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.

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 theory are based on three pillars: ͑i͒ a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotivelike terms, ͑ii͒ a set of gyrokinetic Maxwell ͑Poisson-Ampère͒ equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization ͑Poisson͒ and magnetization ͑Ampère͒ terms, and ͑iii͒ an exact energy conservation law for the gyrokinetic Vlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on rigorous application of Lagrangian and Hamiltonian Lie-transform perturbation methods in the variational derivation of nonlinear gyrokinetic Vlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations that describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry are discussed.

The dynamics of turbulence-driven E B zonal ows has been systematically studied in fully 3-dimensional gyrokinetic simulations of microturbulence in magnetically-conned toroidal plasmas using recently available massively parallel computers. Linear ow damping simulations exhibit an asymptotic residual ow in agreement with recent analytic calculations. Nonlinear global simulations of instabilities driven by temperature gradients in the ion component of the plasma provide key rst principles results supporting the physics picture that turbulence-driven uctuating E B zonal ows can signicantly reduce turbulent transport.Turbulence shear suppression by E B ows is the most likely mechanism responsible for the transition to various forms of enhanced connement regimes observed in magnetically-conned plasmas [1]. Understanding the physical mechanisms of turbulence suppression processes [2,3] here is what controls the generation of the ows and how strongly the ows aects the turbulent transport which is believed to arise from electrostatic pressuregradient driven instabilities. These highly complex nonlinear phenomena can be most eectively investigated by n umerical experiments. One of the most promising approaches is gyrokinetic particle-in-cell simulation [6] which suppress the rapid gyromotion of a charged particle about the magnetic eld line. By making use of recent advances of new low-noise numerical algorithms and by taking advantage of the exciting opportunities oered by high-end massively parallel computing power, it has been able to reproduce key features of turbulent transport observed at the core of tokamak plasmas.The present n umerical experiments clearly demonstrate that turbulence-driven uctuating E B zonal ows play a crucial role in regulating nonlinear saturation and transport levels. This is in agreement with previous toroidal gyrokinetic and gyrouid (a uid model with kinetic eect) simulations of instabilities driven by iontemperature-gradient (ITG) in a local geometry which follows a magnetic eld line [7{9]. However, previous global gyrokinetic simulations, which treat the whole plasma volume, either did not include [10] or did not observe [11,12] signicant eects of these self-generated ows. Since local simulations are restricted to a uxtube domain with radially periodic boundary conditions and since they rely on the assumption of scale separation between the turbulence and equilibrium proles, the key issues of transport scaling and eects of steep pressure proles in transport barriers can only be eectively studied in global simulations. In this report, nonlinear simulation results from a newly developed global gyrokinetic code [13] yield the important conclusion that turbulencedriven uctuating E B ows can signicantly reduce the anomalous transport. In order to understand this key process, the dynamics of E B ows have been systematically analyzed. Linear ow damping simulations exhibit a time asymptotic residual ow in agreement with a recent analytic calculation [14]. The present nonlinear global simulation...

The suppression of turbulence by the E x B flow shear and parallel flow shear is studied in an arbitrary shape finite aspect ratio tokamak plasma using the two point nonlinear analysis previously utilized in a high aspect ratio tokamak plasma [Phys. Plasmas 1, 2940 (1994)]. The result shows that only the E x B flow shear is responsible for the suppression of flute-like fluctuations. This suppression occurs regardless of the plasma rotation direction and is, therefore, relevant for the VH mode plasma core as well as for the H mode plasma edge. Experimentally observed in-out asymmetry of fluctuation reduction behavior can be addressed in the context of flux expansion and magnetic field pitch variation on a given flux surface. The adverse effect of neutral particles on confinement improvement is also discussed in the context of the charge exchange induced parallel momentum damping.

A nonlinear electrostatic gyrokinetic Vlasov equation as well as a Poisson equation have been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov–Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits.

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Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.

Theory of E؋B shear suppression of turbulence in toroidal geometry ͓Phys. Plasmas 2, 1648 ͑1995͔͒ is extended to include fast time variations of the E؋B flows often observed in nonlinear simulations of tokamak turbulence. It is shown that the quickly time varying components of the E؋B flows, while they typically contribute significantly to the instantaneous E؋B shearing rate, are less effective than the slowly time varying components in suppressing turbulence. This is because the shear flow pattern changes before eddies get distorted enough. The effective E؋B shearing rate capturing this important physics is analytically derived and estimated from zonal flow statistics of gyrofluid simulation. This provides new insights into understanding recent gyrofluid and gyrokinetic simulations that yield a reduced, but not completely quenched, level of turbulence in the presence of turbulence-driven zonal flows.