L.‐J. Zheng

Collisionless kinetic ballooning mode equation in the low-frequency regime

The derivation of the fluid‐like kinetic ballooning mode equation from the gyrokinetic and Maxwell equations in the low‐frequency regime is revisited. The traditional results are pointed out to be ordering‐inconsistent. With the theory developed in the paper, two new effects are discovered: the small parallel‐ion‐velocity (SPIV) and the apparent mass (AM) effects. The SPIV effect is induced by the ions with parallel velocities much smaller than the thermal velocity, while the AM effect mainly by the ions with velocities comparable with the thermal velocity. The trapped‐particle contribution to the SPIV effect can be interpreted as a harmonic trapped‐particle effect, which can be even more important than the traditional bounce‐average trapped‐particle effect. The SPIV effect is shown to be more significant in ordering than the inertia effect with the AM and finite Larmor radius modifications. The application of the equation is discussed. The comparison with the ideal magnetohydrodynamic theory is made.

Collisionless kinetic ballooning equations in the comparable frequency regime

The collisionless kinetic ballooning mode equations are obtained from the gyrokinetic and Maxwell equations for the comparable frequency regime (i.e., for frequency of the modes comparable with the ion acoustic frequency along the field lines). A two length‐scale analysis is made to reduce the kinetic ballooning mode equations to a fluid‐like one. The finite Larmor radius (FLR) effect and particle–wave resonance (Landau damping) are taken into account. The trapped‐particle effect is proven to be negligible in the ordering scheme adopted in this paper. It is emphasized that the non‐FLR kinetic effects, especially the parallel electric field effect, are significant in the comparable frequency regime rather than the intermediate one. The non‐FLR kinetic effects are found to be coupled to the vorticity equation through the apparent mass term, which suggests that the toroidal effect is essential to the kinetic mechanism in the singular layer (i.e., the outer region of ballooning mode representation space). The...

TWO-FLUID EQUATIONS FOR SINGULAR MODES WITH FREQUENCY COMPARABLE TO THE PARALLEL ION ACOUSTIC FREQUENCY

Braginskii’s two‐fluid equations, with collisional effects neglected and electron fluid assumed to be isothermal along the magnetic field lines, are employed to obtain the singular mode equations. While the ordering analysis in the paper indicates that, in the intermediate frequency regime (i.e., that with the frequency of the modes higher than the ion acoustic frequency along the magnetic field lines, but lower than the electron one), the stabilizing finite Larmor radius (FLR) effect on the modes dominates over other collisionless, two‐fluid effects, the paper is devoted to the investigation of the comparable frequency regime (i.e., that with the frequency of the modes comparable to the ion acoustic frequency along the field lines). It is found that, in such a regime, the parallel electric field is excited and the plasma behaves nonadiabatically. The singular mode equations consist of two coupled second‐order partial differential equations, in which the FLR, the plasma compression, and the parallel elect...

Destabilization of singular layer modes in the comparable frequency regime

Abstract The nondissipative two-fluid equations for singular modes are employed to investigate the singular layer modes in the toroidal geometry. By matching the solutions in the outer (ideal magnetohydrodynamics (MHD)) and the inner regions a general dispersion relation is obtained, which is able to describe both the interchange and internal kink modes. The results show that the stable modes predicted by the finite-Larmor-radius modified ideal-MHD theory can become unstable due to the parallel electric field effect. The destabilization prevails in the comparable frequency regime (i.e., for a mode frequency comparable to that of the ion acoustic wave propagating along the magnetic field lines) instead of the intermediate frequency regime. Inclusion of the toroidal effect is found to be essential to the destabilization mechanism.

Construction of Monte Carlo operators in collisional transport theory

A Monte Carlo approach for investigating the dynamics of quiescent collisional magnetoplasmas is presented, based on the discretization of the gyrokinetic equation. The theory applies to a strongly rotating multispecies plasma, in a toroidally axisymmetric configuration. Expressions of the Monte Carlo collision operators are obtained for general v‐space nonorthogonal coordinates systems, in terms of approximate solutions of the discretized gyrokinetic equation. Basic features of the Monte Carlo operators are that they fullfill all the required conservation laws, i.e., linear momentum and kinetic energy conservation, and in addition that they take into account correctly also off‐diagonal diffusion coefficients. The present operators are thus potentially useful for describing the dynamics of a multispecies toroidal magnetoplasma. In particular, strict ambipolarity of particle fluxes is ensured automatically in the limit of small departures of the unperturbed particle trajectories from some initial axisymmet...

Collisional ballooning mode dispersion relation in the banana regime

Collisional ballooning mode theory in the banana regime is developed for tokamak configurations from the gyrokinetic formalism. A general dispersion relation is obtained, which in principle can deal with a collision operator of any type. However, investigation of an approximate Fokker–Planck collision operator developed in recent neoclassical transport theory is detailed. The most significant feature of the present theory as compared to the customary treatment lies in that the distinction between particle and fluid velocities is made in the ordering analyses. This reveals that the eigenfrequency of modes is determined by balancing the small‐parallel‐ion‐velocity (SPIV) effect [L.‐J. Zheng and M. Tessarotto, Phys. Plasmas 1, 3928 (1994)], instead of the fluid inertia one, with the instability drives. Since the parallel‐electric‐field effect is found to be negligible as compared to the SPIV effect, in contrast to the customary resistive ballooning mode picture, the leading collisional effect is demonstrated...

Monte Carlo approach to collisional transport

A linearized discretization approach is proposed for the numerical investigation of the gyrokinetic equation describing the dynamics of a strongly rotating plasma in a toroidally axisymmetric configuration. The discretization scheme allows the numerical evaluation of the neoclassical transport matrix in terms of a suitably discretized distribution function. A basic feature of the discretization scheme here developed is that it permits the introduction of Monte Carlo collision operators to advance in time the gyrokinetic state of a suitable set of test particles. Such operators, which apply to general non‐normal gyrokinetic coordinates, are constructed in such a way as to conserve exactly the collisional invariants. A fundamental consequence is that the neoclassical transport coefficients determined in this way fulfill exactly both Onsager’s symmetry relations and the condition of strict ambipolarity for the particle‐flux transport coefficients. Such properties are proven to be satisfied independently of the number of test particles used in the discretization scheme.

Collisional effect on the magnetohydrodynamic modes of low frequency

The kinetic singular layer equation is derived from the ballooning representation formalism. The mode frequency and the ion‐ion collision frequency are assumed to be lower than the ion transit frequency. It is proven that the balance between the instability drives and the small‐parallel‐ion‐velocity (SPIV) effect, instead of inertia effect, has to be imposed on determining the eigenfrequency. Consequently, it is found that the resistive effect is ignorable, while the dominant collisional effect appears to be the collisional contribution to the SPIV effect, leading to the conclusion that the customary resistive magnetohydrodynamic modes, such as the resistive interchange, resistive ballooning, and tearing modes, are nonexistent in this description.

The small‐parallel‐ion‐velocity effect on the ballooning modes with frequency lower than the magnetic drift one

Collisionless kinetic ballooning mode theory is presented in the case of the frequency of modes, ω, being lower than the ion magnetic drift frequency, ωDi, in the outer region of the ballooning representation space. A fluid‐like kinetic ballooning mode equation is obtained. Previous results are pointed out to be inconsistent due to the particle velocities not being distinguished from the fluid ones in the ordering analyses. In our new theory, it is found that the small parallel ion velocity (SPIV) effect, which was first discovered in the regime ωDi≪ω≪ωsi, with ωsi the ion acoustic frequency along the field lines [L.‐J. Zheng and M. Tessarotto, Phys. Plasmas 1, 3928 (1994)], is a dominant effect in ordering with respect to any other kinetic effects previously considered, such as the finite Larmor radius and the bounce‐average trapped particle effects. It is concluded that the eigenfrequency of the modes should be determined by the SPIV effect rather than by the fluid inertia one, which suggests that those...

PROBABILISTIC APPROACH TO MONTE CARLO OPERATORS

A general method for constructing Monte Carlo collision operators is formulated based on the introduction of a stochastic Liouville equation. The approach is applied to the investigation of a suitable discretized gyrokinetic equation describing the dynamics of a strongly rotating multispecies plasma, in a toroidally axisymmetric configuration. Improved expressions are obtained for the Monte Carlo collision operators for general non‐normal (in v space) coordinate systems. As an application, the important case of the bounce‐averaged gyrokinetic equation is discussed, in various cases of interest. Finally, using an approximate collision operator, the friction coefficients are evaluated and expressed in terms of energy and momentum restoring coefficients, thus yielding for the Monte Carlo operators a representation convenient for their numerical implementation.