A Landau fluid model for dissipative trapped electron modes is developed which focuses on an improved description of the ion dynamics. The model is simple enough to allow nonlinear calculations with many harmonics for the times necessary to reach saturation. The model is motivated by a discussion that starts with the gyro-kinetic equation and emphasizes the importance of simultaneously including particular features of magnetic drift resonance, shear, and Landau effects. To ensure that these features are simultaneously incorporated in a Landau fluid model with only two evolution equations, a new approach to determining the closure coefficients is employed. The effect of this technique is to reduce the matching of fluid and kinetic responses to a single variable, rather than two, and to allow focusing on essential features of the fluctuations in question, rather than features that are only important for other types of fluctuations. Radially resolved nonlinear calculations of this model, advanced in time to reach saturation, are presented to partially illustrate its intended use. These calculations have a large number of poloidal and toroidal harmonics to represent the nonlinear dynamics in a converged steady state which includes cascading of energy to both short and long wavelengths.

Several novel theoretical results related to L {yields} H transition physics, VH-mode evolution, Edge Localized Modes and active confinement control are presented. Critical issues are identified, results are discussed and important unresolved questions are listed. The basic physics is discussed in the contexts of current experiments and of ITER.

The results of detailed numerical studies of phenomena in negative compressibility turbulence with sheared perpendicular (i.e., poloidal) flow are presented. The turbulence model is based on the parallel ion flow gradient instability, a representative paradigm for ion drift waves. Studies of coupled turbulence and mean flow evolution indicate the existence of two distinct nonlinear states. In the first state, saturation occurs via nonlinear transfer to damped high-k modes and sheared flow is heavily damped. In the second state, the turbulence level is controlled by the self-consistently generated sheared flow. Transition between these states is determined by the competition between instability growth and damping of rotation. The dynamics of the observed transition is well described and consistent with a simple set of coupled envelope equations. Modulational interaction between small scale turbulence and large scale m≠0 shear flows is observed, as well.

The study of plasma turbulence and transport is a complex problem of critical importance for fusion-relevant plasmas. To this day, the fluid treatment of plasma dynamics is the best approach to realistic physics at the high resolution required for certain experimentally relevant calculations. Core and edge turbulence in a magnetic fusion device have been modeled using state-of-the-art, nonlinear, three-dimensional, initial-value fluid and gyrofluid codes. Parallel implementation of these models on diverse platforms--vector parallel (National Energy Research Supercomputer Center`s CRAY Y-MP C90), massively parallel (Intel Paragon XP/S 35), and serial parallel (clusters of high-performance workstations using the Parallel Virtual Machine protocol)--offers a variety of paths to high resolution and significant improvements in real-time efficiency, each with its own advantages. The largest and most efficient calculations have been performed at the 200 Mword memory limit on the C90 in dedicated mode, where an overlap of 12 to 13 out of a maximum of 16 processors has been achieved with a gyrofluid model of core fluctuations. The richness of the physics captured by these calculations is commensurate with the increased resolution and efficiency and is limited only by the ingenuity brought to the analysis of the massive amounts of data generated.

The universality of the observed characteristics of sheared flows points to a general ingredient to explain the damping/driving mechanisms responsible for the development of these flows in the plasma boundary region of fusion devices. Experiments in the TJ-II stellarator showing that the generation of spontaneous sheared flows at the plasma edge requires a minimum plasma density or density gradient, open a unique possibility to characterize the dynamics of sheared flow development in fusion plasmas.The effective viscosity at the plasma edge can be deduced by means of the decay rate of the perpendicular flow measurement once the driving force has been removed. Changes in the plasma rotation and turbulence have been studied when an electric field is externally applied at the plasma edge. The relaxation of flows and radial electric fields has been compared in the edge plasma region of TJ-II stellarator and CASTOR tokamak showing a striking similarity. The findings can help to test neoclassical and anomalous damping mechanisms in fusion plasmas.Finally, the emergence of the plasma edge sheared flow as a function of plasma density can be explained using a simple second-order phase transition model that reproduces many of the features of the TJ-II experimental data while capturing the qualitative features of the transition near the critical point.

Fast wave poloidal flow generation at the plasma edge of a tokamak is studied with a one-dimensional slab model. In the absence of mode conversion, the poloidal flow can be generated by a spatial change of plasma velocity and current density profiles due to strong minority ion power absorption at the minority ion cyclotron resonance. The electromagnetic force plays a more important role in the flow generation than does the plasma Reynolds stress. In the presence of mode conversion, the flow is mainly generated by interference between the long-wavelength transmitted fast wave and the short-wavelength ion Bernstein wave (IBW) from mode conversion. Flow shear generated in the presence of mode conversion varies spatially with a scale length similar to the IBW wavelength. With mode conversion, the plasma Reynolds stress becomes more important in the flow shear generation than the electromagnetic force. For both cases, the plasma Reynolds stress and the electromagnetic force are out of phase, so that the resultant flow shear is smaller than the larger of the two. The short scale length flow shear enhances the turbulence stabilization.

A simple model of long-wavelength drift waves is used to study the way in which a radial electric-field profile influences the growth and saturation of turbulence. For a fixed external field, the effect of curvature (∂2Er/∂r2) dominates that of shear (∂Er/∂r). In the linear regime, both affect the average k∥ at which ion damping occurs: shear by shifting the eigenmode off the resonant surface and curvature by changing the eigenmode width. Curvature damps more efficiently and also shifts the real frequency of the drift wave, changing the instability drive. In the nonlinear regime, radial trapping at large fluctuation levels limits the ability of an external electric-field profile to affect the spatial structure. Changes in damping are now less effective than the feedback between frequency shift and drive. The importance of the frequency shift caused by electric-field curvature in the presence of finite-amplitude fluctuations has been demonstrated by numerical calculations.

Three‐wave, nonlinear, tearing mode coupling has been measured in the Madison Symmetric Torus (MST) reversed‐field pinch (RFP) [Fusion Technol. 19, 131 (1991)] using bispectral analysis of edge magnetic fluctuations resolved in ‘‘k‐space.’’ The strength of nonlinear three‐wave interactions satisfying the sum rules m1+m2=m3 and n1+n2=n3 is measured by the bicoherency. In the RFP, m=1, n∼2R/a (6 for MST) internally resonant modes are linearly unstable and grow to large amplitude. Large values of bicoherency occur for two m=1 modes coupled to an m=2 mode and the coupling of intermediate toroidal modes, e.g., n=6 and 7 coupled to n=13. These experimental bispectral features agree with predicted bispectral features derived from magnetohydrodynamic (MHD) computation. However, in the experiment, enhanced coupling occurs in the ‘‘crash’’ phase of a sawtooth oscillation concomitant with a broadened mode spectrum suggesting the onset of a nonlinear cascade.

New information on magnetic fluctuations and transport in toroidal devices has been obtained in the MST reversed field pinch through measurement of nonlinear coupling of three waves in k-space, and measurement of current density fluctuations. Measurements of nonlinear coupling of magnetic fluctuations reveals that (1) two poloidal mode number m = 1 modes couple strongly to an m = 2 mode, (2) toroidal mode coupling is broad extending up to n = 20, (3) these features agree with predictions for tearing fluctuations from a nonlinear MHD code, (4) during a sawtooth crash the number of modes involved in nonlinear interactions increases dramatically and the k-spectrum broadens simultaneously. Measurements of current density fluctuations over the outer 20% of the minor radius reveal that (1) low frequency fluctuations are consistent with tearing modes, (2) high frequency fluctuations are localized turbulence which maintains resonance with the equilibrium field as q changes with radius, (3) particle transport from magnetic fluctuations is ambipolar (i.e., = O).

Dissipative trapped electron modes (DTEM) may be one of the causes of deterioration of confinement in tokamak and stellatator plasmas. We have implemented a fluid model to study DTEM turbulence in slab geometry. The electron dynamics include in addition to the adiabatic part, a non-adiabatic piece modeled with an i-delta-type response. The ion dynamics include Landau damping and FLR corrections through Landau fluid approximate techniques and Pade approximants for {Gamma}{sub 0}(b)=I{sub 0}(b)e{sup {minus}b}. The model follows from the gyrokinetic equation. Evolution equations, which closely resemble those used in standard reduced MHD, are presented since these are better suited to non-linear calculations. The numerical results of radially resolved calculations will be discussed. A recently developed hybrid model, which consists of a gyrokinetic implementation for the ions using particles and the same description for the electron dynamics as in the fluid model, will also be presented.

Measurements of edge turbulence and the associated transport are ongoing in the Madison Symmetric Torus (MST) reversed-field pinch [Fusion Technol. 19, 131 (1991)] using magnetic and electrostatic probes. Magnetic fluctuations are dominated by m=1 and n ∼2R/a, tearing modes. Particle losses induced by magnetic field fluctuations have been found to be ambipolar (〈J̃∥ B̃r〉/B0=0). Electrostatic fluctuations are broadband and turbulent, with mode widths Δm∼3–7 and Δn∼70–150. Particle, parallel current, and energy transport arising from coherent motion with the fluctuating Ẽ×B drift have been measured. Particle transport via this channel is comparable to the total particle loss from MST. Energy transport (from 〈P̃Ẽφ 〉/B0) due to electrostatic fluctuations is relatively small, and parallel current transport (from 〈J̃∥ Ẽφ〉/B0) may be small as well.

A simple, paradigmatic model of long-wavelength drift wave turbulence in the presence of a sheared poloidal flow is analyzed in detail. Linear theory predicts that velocity shear induces a strong stabilizing effect by shifting the eigenmode away from the k⋅B=0 resonant surface, thereby enhancing ion damping. However, multiple-helicity numerical calculations indicate that velocity shear has little or no effect on saturated fluctuation levels. Analysis suggests that this result is related to the incidence of a spiky, radially intermittent profile of the turbulent fluctuation levels, induced by low-q mode rational surfaces, and occurs when the turbulent diffusivity exceeds the product of diamagnetic frequency and gyroradius squared. An analytical theory that explains the observed suppression of velocity and magnetic shear damping is presented.

Three-wave nonlinear coupling of spatial Fourier modes is measured in the MST reversed field pinch by applying bispectral analysis to magnetic fluctuations measured at the plasma edge at 64 toroidal locations and 16 poloidal locations, permitting observation of coupling over 8 polodial modes and 32 toroidal modes. Comparison to bispectra predicted by resistive MHD computation indicates reasonably good agreement. However, during the crash phase of the sawtooth oscillation the nonlinear coupling is strongly enhanced, concomitant with a broadened (presumably nonlinearly generated) k spectrum.

Magnetic perturbations caused by external field errors are an important consideration in both the current operation and future design of reversed-field pinch (RFP) devices. Recently, it has been shown that perturbation profiles calculated in the presence of a relaxed RFP equilibrium (J0 =λB0, λ=const) are amplified when compared to those calculated in vacuum. This work extends the approach to include a partially relaxed equilibrium [λ=λ(r)], which is experimentally realistic, and the effects of finite resistivity, viscosity, and flow. The finite gradient in λ produces an amplification of the radial magnetic field perturbation much larger than that predicted for constant λ. Physically, this is due to the presence of an additional perturbed current associated with the gradient in λ which alters the curvature of the perturbation. Finite resistivity, viscosity, and flow act to reduce this amplification.

Magnetic perturbations caused by field errors may pose a threat to magnetic confinement. They can cause the formation of magnetic islands, which may increase transport. The response of a reversed-field pinch (RFP) plasma to field errors has been calculated numerically. A three-dimensional nonlinear magnetohydrodynamic code was used, with field errors specified as boundary conditions. Two types of field errors were studied: those arising from current flowing in the toroidal field coil set and those arising from induced currents in the conducting shell flowing around holes. Design data from two RFPs at the Los Alamos National Laboratory — ZT-40 (currently operating) and ZT-H (being designed) — were used. Results on the effect of the plasma on the radial field perturbation and on island size are given. The effects of the chosen equilibrium current profile, resistivity, and viscosity are discussed.