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

Collisionless electron-temperature-gradient-driven (ETG) turbulence in toroidal geometry is studied via nonlinear numerical simulations To this aim, two massively parallel, fully gyrokinetic Vlasov codes are used, both including electromagnetic effects Somewhat surprisingly, and unlike in the analogous case of ion-temperature-gradient-driven (ITG) turbulence, we find that the turbulent electron heat flux is significantly underpredicted by simple mixing length estimates in a certain parameter regime (ŝ∼1, low α) This observation is directly linked to the presence of radially highly elongated vortices (“streamers”) which lead to very effective cross-field transport The simulations therefore indicate that ETG turbulence is likely to be relevant to magnetic confinement fusion experiments

Electron temperature gradient driven turbulence

PROBLEM TO BE SOLVED: To provide a plasma processing apparatus provided with a gas feed means which feeds gas in pulses into a reaction chamber, where the processing speed and shape of a sample can be set uniform through all its surface, even if an ultrasonic free flow is not established SOLUTION: A plasma processing apparatus is equipped with a reaction chamber 1, which generates plasma by the use of gas to process a sample with it, a gas feed means 9 which feeds gas in pulses to the reaction chamber, and an evacuating means 6 which evacuates the reaction chamber 1, where the feed direction in which gas is fed by the gas feed means 9 and the evacuating direction, in which evacuated gas is discharged by the evacuating means 6 are set identical to each other The reaction chamber 1 is so formed as to be axially symmetrical about its center axis, the gas feed means 9 is arranged on the center axis or symmetrical about a center axis, and the evacuating means 6 is arranged symmetrically about the center axis

Plasma processing apparatus

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

Magnetohydrodynamic (MHD) turbulence is encountered in a wide variety of astrophysical plasmas, including accretion disks, the solar wind, and the interstellar and intracluster medium. On small scales, this turbulence is often expected to consist of highly anisotropic fluctuations with frequencies small compared to the ion cyclotron frequency. For a number of applications, the small scales are also collisionless, so a kinetic treatment of the turbulence is necessary. We show that this anisotropic turbulence is well described by a low-frequency expansion of the kinetic theory called gyrokinetics. This paper is the first in a series to examine turbulent astrophysical plasmas in the gyrokinetic limit. We derive and explain the nonlinear gyrokinetic equations and explore the linear properties of gyrokinetics as a prelude to nonlinear simulations. The linear dispersion relation for gyrokinetics is obtained, and its solutions are compared to those of hot-plasma kinetic theory. These results are used to validate the performance of the gyrokinetic simulation code GS2 in the parameter regimes relevant for astrophysical plasmas. New results on global energy conservation in gyrokinetics are also derived. We briefly outline several of the problems to be addressed by future nonlinear simulations, including particle heating by turbulence in hot accretion flows and in the solar wind, the magnetic and electric field power spectra in the solar wind, and the origin of small-scale density fluctuations in the interstellar medium.

https://iopscience.iop.org/article/10.1086/506172/pdf

Astrophysical Gyrokinetics: Basic Equations and Linear Theory

Kinetic equations for low frequency, short perpendicular wavelength, electromagnetic perturbations in an inhomogeneous, magnetically confined plasma are developed. The analysis makes use of the recently developed high toroidal mode number expansion to reduce the lowest‐order system of equations to a set of ordinary (along the field line) integro‐differential equations. Included in these equations are the effects of finite Larmor radius, magnetic shear, trapped particles, and nonuniform magnetic curvature drifts. Perturbed fields are represented by a scalar potential and two components of the vector potential. Thus, the effects of the compressional component of the perturbed magnetic field are retained and the equations are valid for arbitrary values of the plasma pressure. The extension of the high toroidal mode number expansion to nonaxisymmetric configurations is discussed.

Kinetic equations for low frequency instabilities in inhomogeneous plasmas

A toroidal ECR reactor is described in which a poloidal magnetic field is established in a plasma generating chamber in which a specimen to be processed is disposed on an electrode in a specimen chamber. Microwaves and gaseous reactants are introduced into the plasma generating chamber. A plasma discharge occurs in which high energy electrons are confined in a plasma source region extending between a magnetic mirror formed in the specimen chamber out of line-of-sight to the wafer(s) when disposed on the electrode. A baffle region formed between the two chambers prevents microwaves from entering the specimen chamber. The reactor is particularly suitable for etching or depositing films on semiconductor substrates, since the sensitive substrates are not exposed to the high energy ions and/or photons of the source region.

Toroidal electron cyclotron resonance reactor

Plasma production and heating in the central cell of the Tara tandem mirror [Nucl. Fusion 22, 549 (1982); Plasma Physics and Controlled Nuclear Fusion Research, 1986, Proceedings of the 11th International Conference, Kyoto, Japan (IAEA, Vienna, 1987), Vol. 2, p. 251] have been studied. Using radio‐frequency excitation by a slot antenna in the ion cyclotron frequency range (ICRF), plasmas with a peak β⊥ of 3%, density of 4×1012 cm−3, ion temperature of 800 eV, and electron temperature of 75–100 eV were routinely produced. The plasma radius decreased with increasing ICRF power, causing reduced ICRF coupling and saturation of the plasma beta. About 70% of the applied ICRF power can be accounted for in direct heating of both ions and electrons. Wave field measurements have identified the applied ICRF to be the slow, ion cyclotron wave. In operation without end plugging, the plasma parameters were limited by poor axial confinement and the requirements for maintenance of magnetohydrodynamic stability and micros...

Plasma production and heating in a tandem mirror central cell by radio-frequency waves in the ion cyclotron frequency range

The introduction of a magnetic limiter is proposed for stabilization of MHD interchange modes in the central cell of a tandem mirror. The magnetic limiter creates a ring null in the magnetic field, and electrons which enter the null can stream azimuthally and thereby 'short-circuit' m = 1 fluctuations. This disallows a rigid m = 1 response and introduces finite Larmor radius stabilization effects in much the same way as they appear for higher azimuthal modes. Some pressure gradient can be maintained on the separatrix flux surface by locating the null on a local magnetic maximum. This scheme introduces the possibility of a fully axisymmetric tandem mirror.

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Stabilization of MHD modes in an axisymmetric plasma column through the use of a magnetic divertor

A divertor coil set has been installed on the Tara tandem mirror [Nucl. Fusion 22, 549 (1982); Plasma Physics and Controlled Nuclear Fusion Research 1984 (IAEA, Vienna, 1985), Vol. 2, p. 285] for stabilization of m=1 flutelike modes. The effectiveness of divertor stabilization is discussed in experiments where m=1 modes are driven to instability by plug electron cyclotron heating (ECH) in an ion cyclotron heated (ICH) plasma. The instability onset is characterized by thresholds in ECH power, fueling rate, ICH power, and mapping radius of the divertor null. In general, the stability is enhanced by mapping the null radially inwards into the plasma. The interdependence of these parameters and their effect on equilibrium profiles and stability boundaries are discussed.

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B. Lane

Papers

Kinetic equations for low frequency instabilities in inhomogeneous plasmas

Toroidal electron cyclotron resonance reactor

Plasma production and heating in a tandem mirror central cell by radio-frequency waves in the ion cyclotron frequency range

Stabilization of MHD modes in an axisymmetric plasma column through the use of a magnetic divertor

Experimental studies of divertor stabilization in an axisymmetric tandem mirror