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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.

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.

A discussion of the proper homogeneous Lorentz transformation operator eL=exp[−ω⋅S−ξ⋅K] is given where eL transforms coordinates of an observer O to those of an observer O′. Two methods of evaluation are presented. The first is based on a dynamical analog. It is shown that the transformation can be evaluated from the set of equations that are identical to the set of equations that determine the four-velocity of a charged particle in response to a combined spatially uniform and temporally constant electric field E and magnetic field B, where E is parallel to ξ and B is antiparallel to ω, and E/B=ξ/ω. The principal difference in the two problems is that in the dynamics problem, the initial conditions for the four-velocity u must satisfy the constraint, uu=1, whereas the inner product of the coordinates acted on by eL can have any real value. In order to evaluate eL, one can then apply the simplifying techniques of transforming to the frame where E is parallel or antiparallel to B, whereupon the transformation eL in this special frame is trivially evaluated. Then we transform back to the original frame. We determine the β and the rotation Ω that results from a successive boost and rotation that the operator eL produces. A second method is based on a direct summation of the power series of the matrix elements of eL that has been used in relativistic quantum theory. The summation is facilitated by observing that the operators J±≡K±iS commute with each other, and can be represented in terms of the Pauli spin matrices. Indeed, we can reduce the Lorentz transformation to the product of spinor operators to give a compact way to compute the elements of the Lorentz operator eL.

Stability criteria for a weak relativistic beam-plasma interaction in a strong magnetic field are found. Two beam modes occur,ω≈k zC andω≈k zC-ωc. The dispersion equation of electrostatic two-stream is exactly solved with analytical method.

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.

A generic model of a kinetic plasma formed from a source and sink is presented which without instability would form a strongly unstable state due to a single mode. Instead, the resulting wave-particle resonant interaction maintains the distribution near a marginally stable state through the continual production of fast frequency-sweeping modes that sweep unidirectionally (upward in our case) throughout the energy-inverted region of the distribution function. The energy of these modes can be channeled to the background plasma through wave dissipation and, in our particular example, one quarter of the injected energy is available to be channeled.

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.

An overview is presented of the principles of magnetic confinement of plasmas for the purpose of achieving controlled fusion conditions. Sec. 1 discusses the different nuclear fusion reactions which can be exploited in prospective fusion reactors and explains why special technologies need to be developed for the supply of tritium or {sup 3}He, the probable fuels. In Sec. 2 the Lawson condition, a criterion that is a measure of the quality of confinement relative to achieving fusion conditions, is explained. In Sec. 3 fluid equations are used to describe plasma confinement. Specific confinement configurations are considered. In Sec. 4 the orbits of particle sin magneti and electric fields are discussed. In Sec. 5 stability considerations are discussed. It is noted that confinement systems usually need to satisfy stability constraints imposed by ideal magnetohydrodynamic (MHD) theory. The paper culminates with a summary of experimental progress in magnetic confinement. Present experiments in tokamaks have reached the point that the conditions necessary to achieve fusion are being satisfied.

A study is presented for the stability of alpha-particle driven shear Alfven Eigenmodes (AE) for the normal parameters of the three major burning plasma proposals, ITER (International Thermonuclear Experimental Reactor), FIRE (Fusion Ignition Research Experiment), and IGNITOR (Ignited Torus). A study of the JET (Joint European Torus) plasma, where fusion alphas were generated in tritium experiments, is also included to attempt experimental validation of the numerical predictions. An analytic assessment of Toroidal AE (TAE) stability is first presented, where the alpha particle beta due to the fusion reaction rate and electron drag is simply and accurately estimated in 7-20 keV plasma temperature regime. In this assessment the hot particle drive is balanced against ion-Landau damping of the background deuterons and electron collision effects and stability boundaries are determined. Then two numerical studies of AE instability are presented. In one the High-n stability code HINST is used . This code is capable of predicting instabilities of low and moderately high frequency Alfven modes. HINST computes the non-perturbative solution of the Alfven eigenmodes including effects of ion finite Larmor radius, orbit width, trapped electrons etc. The stability calculations are repeated using the global code NOVAK. We show that for these tokamaks themore » spectrum of the least stable AE modes are TAE that appear at medium-/high-n numbers. In HINST TAEs are locally unstable due to the alphas pressure gradient in all the devices under the consideration except IGNITOR. However, NOVAK calculations show that the global mode structure enhances the damping mechanisms and produces stability in all configurations considered here. A serious question remains whether the perturbation theory used in NOVAK overestimates the stability predictions, so that it is premature to conclude that the nominal operation of all three proposals are stable to AEs. In addition NBI ions produce a strong stabilizing effect for JET. However, in ITER the beam energies needed to penetrate to the core must be high so that a diamagnetic drift frequency comparable to that of the alpha particles is produced by the beam ions which induces a destabilizing effect.« less

A model is presented for determining the hyper-resistivity coefficient that arises due to the presence of magnetic structures that appear in plasma configurations such as the reversed field pinch and spheromak. Emphasis is placed on modeling cases where magnetic islands pass from non-overlap to overlap regimes. Earlier works have shown that a diffusion-based model can give realistic transport scalings when magnetic islands are isolated, and this formalism is extended to apply to the hyper-resistivity problem. In this case electrons may either be in long or short mean-free-path regimes and intuitively-based arguments are presented of how to extend previous theories to incorporate this feature in the presence of magnetic structures that pass from laminar to moderately chaotic regimes.

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.

A nonlinear theory of kinetic instabilities near threshold [H.L. Berk, et al., Plasma Phys. Rep. 23, (1997) 842] is applied to calculate the saturation level of Toroidicity-induced Alfven Eigenmodes (TAE) and be compared with the predictions of (delta)f method calculations [Y. Chen, Ph.D. Thesis, Princeton University, 1998]. Good agreement is observed between the predictions of both methods and the predicted saturation levels are comparable with experimentally measured amplitudes of the TAE oscillations in TFTR [D.J. Grove and D.M. Meade, Nucl. Fusion 25, (1985) 1167].

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...

An analytical model and a numerical procedure are presented which give a kinetic nonlinear description of the Alfven‐wave instabilities driven by the source of energetic particles in a plasma. The steady‐state and bursting nonlinear scenarios predicted by the analytical theory are verified in the test numerical simulation of the bump‐on‐tail instability. A mathematical similarity between the bump‐on‐tail problem for plasma waves and the Alfven wave problem gives a guideline for the interpretation of the bursts in the wave energy and fast particle losses observed in the tokamak experiments with neutral beam injection.

An overview is presented of the principles of magnetic confinement of plasmas for the purpose of achieving controlled fusion conditions. Sec. 1 discusses the different nuclear fusion reactions which can be exploited in prospective fusion reactors and explains why special technologies need to be developed for the supply of tritium or {sup 3}He, the probable fuels. In Sec. 2 the Lawson condition, a criterion that is a measure of the quality of confinement relative to achieving fusion conditions, is explained. In Sec. 3 fluid equations are used to describe plasma confinement. Specific confinement configurations are considered. In Sec. 4 the orbits of particle sin magneti and electric fields are discussed. In Sec. 5 stability considerations are discussed. It is noted that confinement systems usually need to satisfy stability constraints imposed by ideal magnetohydrodynamic (MHD) theory. The paper culminates with a summary of experimental progress in magnetic confinement. Present experiments in tokamaks have reached the point that the conditions necessary to achieve fusion are being satisfied.

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 WKB theory for vector systems of integral equations is developed herein. A variational technique is used to derive the equations for the WKB amplitudes in x‐space or its dual k‐space. Compact, explicit solutions are obtained in one dimension. When a solution breaks down at a turning point, the dual‐space representation can be used to derive the connection formulas between WKB solutions. These connection formulas are equivalent to the rules of the Furry method. The Furry method is used to show how general golbal‐dispersion relations can be constructed.

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 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...

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.

Calculations of a two-stream instability have been made by following the motion of the phase-space boundaries of an incompressible and constant-density phase-space fluid. Because of the condensation of holes, which to a good approximation act as gravitational particles, large-scale nonlinear pulses develop.

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

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)

A status report of microinstability as it applies to 2XIIB and MX theory for mirror machines is presented. It is shown that quasilinear computations reproduce many of the parameters observed in the 2XIIB experiment. In regard to large mirror machines, there are presented detailed calculations of the linear theory of the drift cyclotron loss-cone mode, with inhomogeneous geometry and nonlinear diffusive effects. Further, the stability of a mirror machine to the Alfven ion-cyclotron instability is assessed, and the Baldwin- Callen diffusion is estimated for a spatially varying plasma. (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 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.

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.

In this paper the low‐frequency quadratic form governing the linear response of electromagnetic trapped particle modes is applied to a tandem mirror geometry and several problems are treated. It is shown that the long‐wavelength magnetohydrodynamic (MHD) wall stabilization mechanism persists even when the line bending energy is suppressed by allowing an electrostatic response. In another problem, the amount of charge uncovering needed to suppress electrostatic trapped particle modes is determined as the beta of the plasma rises to its critical MHD beta value βcr and the required charge uncovering is shown to increase substantially as beta approaches βcr. A third problem treats trapped particle instabilities driven by rotation and field line curvature in diffuse and steep boundary models. The steep boundary model offers the possibility of an enhanced ‘‘robust’’ finite Larmor radius (FLR) stabilization due to an amplification of the finite Larmor radius magnetic compressional term when steep pressure gradie...

A nonlinear system of equations is derived for drift waves in a high‐beta plasma (β≫1). The magnetic field pressure is taken small compared to the particle pressure. Pressure balance is established by having a uniform particle pressure with the density and temperature gradients in opposite directions. The primary purpose of the magnetic field is to inhibit radial heat flux. This is the principle of such plasma fusion systems as the wall sustained multiple mirror, compressed liner, and magnetic‐insulated inertial fusion, where the heat is contained over a relatively short radial scale length and a long axial scale length. The nonlinear equations for the mathematical model contain drift instabilities that give rise to radial heat and particle fluxes that can enhance the losses expected from classical collisional effects. The linear and nonlinear evolution of the model is studied here.