Gyrokinetic perpendicular dynamics

Gyrokinetic perpendicular dynamics, an important component not systematically considered in previous gyrokinetic theories, is identified and developed. A "distribution function" S and its governing gyrokinetic equation are introduced to describe the gyrokinetic perpendicular dynamics. The complete treatment of the perpendicular current rendered by the gyrokinetic perpendicular dynamics enables one to recover the compressional Alfvén wave from the gyrokinetic model. From the viewpoint of gyrokinetic theory, the physics of the compressional Alfvén wave is the polarization current at second order. Therefore, in a low frequency gyrokinetic system, the compressional Alfvén wave is naturally decoupled from the shear Alfvén wave and drift wave. In the gyrocenter coordinates, the gyrophase dependent parts of the distribution function S and f are decoupled from the gyrophase independent part f. Introducing the gyrokinetic perpendicular dynamics also extends the gyrokinetic model to arbitrary frequency modes. As an example, the Bernstein wave is recovered from the gyrokinetic model. The gyrokinetic perpendicular dynamics uncovered here emphasizes that the spirit of gyrokinetic reduction is to decouple the gyromotion from the particle's gyrocenter orbit motion, instead of averaging out the gyromotion.