Pullback transformations in gyrokinetic theory

Abstract

The pullback transformation of the distribution function is a key component of gyrokinetic theory. In this paper, a systematic treatment of this subject is presented, and results from applications of the uniform framework developed are reviewed. The focus is on providing a clear exposition of the basic formalism which arises from the existence of three distinct coordinate systems in gyrokinetic theory. The familiar gyrocenter coordinate system, where the gyromotion is decoupled from the rest of particle’s dynamics, is noncanonical and nonfibered. For the phase space (cotangent bundle T*M) associated with a configuration space M, a nonfibered coordinate system (X, V) is a coordinate system where X is not necessarily the coordinates for the configuration space M, and V is not necessarily the coordinates for the cotangent fiber Tx*M at each x. On the other hand, Maxwell’s equations, which are needed to complete a kinetic system, are initially only defined in the fibered laboratory phase space coordinate system. The pullback transformations provide a rigorous connection between the distribution functions in gyrocenter coordinates and Maxwell’s equations in laboratory phase space coordinates. This involves the generalization of the usual moment integrals originally defined on the cotangent fiber of the phase space to the moment integrals on a general six-dimensional symplectic manifold. The resultant systematic treatment of the moment integrals enabled by the pullback transformation is shown to be an important step in the proper formulation of gyrokinetic theory. Without this vital element, a number of prominent physics features, such as the presence of the compressional Alfvén wave and a proper description of the gyrokinetic equilibrium, cannot be readily recovered.