Fundamental statistical descriptions of plasma turbulence in magnetic fields

Abstract

A pedagogical review of the historical development and current status (as of early 2000) of systematic statistical theories of plasma turbulence is undertaken. Emphasis is on conceptual foundations and methodology, not practical applications. Particular attention is paid to equations and formalism appropriate to strongly magnetized, fully ionized plasmas. Extensive reference to the literature on neutral-fluid turbulence is made, but the unique properties and problems of plasmas are emphasized throughout. Discussions are given of quasilinear theory, weak-turbulence theory, resonance-broadening theory, and the clump algorithm. Those are developed independently, then shown to be special cases of the direct-interaction approximation (DIA), which provides a central focus for the article. Various methods of renormalized perturbation theory are described, then unified with the aid of the generating-functional formalism of Martin, Siggia, and Rose. A general expression for the renormalized dielectric function is deduced and discussed in detail. Modern approaches such as decimation and PDF methods are described. Derivations of DIA-based Markovian closures are discussed. The eddy-damped quasinormal Markovian closure is shown to be nonrealizable in the presence of waves, and a new realizable Markovian closure is presented. The test-field model and a realizable modification thereof are also summarized. Numerical solutions of various closures for some plasmaphysics paradigms are reviewed. The variational approach to bounds on transport is developed. Miscellaneous topics include Onsager symmetries for turbulence, the interpretation of entropy balances for both kinetic and fluid descriptions, self-organized criticality, statistical interactions between disparate scales, and the roles of both mean and random shear. Appendices are provided on Fourier transform conventions, dimensional and scaling analysis, the derivations of nonlinear gyrokinetic and gyrofluid equations, stochasticity criteria for quasilinear theory, formal aspects of resonance-broadening theory, Novikov's theorem, the treatment of weak inhomogeneity, the derivation of the Vlasov weak-turbulence wave kinetic equation from a fully renormalized description, some features of a code for solving the direct-interaction approximation and related Markovian closures, the details of the solution of the EDQNM closure for a solvable three-wave model, and the notation used in the article.

Non‐Gaussian Radio‐Wave Scattering in the Interstellar Medium

“…This happens because in the presence of relatively strong electron density fluctuations and of the corresponding perpendicular electric field (see the above estimate of n k /n), the ion polarization drift velocity becomes comparable to the velocity of fluid fluctuations, and so ions can drift compressively across the magnetic field lines. This leads to the coupling of density fluctuations to the Alfvénic fluctuations; a similar effect is at work in the so-called electromagnetic drift wave (or drift Alfvén) turbulence, which has been investigated in the context of plasma fusion devices (e.g., Hasegawa & Mima 1978;Hasegawa & Wakatani 1983;Terry & Horton 1982;Hazeltine 1983;Camargo et al 1996;Krommes 2002). Interestingly, numerical simulations of such turbulence have revealed a rather high level of density fluctuations (in equipartition with kinetic fluctuations) and statistical properties that are clearly non-Gaussian (e.g., Terry et al 2001;Graddock et al 1991).…”

Quasilinear transport modelling at low magnetic shear

“…This can be quantified by the Kubo number K, 21,22 the ratio between the Eulerian autocorrelation time, s ac , of the fluctuating potential and the particle flight time, s f :s f is defined as k x =hv x i, where k x is the radial correlation decay length, and hv x i the root mean square radial velocity of a test particle in the turbulent field. For K < 1, the autocorrelation time of the turbulent field is shorter than the transit time of a test particle around a turbulent eddy, justifying the use of a random-walk diffusive model.…”

Applications of noise theory to plasma fluctuations

“…Various methods have been developed for solving fluctuation problems [4,5,6,7,8,9,10,11]. In this work, we treat random fluctuations as noise and use noise theory to study the statistical properties of stationary Markovian random fluctuations.…”