On the singularity of Lie-transform perturbation approach to the guiding-center problem

We present a novel scheme of carrying out the Lie-transform perturbation for the guiding-center motion, with an aim at addressing directly the problem of singularity, which exists intrinsically in the determining equation for the generating vector, and which gives rise to the formidable gauge functions in the pure oscillating part of the Lie transformation. While such gauge functions must be approximately solved from some partial differential equations in most applications of Lie-transform perturbation, it can be naturally produced through explicit integral over the gyro-angle in the present scheme, which is characterized by a staggered determination of the generating vectors and leaves no unaccountable error of high order in all the succeeding transformation. Based on such scheme, a formalism of guiding-center transformation has been derived in a unified manner retaining the effects of the strong E×B shearing as well as those of electromagnetic fluctuations.