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. ^'^

The maximum normalized beta achieved in long-pulse tokamak discharges at low collisionality falls significantly below both that observed in short pulse discharges and that predicted by the ideal MHD theory. Recent long-pulse experiments, in particular those simulating the International Thermonuclear Experimental Reactor ͑ITER͒ ͓M. Rosenbluth et al., Plasma Physics and Controlled Nuclear Fusion ͑International Atomic Energy Agency, Vienna, 1995͒, Vol. 2, p. 517͔ scenarios with low collisionality e * , are often limited by low-m/n nonideal magnetohydrodynamic ͑MHD͒ modes. The effect of saturated MHD modes is a reduction of the confinement time by 10%-20%, depending on the island size and location, and can lead to a disruption. Recent theories on neoclassical destabilization of tearing modes, including the effects of a perturbed helical bootstrap current, are successful in explaining the qualitative behavior of the resistive modes and recent results are consistent with the size of the saturated islands. Also, a strong correlation is observed between the onset of these low-m/n modes with sawteeth, edge localized modes ͑ELM͒, or fishbone events, consistent with the seed island required by the theory. We will focus on a quantitative comparison between both the conventional resistive and neoclassical theories, and the experimental results of several machines, which have all observed these low-m/n nonideal modes. This enables us to single out the key issues in projecting the long-pulse beta limits of ITER-size tokamaks and also to discuss possible plasma control methods that can increase the soft ␤ limit, decrease the seed perturbations, and/or diminish the effects on confinement.

It is shown that the widely used equilibrium in which the nighttime F region is supported by E×B drifts is unstable if, in addition to the supporting eastward field, a north‐south electric field component exists. The instability, which takes the form of rising and falling sheets of ionization, has a growth rate γ ≈ 3 × 10−4 νin−1 sec−1, where νin is ion‐neutral collision frequency at the F region peak. These conclusions are based on a set of new moment equations that govern the time evolution of the Pedersen conductance and the plasma content of a flux tube. It is argued that temperate‐ and high‐latitude spread F are a result of this instability. The new moment equations also predict: (1) that there is a maximum height‐integrated Pedersen current that the F region can carry; and (2) that barium clouds deform in agreement with observations of recent releases.

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