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
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$${\mathcal{D}}$$. 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. In conclusion, various historical research threads are mentioned and representative entry points to the literature are given. In addition, some outstanding conceptual issues are enumerated.
- Authors:
-
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
- Publication Date:
- Research Org.:
- Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1256594
- Report Number(s):
- PPPL-5178
Journal ID: ISSN 0022-3778; applab; PII: S0022377815000756
- Grant/Contract Number:
- ACO2-09CH11466
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Plasma Physics
- Additional Journal Information:
- Journal Volume: 81; Journal Issue: 06; Journal ID: ISSN 0022-3778
- Publisher:
- Cambridge University Press
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; direct-interaction approximation; drift-wave turbulence; realizable; markovian closure; linear gyrokinetic equations; non-gaussian; statistics; weakly unstable plasma; magnetic-field; isotropic; turbulence; kinetic-theory; adiabatic modifications
Citation Formats
Krommes, John A. 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. United States: N. p., 2015.
Web. doi:10.1017/s0022377815000756.
Krommes, John A. 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. United States. https://doi.org/10.1017/s0022377815000756
Krommes, John A. Mon .
"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". United States. https://doi.org/10.1017/s0022377815000756. https://www.osti.gov/servlets/purl/1256594.
@article{osti_1256594,
title = {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},
author = {Krommes, John A.},
abstractNote = {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. In conclusion, various historical research threads are mentioned and representative entry points to the literature are given. In addition, some outstanding conceptual issues are enumerated.},
doi = {10.1017/s0022377815000756},
journal = {Journal of Plasma Physics},
number = 06,
volume = 81,
place = {United States},
year = {Mon Sep 21 00:00:00 EDT 2015},
month = {Mon Sep 21 00:00:00 EDT 2015}
}
Web of Science
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- Physics of Fluids, Vol. 29, Issue 8
Geometric view on noneikonal waves
journal, April 2014
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Bifurcation theory of the transition to collisionless ion-temperature-gradient-driven plasma turbulence
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Turbulence theories and modelling of fluids and plasmas
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Astrophysical Fluid Dynamics via Direct Statistical Simulation
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The Gyrokinetic Description of Microturbulence in Magnetized Plasmas
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Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics
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Energy Transfer in a Normally Distributed and Isotropic Turbulent Velocity Field in Two Dimensions
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Turbulent Diffusion, Particle Orbits, and Field Fluctuations in a Plasma in a Magnetic Field
journal, January 1968
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Magnetic fluctuations can contribute to plasma transport, ‘‘self-consistency constraints’’ notwithstanding
journal, January 1988
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Transition in shear flows. Nonlinear normality versus non‐normal linearity
journal, December 1995
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Fractional diffusion in plasma turbulence
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Direct Statistical Simulation of Out-of-Equilibrium Jets
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The mechanism of self‐sustainment in collisional drift wave turbulence
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Numerical study of drift wave turbulence with simple models for wave–wave nonlinear coupling
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Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence
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Fundamental statistical descriptions of plasma turbulence in magnetic fields
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Dynamics of Nonlinear Stochastic Systems
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Contributions to Non-Equilibrium Thermodynamics. I. Theory of Hydrodynamical Fluctuations
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Random shearing by zonal flows and transport reduction
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Plasma Edge Turbulence
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A review of zonal flow experiments
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Adiabatic modifications to plasma turbulence theories
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Pseudo-three-dimensional turbulence in magnetized nonuniform plasma
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Gyrokinetic simulations of turbulent transport
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Zonal flow as pattern formation
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Modulational instability of Rossby and drift waves and generation of zonal jets
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Suppression of an instability by the introduction of external turbulence
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Signature of a universal statistical description for drift-wave plasma turbulence
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Model Equations for Strong Turbulence in a Vlasov Plasma
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Nonlinear Self-Sustained Drift-Wave Turbulence
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Comments on ‘‘Adiabatic modifications to plasma turbulence theory’’
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Generalized Cumulant Expansion Method
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A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence
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Turbulent Diffusion, Particle Orbits, and Field Fluctuations in a Plasma in a Magnetic Field
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Turbulent Plasmas in a Magnetic Field—A Statistical Theory
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Pseudo-three-dimensional turbulence in magnetized nonuniform plasma
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Renormalized dielectric function for collisionless drift wave turbulence
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Aspects of a renormalized weak plasma turbulence theory
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Suppression of an instability by the introduction of external turbulence
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Theory of two-point correlation function in a Vlasov plasma
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Direct interaction approximation for Vlasov turbulence from the Kadomtsev weak coupling approximation
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Equilibrium fluctuation energy of gyrokinetic plasma
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- Physics of Fluids, Vol. 29, Issue 8
Transition in shear flows. Nonlinear normality versus non‐normal linearity
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On a self-sustaining process in shear flows
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Mapping closure for probability distribution function in low frequency magnetized plasma turbulence
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Statistical theory of resistive drift-wave turbulence and transport
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Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics
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Imbalanced Weak Magnetohydrodynamic Turbulence
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Gyrokinetic simulations of turbulent transport
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Direct interaction approximation and plasma turbulence theory
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Generation of zonal flows through symmetry breaking of statistical homogeneity
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Non-Gaussian statistics, classical field theory, and realizable Langevin models
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Experimental Signatures of Critically Balanced Turbulence in MAST
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Two-Dimensional Turbulence
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Statistical properties of an ensemble of vortices interacting with a turbulent field
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Zonal flow generation by modulational instability
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Works referencing / citing this record:
Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations
journal, November 2018
- Krommes, John A.
- Journal of Plasma Physics, Vol. 84, Issue 6
Extensions of the Novikov–Furutsu theorem, obtained by using Volterra functional calculus
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- Athanassoulis, Gerassimos A.; Mamis, Konstantinos I.
- Physica Scripta, Vol. 94, Issue 11
S3T stability of the homogeneous state of barotropic beta-plane turbulence
text, January 2014
- Bakas, Nikolaos A.; Constantinou, Navid C.; Ioannou, Petros J.
- arXiv
Statistical State Dynamics: a new perspective on turbulence in shear flow
text, January 2014
- Farrell, Brian F.; Ioannou, Petros J.
- arXiv
Emergence of non-zonal coherent structures
preprint, January 2015
- Bakas, Nikolaos A.; Ioannou, Petros J.
- arXiv