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A method is developed to analytically determine the resonance broadening function in quasilinear theory from first principles, due to either Krook or Fokker-Planck scattering collisions of marginally unstable plasma systems where discrete resonance instabilities are excited without any mode overlap. It is demonstrated that a quasilinear system that employs the calculated broadening functions reported here systematically recovers the growth rate and mode saturation levels for near-threshold plasmas previously calculated from nonlinear kinetic theory. The distribution function is also calculated, which enables precise determination of the characteristic collisional resonance width.

Modulation of mode amplitude and frequency of TAE modes, observed experimentally and referred to as chirping, is investigated using a guiding center code and a δf formalism. Chirping is observed as the development in time of Fourier sidebands that move above and below the nominal mode frequency. Subsequent doubling of the sidebands is also sometimes observed. Equilibria with conventional positive magnetic shear are used, as well as NSTX reversed shear cases. The onset of chirping can be triggered by a sudden increase in mode damping, as can occur by the mode contacting the continuum.

The linear growth rates of TAE (toroidal Alfven eigenmode) modes destabilized by ICRF (ion cyclotron range of frequency) heating are calculated over a range of plasma parameters. Nonlinear saturation of a single unstable mode is investigated both analytically and numerically when wave–particle trapping is the dominant saturation mechanism. A numerical code has been developed based on a reduced resonance description of the wave–particle interaction (using a Hamiltonian formalism). A delta-f algorithm was incorporated to allow a low-noise description of mode evolution with particle sources and sinks present. The numerically observed saturation amplitudes correlate well with theoretical predictions to within 20%. Self-excited frequency sweeping resulting from the excitation of many simultaneous wave–particle resonances at different energies is demonstrated and explained as an extension of previous published theory [Berk et al., Phys. Lett. A 234, 213 (1997)].

Remarkable progress has been made in diagnosing energetic particle instabilities on present-day machines and in establishing a theoretical framework for describing them. This overview describes the much improved diagnostics of Alfven instabilities and modelling tools developed world-wide, and discusses progress in interpreting the observed phenomena. A multi-machine comparison is presented giving information on the performance of both diagnostics and modelling tools for different plasma conditions outlining expectations for ITER based on our present knowledge., 37 pages, 20 figures

The theory of flute-mode stability is developed for a two-energy-component plasma partially terminated by a conducting limiter. The formalism is developed as a preliminary study of the effect of end loss in open-ended mirror machines, where large-Larmor-radius effects are important.

The theory of toroidicity‐induced Alfven eigenmodes (TAE) and kinetic TAE (KTAE) is generalized to arbitrary mode numbers for a large aspect ratio low‐beta circular tokamak. The interaction between nearest neighbors is described by a three‐term recursion relation that combines elements from an outer region, described by the ideal magnetohydrodynamic equations of a cylinder, and an inner region, which includes the toroidicity and the nonideal effects of finite ion Larmor radius, electron inertia, and collisions. By the use of quadratic forms, it is proven that the roots of the recursion relation are stable and it is shown how perturbation theory can be applied to include frequency shifts due to other kinetic effects. Analytic forms are derived which display the competition between the resistive and radiative damping, where the radiation is carried by kinetic Alfven waves. When the nonideal parameter is small, the KTAE modes appear in pairs. When this parameter is large, previously found scaling for the single gap case is reproduced analytically.

Recently a new point of view has been developed for describing saturation of discrete modes excited by weak sources. The method applies to the evolution of energetic particles in the beam plasma instability as well as to the description of how alpha particles evolve when they destabilize Alfven waves under reactor conditions. Over a wide range of parameters the system produces pulsations, where there are relatively brief bursts of wave energy separated by longer intervals of quiescence. There are two types of pulsations: benign and explosive. In the benign phase, valid when particle motion is not stochastic, the distribution function is close to that predicted by classical transport theory, and the instability saturates when the wave trapping frequency equals the expected linear growth rate. If the field amplitude in a burst reaches the level where orbit stochasticity occurs, the quasilinear diffusion causes rapid transfer of particle energy to wave energy and rapid flattening of the particle distribution function. The bursting phase is followed by a relatively long quiescent time interval, where the source provides the necessary free energy to regenerate the cycle. The critical issue is whether the instability develops to a high enough level to produce stochastic diffusion. In general, this question can be assessed by using mapping methods to obtain criteria of overlapping of orbit resonances. If overlap occurs, then the modes will saturate at a high level, which will result in significant anomalous transport effects. This picture is consistent with recent observations of energetic beam losses in tokamak experiments due to Alfven mode excitation.

The variational formulation for a fluid plasma including the parallel and gyroviscosities is developed using the basic approach of Berk et al. [Phys. Fluids 24, 2245 (1981)]. The equivalence of the variational problem to the original viscous fluid equations of motion is shown. The theory is developed for an axisymmetric plasma with no magnetic field in the azimuthal direction and therefore applies to field‐reversed configurations and axisymmetric mirrors. This theory offers the advantage of describing both parallel and transverse ion kinetic effects within the simplicity afforded by a variational fluid model.

The description of stability of the gas dynamic trap is shown to be extremely sensitive to the nature of the boundary conditions. Two model boundary conditions in a moderately long mean‐free‐path limit are considered: insulating boundary conditions and conducting boundary conditions. The former boundary condition reproduces the magnetohydrodynamic (MHD) results of previous studies, where the outflow of ions contribute to the MHD stabilizing properties of the system. However, with conducting boundary conditions it is shown that the outflowing ions do not contribute to the system’s stability. In this case, which is likely to be physically relevant, the MHD stabilizing term only comes from the electron pressure in the expansion region, and the gas dynamic trap would not be as stable as previously envisioned. The physical difference in these two boundary conditions is attributed to a passive feedback mechanism from the oscillating edge potentials that is only present in the insulating boundary case.

H. L. Berk, Physics of Fluids B: Plasma Physics 2, 3212 (1990) https://doi.org/10.1063/1.859232, The general expression of the power transfer from a high-energy ion beam to a background electrostatic plasma wave is obtained for arbitrary wave amplitude. It is verified that phase space gradients produced by a finite amplitude wave enhance the power transfer significantly. © 1990 American Institute of Physics.

The damping of Toroidal Alfven Eigenmodes in Joint European Torus (JET) [P. H. Rebut and B. E. Kenn, Fusion Technol.FUSTE811, 13 (1987)] plasmas is investigated by using a reduced kinetic model. Typically no significant damping is found to occur near the center of the plasma due to mode conversion to kinetic Alfven waves. In contrast, continuum damping from resonance near the plasma edge may be significant, and when it is, it gives rise to damping rates that are compatible with the experimental observations.

Ya. I. Kolesnichenko and his cowerkers observed that the alpha particle energy losses due to Alfven wave excitation in an ignited plasma did not exceed 25% of the total alpha particle energy. (AIP)

A variational method is presented to derive the electromagnetic response for waves of a model plasma with arbitrary β in a magnetic field whose curvature simulated by a gravitational field. The variational method is particularly well suited for deriving the electromagnetic local dispersion relation and differential equations of long-wavelength modes. In particular, the result is applied to finding the critical β needed for stabilization of low-frequency drift waves with density and temperature gradients.

The diochotron instability of a thin, tenuous, cylindrical layer of charged particles, whose gyro-radius is of the order of the mean radius of the cylindrical layer, is investigated using the Vlasov equation. Two distribution functions are considered which give practically the same density in the physical space, but are quite different in the velocity space: in the first the velocity spread is practically zero; in the second the particles oscillate around the mean radius of the cylindrical layer. It is shown that the first distribution exhibits the usual diochotron instability, while the second one does not. It is also shown that, however small the velocity spread, the thickness of the cylindrical layer cannot go to zero. Its minimum value is roughly determined by equating the plasma frequency to the cyclotron frequency.

The problem of the stability of inhomogeneous, electrostatic, multiple water-bag plasmas is considered. Equations are derived for general stationary water-bag equilibria, as well as for the corresponding perturbations. Particular attention is directed to systems with trapped particles in periodic equilibria, and special boundary conditions for the perturbation equations at the trapped-particle turning points are introduced. A normal-mode analysis is carried out for a configuration involving trapped particles occupying a finite region in the vicinity of the trough of an equilibrium wave (BGK mode). The results confirm the validity of the bunched-beam approximation.

This paper treats topics in the linear theory of dissipative trapped- electron instabilities, and in the nonlinear theory of the dissipative trapped- ion instabilities. Stability criteria are obtained for dissipative trapped- electron modes in a tokamak-type magnetic field with shear. In the important case where the mode is driven by the electron temperature gradient, and neglecting ''ballooning'' effects, shear is found to stabilize the mode if typically r/sub n//L/sub s/ greater than 0.1 (r/2R)/sup 1/2/ dln T/sub e//dln n, a condition that can be attained. With weak shear, it is shown, however, that the mode-structure along the field can exhibit substantial ''ballooning''. The dissipative trapped-ion mode, which is not shear-stabilized, is considered in its nonlinear regime. Including nonlinear E x B motions, exact stationary trapped- ion-mode solitary-wave solutions are found, in which the wave energy in longer azimuthal wavelengths passes to shorter-wavelength components that are strongly damped by ion bounce resonances; the amplitudes are sufficiently reduced that a diffusion lower than Kadomtsev's is indicated. As a second nonlinear mechanism, anomalous velocity-space scattering due to the development of an unstable loss- cone-type ion distribution function is considered; this mechanism raises the effective ion collision frequency and increases the stability of themore » longer- azimuthal-wavelength modes. (auth)« less

The marginal stability of long thin field‐reversed plasmas to axisymmetric tearing modes is examined. It is found that stability improves greatly as flux is excluded from the plasma. Such naturally occurring states may account for the lack of tearing mode activity in many experiments. Consideration is also given as to whether, in three dimensions, the tearing mode is an instability or a condition for the breakdown of equilibrium. An example is given that demonstrates that it is an instability. Idealized analytic results and more realistic numerical results are presented.