symmetry property of the magnetohydrodynamic (MHD) wave propagation operator is utilized to express the toroidal eigenmodes as a superposition of the mutually orthogonal cylindrical modes. Because of the degeneracy among cylindrical modes with the same frequency but resonant surfaces of different helicity, the toroidal perturbation produces a zeroth-order mixing of the above modes. The toroidal eigenmodes of frequency S0 have multiple resonant surfaces, with each surface shifted relative to its cylindrical position and carrying a multispectral content. Thus a single helicity toroidal antenna of frequency 0 couples strongly to all different helicity resonant surfaces with matching local Alfven frequency. Zeroth-order coupling between modes in the continuum and global Alfven modes also results from toroidicity and degeneracy. The perturbation technique used in this study is the MHD counterpart of the quantum-mechanical methods and is applicable through the entire range of the MHD spectrum.

**Cyclotron resonance heating induced diffusion **

S. Riyopoulos, T. Tajima, T. Hatori, D. Pfirsch

**Abstract**

The wave induced particle transport during the ion cyclotron resonance heating is studied in collisionless toroidal plasmas. It is shown that the previously neglected non-conservation of the toroidal angular momentum IP sub phi caused by the toroidal wave component E sub phi is necessary to allow particle diffusion and yields the leading diffusive contribution. While the induced ion transport for the RF power in contemporary experiments is of the order of the neoclassical value, that of fast alpha particles is quite large if resonance is present.

**Fine structure of the genetic code **

S.M. Mahajan

**Abstract**

Simple physical principles force the removal of degeneracy in the genetic code, giving it a fine structure; important biological consequences will probably follow.

**Proceedings of US-Japan workshop on magnetic reconnection **

P.H. Diamond

**Semicollisional drift-tearing modes in toroidal plasmas **

T.S. Hahm, L. Chen

**Abstract**

Semicollisional drift-tearing modes are studied analytically in toroidal plasmas. Corresponding differential equations for the eigenmodes are derived via the ballooning mode representation and flux-surface averaging. A dispersion relation is then obtained using asymptotic matching. It is found that the stabilizing effects of good average curvature and finite plasma compression lead to a tearing instability threshold δ_{c}, which is independent of the resistivity.

**Impurity gradient driven turbulence and particle confinement in tokamaks **

T.S. Hahm, P.H. Diamond, P.W. Terry, L. Garcia, B.A. Carreras

**Abstract**

The role of impurity dynamics in resistivity gradient driven turbulence is investigated in the context of modeling tokamak edge plasma phenomena. The effects of impurity concentration fluctuations and gradients on the linear behavior of rippling instabilities and on the nonlinear evolution and saturation of resistivity gradient driven turbulence are studied both analytically and computationally. At saturation, fluctuation levels and paraticle and thermal diffusivities are calculated. In particular, the mean-square turbulent radial velocity is given by <ν_{r}^{2}> = (E_{0}L_{s}/B_{z})^{2}(L_{η}^{-1} + L_{z}^{-1})^{2}. Thus, edged peaked impurity concentrations tend to enhance the turbulence, while axially peaked concentrations tend to quench it. The theoretical predictions are in semi-quantitative agreement with experimental results from the TEXT [Bull. Am. Phys. Soc. **30**, 1443 (1985)]. Caltech [Phys. Fluids **29**, 309 (1986)], and Tosca [the 12*th *European Conference on Controlled Fusion and Plasma Physics Vol. I, p.311 (European Physical Society, Budapest, 1985)] tokamaks. Finally, a theory of the density clamp observed during CO-NBI on the ISX-B tokamak [Plasma Physics and Controlled Nuclear Fusion Research, Vol. I, p. 377 (JAEA, Vienna, 1981)] is proposed.

**Scale separation closure and Alfven wave turbulence **

C.Y. Chen, S.M. Mahajan

**Abstract**

Based on the concept of scale separation between coherent response function and incoherent source for renormalized turbulence theories, a closure scheme is proposed. A model problem dealing with shear-Alfven wave turbulence is numerically solved; the solution explicitly shows expected turbulence features such as frequency shift from linear modes, band-broadening, and a power law dependence for the turbulence spectrum.

**Particle simulation of low frequency fluctuations in a sheared magnetic field **

R.D. Sydora

**Abstract**

The tools of particle simulation are used to investigate the physics of density gradient-driven low frequency fluctuations in a sheared magnetic field. This is important for the understanding of anomalous plasma losses in toroidal confinement devices. Two types of fluctuations are studied. The first is the electrostatic drift wave in a sheared magnetic field. The existence of absolutely stable drift wave eigenmodes, localized about a single mode rational surface, is established in the sheared slab geometry. From the simulation results in the strong shear limit the eigenmode frequency and mode structure agree with the linear theory. In the weak shear regime localized, convective transient growths are observed before the eigenmodes have developed. An extension of the strong shear case to three dimensions is made and the eigenmode stability is examined. The modification of the electron dynamics due to the presence of multiple rational surfaces and thermal noise is shown to affect the eigenmode stability. Enhancements above the thermal level are observed in certain parameter regimes. The second kind of fluctuation studied is the resistive interchange (g) mode in a sheared magnetic field. The saturation and nonlinear evolution of the instability is compared in single and multiple rational surface configurations. It is found that in the single rational surface case the unstable g-modes saturate by density flattening over the eigenmode width. In the presence of multiple rational surfaces localized profile modification is absent and the convective nonlinearities dominate. It is shown that a mixing length theory is adequate to describe the saturated potential fluctuations. The density fluctuations saturate by coupling to small scales.

**Local conservation laws for the Maxwell-Vlasov and collisionless kinetic guiding center theories **

D. Pfirsch, P.J. Morrison

**Abstract**

With use of a recent variational formulation of the Maxwell-Vlasov and guiding-center theories [D. Pfirsch, Z. Naturforsch. A 39, 1 (1984)], the energy-momentum and angular momentum tensors for such theories are derived and the corresponding local conservation laws are proven. The energy-momentum tensor is shown to be symmetric in its spatial components while the angular momentum density is naturally antisymmetric.

**Tearing mode driven by equilibrium parallel electric field **

F. Cozzani, S.M. Mahajan

**Abstract**

The effect of the parallel equilibrium current on the linear stability of the drift-tearing mode in the collisional regime is investigated analytically. In the appropriate parameter regime, a new unstable mode, driven by equilibrium current, is found and its relevance to tokamak discharges is discussed.

**Variational quadratic form for low frequency electromagnetic perturbations; (I) Formalism **

H.L. Berk, B.G. Lane

**Abstract**

A variational formalism is obtained in the limit of large perpendicular wavenumber which simultaneously includes electrostatic and electromagnetic perturbations, finite Larmor radius corrections, equilibrium plasma rotation, and arbitrary particle bounce effects. A tractable final expression is obtained and kinetic integrals are evaluated in special limits. The more accurate noneikonal expression is obtained from the asymptotic matching of the eikonal form to more restrictive noneikonal quadratic forms derived elsewhere.

**Spectrum of resistivity gradient driven turbulence **

P. Terry, K.C. Shaing, P.H. Diamond, L. Garcia, B.A. Carreras

**Abstract**

The resistivity fluctuation correlation function and electrostatic potential spectrum of resistivity-gradient-driven turbulence are calculated analytically and compared to the results of three-dimensional numerical calculations. Resistivity-gradient-driven turbulence is characterized by effective Reynolds' numbers of order unity. Steady-state solution of the renormalized spectrum equation yields an electrostatic potential spectrum <φ^{2}>_{ky}~k_{y}^{- 3.25} . Agreement of the analytically calculated potential spectrum and mean-square radial velocity with the results of multiple helicity numerical calculations is good. This comparison constitutes a quantitative test of the analytical turbulence theory used. The spectrum of magnetic fluctuations is also calculated and agrees well with that obtained from the numerical computations.

**Electromagnetic solitary waves in magnetized plasmas **

R.D. Hazeltine, D.D. Holm, P.J. Morrison

**Abstract**

A Hamiltonian formulation, in terms of noncanonical Poisson bracket, is presented for a nonlinear fluid system that includes reduced magnetohydrodynamics and the Hasegawa-Mima equation as limiting cases. The single-helicity and axisymmetric versions possess three nonlinear Casimir invariants, from which a generalized potential can be constructed. Variation of the generalized potential yields a description of exact nonlinear stationary states. The new equilibria, allowing for plasma flow as well as partial electron adiabaticity, are distinct from those found in conventional magnetohydrodynamic theory. They differ from electrostatic stationary states in containing plasma current and magnetic field excitation. One class of steady-state solutions is shown to provide a simple electromagnetic generalization of drift-solitary waves.

**Compressibility effects on ideal kinetic ballooning modes and elimination of finite larmor radius stabilization **

C.T. Hsu, R.D. Hazeltine, P.J. Morrison

**Abstract**

The dynamics of ideal and kinetic ballooning modes are considered analytically including parallel ion dynamics, but without electron dissipation. For ideal modes, parallel dynamics predominantly determine the growth rate when β is within ≈ 30% of the ideal threshold, resulting in a substantial reduction in growth rate. Compressibility also eliminates the stabilization effects of finite Larmor radius (FLR); FLR effects (when temperature gradients are neglected) can even increase the growth rate above the MHD value. Temperature gradients accentuate this by adding a new source of free energy independent of the MHD drive, in this region of ballooning coordinate corresponding in MHD to the continuum. Analytic dispersion relations are derived demonstrating the effects above; the formalism emphasizes the similarities between the ideal MHD and kinetic cases.

**A generalized reduced fluid model with finite ion-gyroradius **

C.T. Hsu, R.D. Hazeltine, P.J. Morrison

**Abstract**

Reduced fluid models have become important tools for studying the nonlinear dynamics of plasma in a large aspect-ratio tokamak. A self-consistent nonlinear reduced fluid model, with finite ion-gyroradius effects is presented. The model is distinctive in being correct to O((ρ_{i}/a)^{2}) and in satisfying an exact, relatively simple energy conservation law.

**Two-point theory of current-driven ion-cyclotron turbulence **

T. Chiueh, P.H. Diamond

**Abstract**

An analytical theory of current-driven, ion-cyclotron turbulence that treats incoherent phase-space density granulations (clumps) is presented. In contrast to previous investigations, it focuses on the physically relevant regime of weak collective dissipation, where waves and clumps coexist. The threshold current for nonlinear instability is calculated and is found to deviate from the linear threshold by up to 7 percent. A necessary condition for the existence of stationary wave-clump turbulence is derived, and shown to be analogous to the test-particle model, fluctuation-dissipation theorem result. The structure of three-dimensional magnetized clumps is characterized. It is proposed that nonlinear instability is saturated by collective dissipation due to ion-wave scattering. For this wave-clump turbulence regime, it is found that the fluctuation level (eφ/T_{e})rms ≤ 0.1, and that the modification of anomalous resistivity to levels predicted by conventional nonlinear wave theories is moderate. In marked contrast to the quasi-linear prediction, it is also shown that ion heating significantly exceeds electron heating, and that a drag force on electrons sustains a steady current and is set up to counteract the accelerating electric field along field lines.

**Collisional transport for a superthermal ion species in a plasma **

F. Cozzani, W. Horton

**Abstract**

The transport theory of a high-energy ion species injected isotropically in a magnetized plasma is considered for arbitrary ratios of the high-energy ion cyclotron frequency to the collisional slowing down time. The assumptions of (1) low fractional density of the high-energy species and (2) average ion speed faster than the thermal ions and slower than the electrons are used to decouple the kinetic equation for the high-energy species from the kinetic equations for background ions and electrons. The kinetic equation is solved by a Chapman-Enskog expansion in the strength of the gradients; an equation for the first correction to the lowest-order distribution function is obtained without scaling a priori the collision frequency with respect to the gyrofrequency. Various transport coefficients are explicitly calculated for the two cases of a weakly and a strongly magnetized plasma.

**Covariant poisson brackets for classical fields **

J.E. Marsden, R. Montgomery, P.J. Morrison, W.B. Thompson

**Abstract**

Poisson brackets that are covariant under spacetime coordinate changes are presented for relativistic field theories. The formalism described here is an alternative to the sympletic formulation of field theories and has several advantages. It applies to relativistic fluids and plasmas written in Eulerian variables, while the symplectic formulation does not. It is expected to simplify and clarify the transition to the dynamical (or 3+1) Hamiltonian formalism as well.

**Finite orbit analysis for long wavelength modes in a plasma with a hot component **

J. Hammer, H.L. Berk

**Abstract**

The z-pinch model is used to calculate finite Larmor radius effects of a plasma with a hot component plasma annulus. The equations are analyzed for layer modes and the finite Larmor radius stabilization condition is calculated. Stability requires k^{2}ρ^{2}_{h}Rβ_{h}δ ≥1, where k is the wavenumber in the z direction, ρ_{h} the hot species Larmor radius, β_{h} the hot particle beta, and δ the thickness of the pressure profile. In addition a new instability is found, caused by the interaction of the precessional modes associated with inner and outer edges of the hot particle pressure profile.

**Low frequency modes in a rotating plasma with finite axial length **

A. Ishida, W. Horton

**Abstract**

In a rotating plasma with finite axial length which simulates the central cell plasma in tandem mirrors, the dispersion relation for the low frequency electrostatic modes is derived using the full kinetic description. The analysis includes the effects of (1) the diamagnetic and E X B drifts, (2) the lowest order of the ion Larmor radius, (3) the bounce motion of particles along the magnetic field, and (4) the anisotropy of temperature. A new flute mode driven by the combination of the centrifugal force and Τ_{⊥} > Τ_{||} anisotropy is derived which may explain the oscillations observed in the TMX expriment.

**A four-field model for tokamak plasma dynamics **

M. Kotschenreuther, P.J. Morrison, R.D. Hazeltine

**Abstract**

A generalization of reduced magnetohydrodynamics is constructed from moments of the Fokker-Planck equation. The new model uses familiar aspect-ratio approximations but allows for evolution as slow as the diamagnetic drift frequency, thereby including certain finite Larmor radius effects; pressure gradient terms in a generalized Ohm's law, thus making accessible the adiabatic electron limit; and plasma compressibility, including the divergence of both parallel and perpendicular flows. The system is isothermal and surprisingly simple, involving only one additional field variable, i.e., four independent fields replace the three fields of reduced magnetohydrodynamics. It possesses a conserved energy. The model's equilibrium limit is shown to reproduce not only the large aspect-ratio Grad-Shafranov equation, but also such finite Larmor radius effects as the equilibrium ion parallel flow. Its linearized version reproduces, inter alia, crucial physics of the long mean-free-path electron response. Nonlinearly, the four-field model is shown to describe diffusion in stochastic magnetic fields with good qualitative accuracy.

**A stroll through the soliton theory **

S. Eliezer

**Abstract**

The following sections are included: (1) ion plasma waves and the KdV equation, (2) Burger's equation, (3) symmetries, (4) nonlinearity, dissipation, and dispersion, (5) solitons and cnoidal waves, (6) Miura, Gardner, and Backlund transformations, (7) conservation laws, (8) multisoliton solutions and the superposition principle, (9) time independent Schroedinger equation, and (10) inverse scattering theory. (MOW)