This is a review of what is known about fluctuations and anomalous transport processes in tokamaks. It mostly considers experimental results obtained after, and not included in, the reviews of Liewer [Nucl. Fusion 25, 543 (1985)], Robinson [in Turbulence and Anomalous Transport in Magnetized Plasmas (Ecole Polytechnique, Palaiseau, France, 1986), p. 21], and Surko [in Turbulence and Anomalous Transport in Magnetized Plasmas (Ecole Polytechnique, Palaiseau, France, 1986), p. 93]. Therefore much of the pioneering work in the field is not covered. Emphasis is placed on results where comparisons between fluctuations and transport properties have been attempted, particularly from the tokamak TEXT [Nucl. Technol./Fusion 1, 479 (1981)]. A brief comparison of experimentally measured total fluxes with the predictions of neoclassical theory demonstrates that transport is often anomalous; fluctuations are thought to be the cause.The measurements necessary to determine any such fluctuation‐driven fluxes are described. The diagnostics used to measure these quantities, together with some of the statistical techniques employed to analyze the data, are outlined. In the plasma edge detailed measurements of the quantities required to directly determine the fluctuation‐driven fluxes are available. The total and fluctuation‐driven fluxes are compared: the result emphasizes the importance of edge turbulence. No model adequately describes all the measured properties. In the confinement region experimental observations are presently restricted to measurements of density and potential fluctuations and their correlations. Various distinct turbulence features that have been observed are described, and their characteristics compared with the predictions of various models. Correlations observed between these fluctuations and plasma transport properties are summarized. A separate section on magnetic fluctuations shows there is very little information available inside the plasma, generally prohibiting quantified comparisons between fluctuation levels and transport. Both coherent and turbulent magnetic fluctuations are addressed, and the differences between low and high plasma pressure (low and high beta) are noted. The contributions of alternate confinement devices, such as stellarators and reversed field pinches, to understanding tokamak anomalous transport are discussed. Finally, future directions are proposed.

Fluctuations and anomalous transport in tokamaks

Nonlinear gyrofluid equations are derived by taking moments of the nonlinear, electrostatic gyrokinetic equation. The principal model presented includes evolution equations for the guiding center n, u∥, T∥, and T⊥ along with an equation expressing the quasineutrality constraint. Additional evolution equations for higher moments are derived that may be used if greater accuracy is desired. The moment hierarchy is closed with a Landau damping model [G. W. Hammett and F. W. Perkins, Phys. Rev. Lett. 64, 3019 (1990)], which is equivalent to a multipole approximation to the plasma dispersion function, extended to include finite Larmor radius effects (FLR). In particular, new dissipative, nonlinear terms are found that model the perpendicular phase mixing of the distribution function along contours of constant electrostatic potential. These ‘‘FLR phase‐mixing’’ terms introduce a hyperviscositylike damping ∝k⊥2‖Φkk×k’‖, which should provide a physics‐based damping mechanism at high k⊥ρ which is potentially as important as the usual polarization drift nonlinearity. The moments are taken in guiding center space to pick up the correct nonlinear FLR terms and the gyroaveraging of the shear. The equations are solved with a nonlinear, three‐dimensional initial value code. Linear results are presented, showing excellent agreement with linear gyrokinetic theory.

/pdf/gyrofluid-turbulence-models-with-kinetic-effects-4u25afimnn.pdf

Gyrofluid turbulence models with kinetic effects

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.

http://iopscience.iop.org/article/10.1088/0029-5515/18/8/006/pdf

Microinstability theory in tokamaks

The stability of the ion temperature gradient modes is investigated using the kinetic ion response without expansions in ωD/ω. A systematic parameter study is carried out using a low‐beta circular flux surface equilibrium in order to determine the stability boundaries in ηi vs en space (ηi=d ln Ti/ d ln n, en=Ln/R). Particular attention is devoted to the consequences of the presence of these modes for anomalous ion transport.

Ion temperature-gradient-driven modes and anomalous ion transport in tokamaks

Recent results in the theory of turbulent momentum transport and the origins of intrinsic rotation are summarized. Special attention is focused on aspects of momentum transport critical to intrinsic rotation, namely the residual stress and the edge toroidal flow velocity pinch. Novel results include a systematic decomposition of the physical processes which drive intrinsic rotation, a calculation of the critical external torque necessary to hold the plasma stationary against the intrinsic residual stress, a simple model of net velocity scaling which recovers the salient features of the experimental trends and the elucidation of the impact of the particle flux on the net toroidal velocity pinch. Specific suggestions for future experiments are offered.

/pdf/physics-of-non-diffusive-turbulent-transport-of-momentum-and-2w6s7so2kq.pdf

Physics of non-diffusive turbulent transport of momentum and the origins of spontaneous rotation in tokamaks

An integral equation governing an instability due to ion temperature gradients is derived. In the presence of magnetic shear, localized non‐convective normal modes of instability are shown to exist if the relative temperature gradient is larger than that of density, unless the shear is exceedingly strong, i.e., the field shears through a large angle in the distance in which the temperature drops. Quasi‐modes which are less localized in the direction of the gradient can be constructed from these normal modes and a large thermal diffusion may be expected. Conversely the mass diffusion is shown to be rather slow so that it is reasonable to assume that an effective ``divertor'' should keep the actual heat loss quite small.

/pdf/instabilities-due-to-temperature-gradients-in-complex-1v5qquuf72.pdf

B. Coppi

Papers

Instabilities due to Temperature Gradients in Complex Magnetic Field Configurations

Drift Instability Due to Impurity Ions

Stabilization by shear and negative V″ (CN-21/106)