A tutorial introduction to the statistical theory of turbulent plasmas, a half-century after Kadomtsev’sPlasma Turbulenceand the resonance-broadening theory of Dupree and Weinstock

Abstract

In honour of the 50th anniversary of the influential review/monograph on plasma turbulence by B. B. Kadomtsev as well as the seminal works of T. H. Dupree and J. Weinstock on resonance-broadening theory, an introductory tutorial is given about some highlights of the statistical-dynamical description of turbulent plasmas and fluids, including the ideas of nonlinear incoherent noise, coherent damping, and self-consistent dielectric response. The statistical closure problem is introduced. Incoherent noise and coherent damping are illustrated with a solvable model of passive advection. Self-consistency introduces turbulent polarization effects that are described by the dielectric function D. Dupree's method of using D to estimate the saturation level of turbulence is described; then it is explained why a more complete theory that includes nonlinear noise is required. The general theory is best formulated in terms of Dyson equations for the covariance C and an infinitesimal response function R, which subsumes D. An important example is the direct-interaction approximation (DIA). It is shown how to use Novikov's theorem to develop an x-space approach to the DIA that is complementary to the original k-space approach of Kraichnan. A dielectric function is defined for arbitrary quadratically nonlinear systems, including the Navier-Stokes equation, and an algorithm for determining the form of D in the DIA is sketched. The independent insights of Kadomtsev and Kraichnan about the problem of the DIA with random Galilean invariance are described. The mixing-length formula for drift-wave saturation is discussed in the context of closures that include nonlinear noise (shielded by D). The role of R in the calculation of the symmetry-breaking (zonostrophic) instability of homogeneous turbulence to the generation of inhomogeneous mean flows is addressed. The second-order cumulant expansion and the stochastic structural stability theory are also discussed in that context. Various historical research threads are mentioned and representative entry points to the literature are given. Some outstanding conceptual issues are enumerated.

Extensions of the Novikov–Furutsu theorem, obtained by using Volterra functional calculus

“…In their works (Furutsu, 1963;Novikov, 1965), on electromagnetic waves and turbulence respectively, K. Furutsu ( ; )   at s . The above equation, called henceforth the Novikov-Furutsu (NF) theorem, has been extensively used in turbulent diffusion (Cook, 1978;Hyland, McKee and Reeks, 1999;Shrimpton, Haeri and Scott, 2014;Krommes, 2015;Martins Afonso, Mazzino and Gama, 2016), random waves (Sobczyk, 1985;Rino, 1991;Bobryk, 1993;Konotop and Vazquez, 1994;Creamer, 2008), and stochastic dynamics (Protopopescu, 1983;Klyatskin, 2005Klyatskin, , 2015Scott, 2013;Kliemann and Sri Namachchivaya, 2018). One particular use of the NF theorem lies in the formulation of partial differential equations, called generalized Fokker-Planck-Kolmogorov (genFPK) equations, that govern the evolution of the probability density function (pdf) of the response to a random differential equation (RDE) excited by coloured noise (Sancho et al, 1982;Cetto and de la Peña, 1984;Fox, 1986;Hyland, McKee and Reeks, 1999;Venturi et al, 2012).…”

A solvable model of Vlasov-kinetic plasma turbulence in Fourier–Hermite phase space

“…1968). Fast-forwarding over several decades of plasma-turbulence theory (see Krommes 2015 and Laval, Pesme & Adam 2016 for review and references), the notion of ‘phase-space turbulence’, pioneered by Dupree (1972), has, in the recent years, again become a popular object of study, treated either, Dupree-style, in terms of formation of phase-space structures and their effect on the transport properties of the plasma (Kosuga & Diamond 2011; Lesur, Diamond & Kosuga 2014 a , b ; Kosuga et al. 2014, 2017) or in terms of a kinetic cascade carrying free energy to collisional scales in velocity space (Watanabe & Sugama 2004; Schekochihin et al.…”

Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

“…Transport coefficients enter the fluid equations because the dynamics of the hydrodynamic and non-hydrodynamic subspaces are coupled by the evolution equation for . That coupling is a special case of the statistical closure problem (Krommes 2015) for passive equations with random coefficients. (Such equations are said to possess a stochastic nonlinearity.)…”

“…An example of a projection operator is the ensemble average 2 . Thus, all of statistical closure theory (Krommes 2002, 2015) can be said to involve projection operators. However, this is stretching the point.…”

“…2Indeed, any linear, properly normalized averaging operation possesses the property and is thus a projection operator. Weinstock (1969, 1970) employed such operators in his approach to Vlasov turbulence; see some discussion by Krommes (2002) and Krommes (2015). Weinstock’s work was pursued, for example, by Misguich & Balescu (1975) and Misguich (1975).…”