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

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$${\mathcal{D}}$$. Dupree’s method of using$${\mathcal{D}}$$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$${\mathcal{D}}$$. An important example is the direct-interaction approximation (DIA). It is shown how to use Novikov’s theorem to develop an$$\boldsymbol{x}$$-space approach to the DIA that is complementary to the original$$\boldsymbol{k}$$-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$${\mathcal{D}}$$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$${\mathcal{D}}$$). 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. In addition, some outstanding conceptual issues are enumerated.