A comprehensive review of zonal flow phenomena in plasmas is presented. While the emphasis is on zonal flows in laboratory plasmas, planetary zonal flows are discussed as well. The review presents the status of theory, numerical simulation and experiments relevant to zonal flows. The emphasis is on developing an integrated understanding of the dynamics of drift wave–zonal flow turbulence by combining detailed studies of the generation of zonal flows by drift waves, the back-interaction of zonal flows on the drift waves, and the various feedback loops by which the system regulates and organizes itself. The implications of zonal flow phenomena for confinement in, and the phenomena of fusion devices are discussed. Special attention is given to the comparison of experiment with theory and to identifying directions for progress in future research.

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Zonal flows in plasma—a review

The dissipation induced by coulomb-collisional scattering provides an irreducible minimum, and thus a useful standard for comparison, for transport processes in a hot, magnetically confined plasma. The kinetic description of this dissipation is provided by an equation of the Fokker-Planck form. As in the standard transport theory for a neutral gas, approximate solution of the Fokker-Planck equation permits the calculation of transport coefficients, which linearly relate the fluxes of particles, energy, and electric charge, to the density and temperature gradients, and to the electric field. The transport relations are useful in studying the confinement properties of present and future experimental devices for research in controlled thermonuclear fusion. The transport theory for a magnetized plasma (in which the Larmor radius is much smaller than gradient scale lengths describing the plasma fluid) departs from the theory for a neutral gas in several fundamental ways. Thus, transport coefficients for a magnetized plasma can be calculated even when the collisional mean free path is much longer than the gradient scale length (as would pertain in thermonuclear regimes). Such transport coefficients are generally nonlocal, being defined in terms of averages over surfaces with macroscopic dimensions. Furthermore, when the mean free path is long, the magnetized-plasma transport coefficients depend crucially upon the magnetic field geometry, the effects of which must be treated at the kinetic level of the Fokker-Planck equation. The results display several novel couplings between collisional dissipation and the electromagnetic field. The present review of magnetized-plasma transport theory is intended to be as widely accessible as possible. Thus the relevant features of magnetic confinement in closed (toroidal) systems, and of charged particles in spatially varying fields, are derived, at least in outline, from first principles. Although consideration is given to "classical" transport in which most field geometric effects are omitted, major emphasis is placed on the "neoclassical" theory which has been developed over the last decade. Neoclassical transport coefficients are specifically relevant to a magnetically confined plasma, rather than to just a magnetized plasma; their unusual features, such as nonlocality and geometry dependence, become particularly important in the high temperature regime of proposed thermonuclear reactors. The area of neoclassical theory which seems most complete---its application to axisymmetric tokamak-type confinement systems---is correspondingly stressed.

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Theory of plasma transport in toroidal confinement systems

Tokamak plasmas are inherently comprised of multiple ion species. This is due to wall-bred impurities and, in future reactors, will result from fusion-born alpha particles. Relatively small densities nI, of highly charged non-hydrogenic impurities can strongly influence plasma transport properties whenever . The determination of the complete neoclassical Onsager matrix for a toroidally confined multispecies plasma, which provides the linear relation between the surface averaged radial fluxes and the thermodynamic forces (i.e. gradients of density and temperature, and the parallel electric field), is reviewed. A closed set of one-dimensional moment equations is presented for the time evolution of thermodynamic and magnetic field quantities which results from collisional transport of the plasma and two-dimensional motion of the magnetic flux surface geometry. The effects of neutral-beam injection on the equilibrium and transport properties of a toroidal plasma are consistently included.

https://iopscience.iop.org/article/10.1088/0029-5515/21/9/003/pdf

Neoclassical transport of impurities in tokamak plasmas

Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic the- ory are based on three important pillars: (1) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotive-like terms; (2) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov dis- tribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms derived from the quadratic nonlinearities in the gyrocenter Hamiltonian; and (3) an exact energy conservationlaw for the gyrokineticVlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on the rigorous applications of Lagrangian and Hamiltonian Lie-transform perturba- tion methods used in the variationalderivationof nonlineargyrokineticVlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations, which describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry, are also discussed.

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Foundations of nonlinear gyrokinetic theory

This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the inertial range above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-field-strength fluctuations. The former are governed by the reduced magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations become the slow and entropy modes of the conventional MHD). In the dissipation range below ion gyroscale, there are again two cascades: the kinetic-Alfven-wave (KAW) cascade governed by two fluid-like electron reduced magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations both in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was Landau-damped at the ion gyroscale to collisional scales in the phase space and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and converted into electron heat. Kolmogorov-style scaling relations are derived for all of these cascades. The relationship between the theoretical models proposed in this paper and astrophysical applications and observations is discussed in detail.

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Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas

The neoclassical theory of plasma transport in axisymmetric, toroidal confinement systems, is developed by means of a variational principle for the rate of irreversible entropy production. The variational principle derived here employs the full Fokker‐Planck collision operator, including both like and unlike species collisions. Using the variational principle, all the relevant neoclassical transport coefficients are systematically evaluated in the “banana” regime of small collisional frequency, to lowest order in the inverse aspect ratio. These results include both the “diagonal” and “cross” coefficients for the particle fluxes, ion and electron heat flux, and electric current. By combining the transport coefficients with appropriate moments of the drift equation, a closed set of equations which accurately summarize the predictions of neoclassical theory in the banana regime is obtained. The significance of these equations, in particular with regard to recent tokamak experiments, is discussed briefly.

Plasma Transport in Toroidal Confinement Systems

A nonlinear analysis of collisional drift waves is presented in which a systematic expansion is made in powers of the wave amplitude. The two‐fluid equations are used, including the effects of resistivity, viscosity, and thermal transport. The result, for the wave amplitude as a function of magnetic field in the linearly unstable region close to marginal stability, agrees reasonably well with experiment.

Amplitude Limitation of a Collisional Drift Wave Instability

The neoclassical plasma transport coefficients for axisymmetric toroidal magnetic confinement systems are calculated in the regime of low‐to‐intermediate collision frequency. The problem of solving the linearized drift kinetic equations, and calculating the transport coefficients, is formulated as a variational principle. A maximal form of the variational principle, which is valid in the low‐to‐intermediate collision frequency regime, is used in a numerical solution of the finite dfiference equations, to obtain the distribution function at points on a mesh in phase space. A new analytical result is used to check the accuracy of the numerical calculation for small collision frequencies; this is a correction to the asymptotic banana regime result, obtained from a Wiener‐Hopf analysis of the boundary layer in phase space between the trapped and untrapped particle regions. An analytical correction to the plateau regime analtyical result is also obtained. The transport coefficients are found to be monotonic functions of collision frequency, with a gradual transition from the banana regime to the plateau regime. Analytical formulas have been fitted to the numerically computed results.

Transport properties of a toroidal plasma at low‐to‐intermediate collision frequencies

The collisional regime of neoclassical transport theory is investigated, using a moment equation approach as well as a method based on the drift kinetic equation. Allowing for both density and temperature gradients, and an externally induced toroidal electric field, the transport coefficients describing particle and energy flux perpendicular to the magnetic field of an axisymmetric confinement system are derived. Relations between the parallel and perpendicular fluxes, which are exact in the collisional regime, are also derived. Charge neutrality is used to obtain the electrostatic potential variation on a magnetic surface.

F. L. Hinton

Papers

Theory of plasma transport in toroidal confinement systems

Plasma Transport in Toroidal Confinement Systems

Amplitude Limitation of a Collisional Drift Wave Instability

Transport properties of a toroidal plasma at low‐to‐intermediate collision frequencies

Collision-dominated plasma transport in toroidal confinement systems