Abstract
Charged particle motion in axisymmetric toroidal magnetic fields is analysed within the context of the canonical Hamiltonian guiding centre theory. A canonical transformation to variables measuring the drift orbit deviation from a magnetic field line is introduced and an analytical transformation to action-angle variables is obtained, under a zero drift width approximation. The latter is used to provide compact formulas for the orbital spectrum of the drift motion, namely the bounce/transit frequencies as well as the bounce/transit averaged toroidal precession and gyration frequencies. These formulas are shown to have a remarkable agreement with numerically calculated full drift width frequencies and significant differences from standard analytical formulas based on a pendulum-like Hamiltonian description. The analytical knowledge of the orbital spectrum is crucial for the formulation of particle resonance conditions with symmetry-breaking perturbations and the study of the resulting particle, energy and momentum transport.
Highlights
Charged particle dynamics in toroidal magnetic fields has been the key theoretical issue for the study of magnetically confined fusion plasmas for many decades
The guiding centre (GC) motion in an axisymmetric magnetic field is analysed under a Hamiltonian formulation in canonical variables
The zero drift width (ZDW) approximation has been described in the context of the canonical formulation through a canonical transformation to variables measuring the deviation of the GC from a magnetic field line of reference
Summary
Charged particle dynamics in toroidal magnetic fields has been the key theoretical issue for the study of magnetically confined fusion plasmas for many decades. In the action-angle variable set, the topology of the motion is described by multidimensional tori and each orbit is labelled by a distinct set of the invariant values of the three action variables (Kaufman 1972) This simple orbit parametrization allows for an orbit-based analysis of particle, energy and momentum transport which is useful for the study of energetic particle dynamics in fusion plasmas in direct relation to velocity–space tomography techniques (Stagner & Heidbrink 2017; Tholerus, Johnson & Hellsten 2017). The collective particle dynamics in the presence of perturbations is characterized by the modification of the unperturbed particle distribution functions (White 2011, 2012; Podesta, Gorelenkova & White 2014) Another important feature of the action-angle description is that the different time scales of the motion are well separated in different degrees of freedom allowing for a systematic dynamical reduction to a hierarchy of evolution equations for the reduced distribution functions (Brizard 2000; Kominis et al 2010).
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