Improvement of the Gyocenter-Gauge (G-Gauge) algorithm

The gyrocenter-gauge (g-gauge) algorithm was improved to simulate rf waves propagating in the three-dimensional sheared magnetic field. The conventional local gyro-center coordinate system $(X,Y,Z,\mu,\theta,u)$ is constructed on the local magnetic field. When particle travel in a sheared magnetic field, the coordinates of particles must be transformed between different local coordinate systems. To avoid these transformation, a new geometric approach is developed to construct a global Cartesian gyro-center coordinate system $(X,Y,Z,v_{x},v_{y},v_{z})$, where $(X,Y,Z)$ is the coordinate of the gyro-center, and $(v_{x},v_{y},v_{z})$ is the velocity of particle. In the g-gauge theory, the perturbation of distribution function, is obtained from the Lie derivative of gyro-center distribution function F along the perturbing vector field G. The evolution of the first order perturbed distribution contains a term $L_{\tau}L_{G}F=L_{\left[\tau,G\right]}F$ , where $\tau$ is the Hamilton vector field of unperturbed world-line of particles. It is proved that vector field $\left[\tau,G\right]$ may be directly solved from the electromagnetic fields. In the improved algorithm, $L_{G}F$ is calculated by integrating along the unperturbed world-line. The improved g-gauge algorithm has been successfully applied to study the propagation and evolution of rf waves in three-dimensional inhomogeneous magnetic field.