Search

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.

This review examines the nonlinear dynamics of a plasma near marginal stability where there is a balance between the destabilizing resonant kinetic drive and the stabilizing dissipation in the background plasma. Depending on the degree of extrinsic stochasticity of the resonant particles, different saturated states are analytically described. At low enough stochasticity, an explosive self‐similar solution is predicted. Numerical simulation shows this explosive behavior is a precursor to the formation of phase space structures in the form of holes and clumps, which cause up an down shifted frequency sweeping due to plasma dissipation. Experimental evidence for the predicted behavior is discussed.

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)].

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 damping of Toroidal Alfven Eigenmodes in JET 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.

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 problem of a weak source of particles that forms a distribution function that is unstable to a discrete number of modes with the electrostatic bump‐on‐tail instability taken as a paradigm is considered. Over a wide range of parameters the system produces pulsations, where there are relatively brief bursts of waves separated by longer intervals of quiescent behavior. 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 reaches the level where orbit stochasticity occurs, the particle diffusion leads to a further conversion of the distribution’s free energy to wave energy. This leads to a rapid quaslinear relaxation (a phase space explosion) of the distribution function. Hence the overall response of the system is characterized b...

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)

The theory for the dissipative trapped particle modes is developed for a tandem mirror taking into account equilibrium rotational effects, electron temperature gradients, and an axial ambipolar potential, for a broad range of collisional parameters. The electrons are treated as a Maxwellian plasma that occupies the central cell and anchor cell regions. It is assumed that the eigenfunction is piecewise constant with abrupt transitions between the anchor and central cell regions. It is found that when ω/νp≳1, with ω the mode frequency and νp the Pastukhov loss rate, that the energy conservation structure of the collision operator produces important changes to previously developed theories. A solution to the problem is achieved by using the solution for the lifetime of an electron in an ambipolar trap, taking into account the global energy conservation. The energy conservation structure also allows a self‐consistent description of dissipative instabilities when thermal gradient and electric fields are presen...

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

Injection of a superthermal ion component into a toroidal plasma can raise the fusion power density to levels considerably higher than those attainable in ordinary one-component plasmas. As a result, energy break-even in D-T can be attained in bulk plasmas with ntau-values as low as 10$sup 13$ cm$sup - 3$.s, at temperatures of approximately 5 keV. The optimal energy of the superthermal ion component is in the range 100 to 200 keV; injection-energy requirements can be reduced by adiabatic compression prior to or during the reaction phase. The appropriate ratio anti GAMMA of fast-ion pressure to bulk- plasma pressure is about 0.5. The stability of two-component tokamak plasmas was examined, and is found to be favorable for parameters of practical interest. (auth)

Electron bounce modes can occur in mirror plasmas when the spread in the bounce frequency is small. This condition is satisfied in mirror devices when electrons are principally confined by an approximately quadratic electrostatic potential. These modes are examined by numerically solving an integral equation for the perturbed wave potential in a mirror plasma. A long wavelength mode is found that can be destabilized by ions because of the loss-cone nature of their distribution. Threshold densities and maximum growth rates are calculated using a perturbation method. The theoretical stability threshold predictions agree closely with Baseball II measurements.

The integral equation for the magnetic compressional mode, accounting for geometrical effects along the field line and using the eikonal approximation across the field line, is solved numerically for the eigenvalues and eigenfunctions. These results reproduce the analytic estimates when there is strong drift reversal. The representation of the eigenfunction of the form B∥=[C(ψ)/B] ×(dP⊥h/dψ) is found to give accurate growth rates over a large range of parameter values. For typical EBT‐S [Plasma Phys. 25, 597 (1983)] parameters, instability is predicted for all pressure scale lengths just below those needed for drift reversal, i.e., ‖R ∂(Pc) +P⊥h)/2B2 ∂r‖>1 (where P is the article pressure, c and h refer to cold and hot components, B is the midplane magnetic field, and R is the midplane radius of curvature). If larger core densities are present, a wave–particle resonance arises when the particle drifts are not reversed, causing instability up to much larger pressure scale lengths. Stability for all values of the ratio of hot electron density to core density is obtained with ‖R ∂Pc/B2 ∂r‖>1+P∥h/P⊥h.

The fraction of warm plasma needed to stabilize the drift cyclotron loss‐cone mode in a mirror machine is computed. With warm plasma present, a remnant short wavelength instability persists.

Distributions that give rise to energetic trapped particle pressures peaked in the ‘‘good curvature’’ region of a tokamak (sloshing distributions) are examined in an attempt to find stable regimes for both the magnetohydrodynamic (MHD) and precessional modes. It is the precessional drift destabilization of ballooning modes that inhibits bridging the unstable gap to second stability by the use of deeply trapped energetic particles unless the hot particles have an extremely large energy (∼0.35 MeV for a tokamak like PDX [Phys. Rev. Lett. 49, 326 (1982)]). Unfortunately, our calculations indicate that the sloshing particles do not have a significant stabilizing effect. An analytic treatment shows that stability for the precessional mode can be found only if the sign of the energetic particle magnetic drift frequency can be reversed from its value in vacuum bad curvature without hot species diamagnetism. This is difficult to do in a tokamak because of the destabilizing contribution of the geodesic curvature t...

It is shown that for trapped particle modes in tandem mirrors, the pressure of the passing particles in the anchor region introduces a stabilizing term proportional to the sum of the anchor’s field line curvature and total diamagnetic pressure gradient. The theory is applied to the proposed gas dynamic trap experiment.

Difficulties in the stabilization of ideal magnetohydrodynamic (MHD) ballooning modes by finite Larmor radius (FLR) effects are considered, in tandem mirror geometry for azimuthal mode numbers l>1. A kinetic formalism is used to obtain corrections to the long, thin approximation, when keeping terms of quadratic order in the curvature. If ηi =∂ ln Ti /∂ ln ni ≥0, with Ti, ni, the ion temperature and density, ion resonance effects eliminate absolute FLR stability, though the residual growth rates are substantially reduced from the MHD values. However, the residual modes are still important, and mixing length estimates of the confinement degradation from modes with l>l indicate they can still severely limit the achievement of reactor‐grade operation near and above the threshold beta values predicted from the ideal MHD theory. This is most severe if the ion temperature decreases radially (ni>0), whereupon significant instabilities can even arise below the ideal threshold. However, if −2/3

The stability of an axisymmetric magnetic‐mirror cell surrounded by a conducting wall is studied using the sharp‐boundary model. Depending on the ion pressure model p(B), magnetohydrodynamic (MHD) stability can be achieved in certain ranges at either high or low β. Suppression of the Alfven ion‐cyclotron microinstability requires the mirror cell to be short (length≲20 ion Larmor radii). Adiabaticity of ion orbits is then found to be a constraint that is difficult to satisfy. Within the approximations used in this paper, satisfaction of all constraints is achieved in only a small region of parameter space.

A quasi‐linear model to describe the evolution of a mirror plasma is presented. It is shown that, as in the 2XIIB mirror experiment, a plasma stream sufficiently reduces rf turbulence so that a high density steady state mirror plasma can be sustained by a neutral beam. The scaling law for this plasma configuration is derived. The energy lifetime is found to be the Spitzer electron‐ion drag time even when electron thermal conduction to the external plasma can be neglected.

Whereas plasma return currents along a magnetic field arise for an electromagnetic diffusion time, it is found that appreciable plasma return currents perpendicular to a magnetic field are not induced for built‐up times slow compared with an Alfven transit time.

An analytic expression giving the drift‐cone stability criterion of a finite‐beta plasma is derived using a mirror‐ratio‐dependent loss‐cone ion distribution. This is used to determine a beta limit for stability which is found to be insensitive to changes in mirror ratio and plasma temperature.