A new procedure for calculating the nonlinear energy transfer and linear growth/damping rate of fully developed turbulence is derived. It avoids the unphysically large damping rates typically obtained using the predecessor method of Ritz ͓Ch. P. Ritz, E. J. Powers, and R. D. Bengtson, Phys. Fluids B 1, 153 ͑1989͔͒. It enforces stationarity of the turbulence to reduce the effects of noise and fluctuations not described by the basic governing equation, and includes the fourth-order moment to avoid the closure approximation. The new procedure has been implemented and tested on simulated, fully developed two-dimensional ͑2-D͒ turbulence data from a 2-D trapped-particle fluid code, and has been shown to give excellent reconstructions of the input growth rate and nonlinear coupling coefficients with good noise rejection. However, in the experimentally important case where only a one-dimensional ͑1-D͒ averaged representation of the underlying 2-D turbulence is available, this technique does not, in general, give acceptable results. A new 1-D algorithm has thus been developed for analysis of 1-D measurements of intrinsically 2-D turbulence. This new 1-D algorithm includes the nonresonant wave numbers in calculating the bispectra, and generally gives useful results when the width of the radial wave number spectrum is comparable to or less than that of the poloidal spectrum.

The Tokamak Fusion Test Reactor ͑TFTR͒ ͑R. J. Hawryluk, to be published in Rev. Mod. Phys.͒ experiments on high-temperature plasmas, that culminated in the study of deuterium-tritium D-T plasmas containing significant populations of energetic alpha particles, spanned over two decades from conception to completion. During the design of TFTR, the key physics issues were magnetohydrodynamic ͑MHD͒ equilibrium and stability, plasma energy transport, impurity effects, and plasma reactivity. Energetic particle physics was given less attention during this phase because, in part, of the necessity to address the issues that would create the conditions for the study of energetic particles and also the lack of diagnostics to study the energetic particles in detail. The worldwide tokamak program including the contributions from TFTR made substantial progress during the past two decades in addressing the fundamental issues affecting the performance of high-temperature plasmas and the behavior of energetic particles. The progress has been the result of the construction of new facilities, which enabled the production of high-temperature well-confined plasmas, development of sophisticated diagnostic techniques to study both the background plasma and the resulting energetic fusion products, and computational techniques to both interpret the experimental results and to predict the outcome of experiments.

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The application of a new bispectral analysis technique to density fluctuation measurements in the core of the Tokamak Fusion Test Reactor indicates that the peak in the autopower spectrum usually lies in a region of linear stability. Large changes in the linear and nonlinear characteristics of the turbulence are observed as the plasma toroidal rotation and/or confinement properties are varied, while estimates of the turbulence-driven diffusivity varies only slightly with rotation. These observations are consistent with the operation of a global organizing property that may be related to the observation of Bohm-like scaling of ion thermal transport. [S0031-9007(97)03632-6] PACS numbers: 52.55. Fa, 52.25.Gj, 52.35.Ra A central scientific issue for the study of confined hightemperature plasmas in toroidal magnetic confinement systems is to understand and determine the cause and rate of cross-field transport. While a number of theories have been developed to identify the underlying microturbulence thought to cause this anomalous transport, only a limited number of experiments have been performed to directly characterize the properties of the turbulence and thus challenge these identifications [1]. To date, most experimental measurements in tokamak plasmas have concentrated on direct observables of the local turbulence, such as the amplitude of the fluctuations, the autopower spectrum, and correlation lengths. However, a more definitive comparison with theory lies in the determination of the linear growth rate and nonlinear coupling between turbulent modes in the plasma.This Letter reports the first experimental estimation of nonlinear energy transfer between modes of different spatial wave number in fully developed density turbulence in a high temperature magnetically confined plasma. The measurements were made with Beam Emission Spectroscopy (BES) [2] in the core of the Tokamak Fusion Test Reactor (TFTR) [3]. Estimations of the nonlinear energy transfer rate were obtained from this data by employing a novel nonresonant one-dimensional (1D) analysis algorithm [4,5] which has been developed to statistically calculate the amount of the nonlinear energy transfer between modes and the linear growth/damping rate of each mode. The results of these observations and analyses for turbulence in the hot plasma core region contradict the usual assumption that the peak in the turbulence power spectrum lies in the region of maximum linear growth rate.The bispectral analysis algorithm, an extension of the earlier method of Ritz et al. [6,7], utilizes the single-field nonlinear drift wave coupling model [4-7] as a governing equation. This governing equation is appropriate to describe stationary turbulence governed by the balance of linear growth and nonlinear three-wave coupling interactions in wave number space. The algorithm enforces stationarity, or statistical ergodicity, of the turbulence to eliminate the effects of noise and fluctuations not described by the governing equation, which otherwise give unphysically large dampin...