A tutorial introduction to the statistical theory of turbulent plasmas, a half century after Kadomtsev’s Plasma Turbulence and the resonance-broadening theory of Dupree and Weinstock
Authors: John A. Krommes
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 R. H. 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
in-variance 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.
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Submitted to: Plasma Physics
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Download PPPL-5178 (pdf 2.6 MB 80 pp)
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