Second-order nonlinear gyrokinetic theory: from the particle to the gyrocentre

A gyrokinetic reduction is based on a specific ordering of the different small parameters characterizing the background magnetic field and the fluctuating electromagnetic fields. In this tutorial, we consider the following ordering of the small parameters:$\unicode[STIX]{x1D716}_{B}=\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FF}}^{2}$where$\unicode[STIX]{x1D716}_{B}$is the small parameter associated with spatial inhomogeneities of the background magnetic field and$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FF}}$characterizes the small amplitude of the fluctuating fields. In particular, we do not make any assumption on the amplitude of the background magnetic field. Given this choice of ordering, we describe a self-contained and systematic derivation which is particularly well suited for the gyrokinetic reduction, following a two-step procedure. We follow the approach developed in Sugama (Phys. Plasmas, vol. 7, 2000, p. 466): In a first step, using a translation in velocity, we embed the transformation performed on the symplectic part of the gyrocentre reduction in the guiding-centre one. In a second step, using a canonical Lie transform, we eliminate the gyroangle dependence from the Hamiltonian. As a consequence, we explicitly derive the fully electromagnetic gyrokinetic equations at the second order in$\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FF}}$.

Lifting of the Vlasov–Maxwell bracket by Lie-transform method

The Vlasov–Maxwell equations possess a Hamiltonian structure expressed in terms of a Hamiltonian functional and a functional bracket. In the present paper, the transformation (‘lift’) of the Vlasov–Maxwell bracket induced by the dynamical reduction of single-particle dynamics is investigated when the reduction is carried out by Lie-transform perturbation methods. The ultimate goal of this work is to provide an explicit pathway to the Hamiltonian formulations for the guiding-centre and gyrokinetic Vlasov–Maxwell equations, which have found important applications in our understanding of turbulent magnetized plasmas. Here, it is shown that the general form of the reduced Vlasov–Maxwell equations possesses a Hamiltonian structure defined in terms of a reduced Hamiltonian functional and a reduced bracket that automatically satisfies the standard bracket properties.

A First Order Automated Lie Transform

The objective of the present paper is to focus on the problem of the normalization of a Hamiltonian system via the elimination of angle variables involved using the Lie transform technique. The algorithm that we construct assumes that the Hamiltonian is periodic in [Formula: see text] angle variables, with two rates: fast and slow. If the angle variables have the same rate only one transformation is required. The equations needed to evaluate the elements of each transformation and the secular perturbations are constructed.