[en] Catto earlier introduced an interesting and plausible modification of the usual resonance-broadening prescription for obtaining the nonlinear dielectric function. He argued reasonably that one should employ that prescription only for the nonadiabatic response, and that one should treat the adiabatic response essentially exactly. However, Misguich, in a recent Comment on Catto's work, found an apparent divergence in a form for the renormalized dielectric which he argued was equivalent to Catto's. Misguich was thus led to conclude that, at least for stationary turbulence, Catto's form was suspect, and that a more intricate renormalization might have to be used to obtain a sensible, convergent result. It is argued that this conclusion is incorrect, at least for the reasons Misguich gives

[en] Further details are provided of a soon-to-be published dialog [Phys. Fluids 29 (July, 1986)] which discussed the role of the small scales in fluid clump theory. It is argued that the approximation of the clump lifetime which is compatible with exponentially rapid separation of adjacent orbits is inappropriate for the description of the dynamically important large scales. Various other remarks are made relating to the analytic treatment of strong drift-wave-like turbulence

[en] A review is given of modern theories of statistical dynamics as applied to problems in plasma physics. The derivation of consistent renormalized kinetic equations is discussed, first heuristically, later in terms of powerful functional techniques. The equations are illustrated with models of various degrees of idealization, including the exactly soluble stochastic oscillator, a prototype for several important applications. The direct-interaction approximation is described in detail. Applications discussed include test particle diffusion and the justification of quasilinear theory, convective cells, E vector x B vector turbulence, the renormalized dielectric function, phase space granulation, and stochastic magnetic fields

[en] A model stochastic differential equation is considered which describes guiding center electron motion in a statistically specified spectrum of turbulent magnetic fluctuations. The fluctuation intensity is assumed to satisfy the Chirikov criterion (resonance overlap) for onset of stochasticity. In this limit typical lines diffuse and are adequately described by a quasilinear diffusion coefficient D/sub m/. However, quasilinear theory does not describe an important mechanism for loss of particle correlations: particles collisionally diffuse from one line to an adjacent one which diverges rapidly from the first, carrying the particles away. The scale length L/sub K/ for line divergence is related to the inverse of the Kolmogorov-Sinai entropy. An attempt is made to determine L/sub K/ from a simplified Eulerian vertex renormalization. The exponentiation length which emerges is L/sub K/ approximately L/sub s/(anti k²/sub theta/D''/sub m/L/sub s/)/sup -1/3/, where L/sub s/ is the shear length, k bar/sub theta/ is a typical azimuthal wavenumber, and D''/sub m/ is of order D/sub m/. In a particular limit of weak shear, the particle diffusion coefficient can then be estimated as D approximately DELTA r²/tau/sub c/, where Δr² approximately D/sub m/z(tau/sub c/), z(tau) is the distance traveled along the lines in time tau, and for static fluctuations tau/sub c/ approximately tau(L/sub delta/), where L/sub delta/ is L/sub K/ multiplied by a logarithmic factor involving the perpendicular collisional diffusion coefficient

[en] Fusion physics poses an extremely challenging, practically complex problem that does not yield readily to simple paradigms. Nevertheless, various of the theoretical tools and conceptual advances emphasized at the KaufmanFest 2007 have motivated and/or found application to the development of fusion-related plasma turbulence theory. A brief historical commentary is given on some aspects of that specialty, with emphasis on the role (and limitations) of Hamiltonian/symplectic approaches, variational methods, oscillation-center theory, and nonlinear dynamics. It is shown how to extract a renormalized ponderomotive force from the statistical equations of plasma turbulence, and the possibility of a renormalized K-? theorem is discussed. An unusual application of quasilinear theory to the problem of plasma equilibria in the presence of stochastic magnetic fields is described. The modern problem of zonal-flow dynamics illustrates a confluence of several techniques, including (i) the application of nonlinear-dynamics methods, especially center-manifold theory, to the problem of the transition to plasma turbulence in the face of self-generated zonal flows; and (ii) the use of Hamiltonian formalism to determine the appropriate (Casimir) invariant to be used in a novel wave-kinetic analysis of systems of interacting zonal flows and drift waves. Recent progress in the theory of intermittent chaotic statistics and the generation of coherent structures from turbulence is mentioned, and an appeal is made for some new tools to cope with these interesting and difficult problems in nonlinear plasma physics. Finally, the important influence of the intellectually stimulating research environment fostered by Prof. Allan Kaufman on the author's thinking and teaching methodology is described.

[en] A kinetic theory for the nonlinear damping of collisionless drift waves in a shear-free magnetic field is presented. The general formalism is a renormalized version of induced scattering on the ions and reduces correctly to weak turbulence theory. The approximation studied explicitly reduces to Compton scattering, systematizes thee earlier calculations of Dupree and Tetreault (DT) [Phys. Fluids 21, 425 (1978)], and extends that theory to finite ion gyroradius. Certain conclusions differ significantly from those of DT

[en] It is argued that Dupree's procedure (Phys. Fluids 15, 334(1972)) for computing self-sustaining clump spectra is tautological

[en] Some aspects of low-frequency, long-wavelength fluctuations are considered. A stochastic model is used to show that power-law time correlations need not arise from self-organized criticality. A formula for the frequency spectrum of uncorrelated, overlapping avalanches is shown to be a special case of the spectral balance equation of renormalized statistical turbulence theory. It is argued that there need be no contradiction between the presence of long-time correlations and the existence of local transport coefficients

[en] The direct-interaction approximation is used to find statistically steady states of a system of three modes, with complex frequencies, coupled by a quadratic nonlinearity. These states are compared to the exact predictions of an ensemble of realizations with Gaussianly distributed initial conditions. The direct-interaction approximation is shown to be reasonably successful in this context

[en] Several popular theories of the renormalized dielectric are examined and shown to be logically flawed. A recent conclusion that the weak-coupling approximation to the renormalized quasilinear dielectric is divergent is shown to be misleading because of an improper definition of the dielectric. The usual resonance-broadened dielectric is shown to be in error because the approximation neglects subtle correlations of the same order and physical importance as the terms retained. The problem is traced specifically to an erroneous application of statistical averaging and to the often-ignored difference between the infinitesimal response function and the single particle propagator. The procedure of resonance-broadening the non-adiabatic response is discussed, but no justification for the usual form of this approximation is found