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.

A physically comprehensive and theoretically based transport model tuned to three-dimensional (3-D) ballooning mode gyrokinetic instabilities and gyrofluid nonlinear turbulence simulations is formulated with global and local magnetic shear stabilization and E×B rotational shear stabilization. Taking no fit coefficients from experiment, the model is tested against a large transport profile database with good agreement. This model is capable of describing enhanced core confinement transport barriers in negative central shear discharges based on rotational shear stabilization. The model is used to make ignition projections from relative gyroradius scaling discharges.

A closed set of fluid moment equations is developed which represents kinetic Landau damping physics and which takes a simple form in wave-number space. The linear-response function corresponds to a three-pole (or four-pole) approximation to the plasma dispersion function Z. Alternatively, the response is exact for a distribution function which is close to Maxwellian, but which decreases asymptotically as \/v^ (or 1/L'^). Among other applications, these equations should be useful for nonlinear studies of turbulence driven by the ion-temperature-gradient instability or other drift-wave microinstabilities.PACS numbers: 52.35.Qz, 02.60.+y, 52.25.Kn Because of their relative simplicity, fluid moment equations have been used in a number of recent nonlinear studies of turbulence driven by the ion-temperature-gradient (ITG) instability'"^ and other microinstabilities. This turbulence is of interest because it cancause transport in tokamaks and other plasmas. Moment equations must be closed by an approximation scheme. The classic method of Braginskii"^ is rigorous in the short-mean-free-path regime, but inapplicable to coUisionless plasmas. This Letter proposes a closure method which (1) ensures particle, momentum, and energy conservation, (2) takes on a simple form in wavenumber space, and (3) has a linear-response function very close to that of a coUisionless, Maxwellian plasma. This closure method successfully models kinetic resonances (such as Landau damping) not only in one dimension but also in slab geometry where it reproduces the correct marginal stability behavior of the ITG mode.Several authors have suggested that the effects of kinetic Landau damping may be modeled in fluid moment equations by adding dissipative terms. Lee and Diamond' set the parallel momentum viscosity to AIH^UU/ I G> I, where co is the mode frequency, and vu^iTi/rrii)^^^ is the thermal ion speed. Hamaguchi and Horton^ suggest modifying both /in and the parallel heat conductivity x\u although their simulations use values which are constant for all modes independent of wave frequency and wavelength. Waltz^ has proposed setting /i||=;tii^min(2'^^i?ti/Uiil,2yti/|ft^r|), where cOr is the real part of an instantaneous estimate of the mode frequency. However, no comparison of any of these models with exact kinetic Landau damping has been published. We shall show that any model with a nonzero /xii faces difficulties of interpretation and yields inaccurate thresholds and growth rates for the ITG instability.It has been suggested' that Landau damping for ITG modes can be ignored well above marginal stability so that Braginskii-based fluid equations can be used. However, kinetic effects cannot be ignored for higher radial eigenmodes which appear to cause more transport. ^'^