The optimal pitch angle which maximizes the penetration distance, along the magnetic field, of relativistic charged particles injected from the midplane of an axisymmetric field is investigated analytically and numerically. Higher-order terms of the magnetic moment invariant are necessary to correctly determine the mirror point of trapped energetic particles, and therefore the loss cone. The modified loss cone resulting from the inclusion of higher-order terms is no longer entirely defined by the pitch angle but also by the phase angle of the particle at the point of injection. The optimal orientation of the injection has a nonzero component perpendicular to the magnetic field line, and is in the plane tangential to the flux surface. Numerical integration of particle orbits were carried out for a relativistic electron in a dipole field, showing agreement with analytic expressions. The results are relevant to experiments, which are concerned with injection of relativistic beams into the atmosphere from aboard a spacecraft in the magnetosphere.

The compressional component of magnetic perturbation dB k can play an important role in drift-Alfvenic instabilities in tokamaks, especially as the plasma b increases (b is the ratio of kinetic pressure to magnetic pressure). In this work, we have formulated a gyrokinetic particle simulation model incorporating dB k , and verified the model in kinetic Alfven wave simulations using the Gyrokinetic Toroidal Code in slab geometry. Simulations of drift-Alfvenic instabilities in tokamak geometry shows that the kinetic ballooning mode (KBM) growth rate decreases more than 20% when dB k is neglected for b e ¼ 0:02, and that dB k has stabilizing effects on the ion temperature gradient instability, but negligible effects on the collisionless trapped electron mode. The KBM growth rate decreases about 15% when equilibrium current is neglected.

A method for gyrokinetic simulation of low frequency (lower than the cyclotron frequency) magnetic compressional modes in general geometry is presented. The gyrokinetic-Maxwell system of equations is expressed fully in terms of the compressional component of the magnetic perturbation, δB∥, with finite Larmor radius effects. This introduces a “gyro-surface” averaging of δB∥ in the gyrocenter equations of motion, and similarly in the perpendicular Ampere’s law, which takes the form of the perpendicular force balance equation. The resulting system can be numerically implemented by representing the gyro-surface averaging by a discrete sum in the configuration space. For the typical wavelength of interest (on the order of the gyroradius), the gyro-surface averaging can be reduced to averaging along an effective gyro-orbit. The phase space integration in the force balance equation can be approximated by summing over carefully chosen samples in the magnetic moment coordinate, allowing for an efficient numerical implementation.