W. M. Tang

Microinstability theory in tokamaks

Significant investigations in the area of tokamak microinstability theory are reviewed. Special attention is focused on low-frequency electrostatic drift-type modes, which are generally believed to be the dominant tokamak microinstabilities under normal operating conditions. The basic linear formalism including electromagnetic (finite-beta) modifications is presented along with a general survey of the numerous papers investigating specific linear and non-linear effects on these modes. Estimates of the associated anomalous transport and confinement times are discussed, and a summary of relevant experimental results is given. Studies of the non-electrostatic and high-frequency instabilities associated with the presence of high-energy ions from neutral-beam injection (or with the presence of alpha-particles from fusion reactions) are also surveyed.

Toroidal drift modes driven by ion pressure gradients

Ion pressure gradient‐driven drift modes are analyzed for their parametric dependence on the shear, the toroidal aspect ratio, and the pressure gradient using the ballooning toroidal mode theory. An approximate formula for the anomalous ion thermal conductivity is derived for the turbulent regime.

Non-linear saturation of the dissipative trapped-ion mode by mode coupling

The non-linear saturation of the dissipative trapped-ion mode is analysed. The basic mechanism considered is the process whereby energy in long-wavelength unstable modes is non-linearly coupled via × convection to shortwavelength modes stabilized by Landau damping due to both circulating and trapped ions. In the usual limit of the mode frequency being small relative to the effective electron collision frequency, a one-dimensional non-linear partial differential equation for the potential can be derived, as was first shown by LaQuey, Mahajan, Tang, and Rutherford. The stability and accessibility of the possible equilibria for this equation are examined in detail, both analytically and numerically. The equilibrium emphasized by LaQuey et al. is shown to be unstable. However, a class of non-linear saturated states which are stable to linear perturbations is found. Included in the analysis are the effects of both ion collisions and dispersion due to finite-ion-banana-width effects. Cross-field transport is estimated and the scaling of the results is considered for tokamak parameters (specifically those for the Princeton Large Torus). It is concluded that the anomalous cross-field transport can be much lower than the estimate of Kadomtsev and Pogutse, for relevant parameters near marginal stability for the linear modes.

Shearing rate of time-dependent E×B flow

Theory of E×B shear suppression of turbulence in toroidal geometry [Phys. Plasmas 2, 1648 (1995)] 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.

Toroidal electron temperature gradient driven drift modes

The electron temperature gradient in tokamak geometry is shown to drive a short wavelength lower hybrid drift wave turbulence resulting from the unfavorable magnetic curvature on the outside of the torus. Ballooning mode theory is used to determine the stability regimes and the complex eigenfrequencies. At wavelengths of the order of the electron gyroradius, the polarization is electrostatic and the growth rate is greater than the electron transit time around the torus. At longer wavelengths of the order of the collisionless skin depth, the polarization is electromagnetic with electromagnetic vortices producing the dominant transport. The small scale electrostatic component of the turbulence produces a small, of order (me/mi)1/2, drift wave anomalous transport of both the trapped and passing electrons while the c/ωpe scale turbulence produces a neo‐Alcator [Nucl. Fusion 25, 1127 (1985)] type transport from the stochastic diffusion of the trapped electrons.

Kinetic-ballooning-mode theory in general geometry

A systematic procedure for studying the influence of kinetic effects on the stability of MHD ballooning modes is presented. The ballooning mode formalism, which is particularly effective for analysing high-mode-number perturbations of a plasma in toroidal systems, is used to solve the Vlasov-Maxwell equations for modes with perpendicular wavelengths on the scale of the ion gyroradius. The local stability on each flux surface is determined by the solution of three coupled integro-differential equations which include effects due to finite gyroradius, trapped particles, and wave-particle resonances. More tractable forms of these equations are then obtained in the low (ω < ωbi, ωti) and intermediate- (ωbi, ωti < ω < ωbe, ωte) frequency regimes with ωbj and ωtj being the average bounce and transit frequencies of each species. After further simplifying approximations, the kinetic results here are shown to be reducible to the MHD-ballooning-mode equations in the analogous limits, ω ωs where ωs = cs/Lc, with cs being the acoustic speed and Lc the connection length.

Electromagnetic kinetic toroidal eigenmodes for general magnetohydrodynamic equilibria

A comprehensive analysis of low‐frequency, high‐toroidal‐mode‐number linear eigenmodes for tokamaks is presented. The most significant new features of this stability study are that the calculation is interfaced with a general numerical magnetohydrodynamic equilibrium, and that it is fully electromagnetic. The ballooning formalism is employed and all important kinetic effects, including those of trapped particles, are retained. In particular, the familiar trapped‐electron drift‐wave frequency regime is considered and results are presented for three sequences of artificial equilibria; one of increasing β(≡plasma pressure/magnetic pressure) values; one of varying equilibrium shape, from inverse‐D to circular to normal‐D; and one of increasing vertical ellipticity. The analysis is then applied to a realistic (self‐consistent) high‐β equilibrium generated with data obtained from the ISX‐B tokamak experiment. Here it is found that for the usual trapped‐electron branch, kinetic microinstabilities appear to be pr...

Gyro-kinetic simulation of global turbulent transport properties in tokamak experiments

A general geometry gyro-kinetic model for particle simulation of plasma turbulence in tokamak experiments is described. It incorporates the comprehensive influence of noncircular cross section, realistic plasma profiles, plasma rotation, neoclassical (equilibrium) electric fields, and Coulomb collisions. An interesting result of global turbulence development in a shaped tokamak plasma is presented with regard to nonlinear turbulence spreading into the linearly stable region. The mutual interaction between turbulence and zonal flows in collisionless plasmas is studied with a focus on identifying possible nonlinear saturation mechanisms for zonal flows. A bursting temporal behavior with a period longer than the geodesic acoustic oscillation period is observed even in a collisionless system. Our simulation results suggest that the zonal flows can drive turbulence. However, this process is too weak to be an effective zonal flow saturation mechanism.

Gyrokinetic particle simulation of ion temperature gradient drift instabilities

Ion temperature gradient drift instabilities have been investigated using gyrokinetic particle simulation techniques for the purpose of identifying the mechanisms responsible for their nonlinear saturation as well as the associated anomalous transport. For simplicity, the simulation has been carried out in a shear‐free slab geometry, where the background pressure gradient is held fixed in time to represent quasistatic profiles typical of tokamak discharges. It is found that the nonlinearly generated zero‐frequency responses for the ion parallel momentum and pressure are the dominant mechanisms giving rise to saturation. This is supported by the excellent agreement between the simulation results and those obtained from mode‐coupling calculations, which give the saturation amplitude as ‖eΦ/Te‖ ≂(‖ωl+iγl‖/Ωi)/(k⊥ ρs)2, and the quasilinear thermal diffusivity as χi ≂γl/k2⊥, where ωl and γl are the linear frequency and growth rate, respectively, for the most unstable mode of the system. In the simulation, the ...

Collisional effects on kinetic electromagnetic modes and associated quasilinear transport

The general procedure for the analysis of low‐frequency electrostatic and electromagnetic modes in toroidal geometry is now well known. In the collisionless limit, the relevant dynamics (e.g., trapped particles, resonances, etc.) can be treated appropriately. However, with the introduction of collisional effects it is customary, for tractability, to employ model collision operators of varying degrees of complexity. Guided by results of earlier studies of alternative collision operators in unsheared slab geometry and in toroidal geometry, an improved model collision operator is introduced here for calculating toroidal eigenmodes. Analytic and numerical results are presented to support its relevance and to demonstrate its improvement over earlier models. The associated quasilinear particle and energy transport coefficients for each species are also calculated, and compared with the usual Dj≂κj≂γ/k2⊥ estimate.