Testing the conservative character of particle simulations: I. Canonical and noncanonical guiding center model in Boozer coordinates

Abstract

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating terms containing the covariant component BΨP of the magnetic field vector with respect to the poloidal flux function ΨP. In an unperturbed plasma, BΨP can be eliminated via exact coordinate transformations, but, in general, one relies on approximations, assuming that the effect of BΨP is small. Here, we are interested in the question whether Hamiltonian conservation laws are still satisfied when BΨP is retained in the presence of fluctuations. Considering fast ions in the presence of a shear Alfvén wave field with fixed amplitude, fixed frequency, and a single toroidal mode number n, we show that simulations using the code ORBIT with and without BΨP yield practically the same resonant and nonresonant GC orbits. The numerical results are consistent with theoretical analyses (presented in the appendix), which show that the unabridged GC Lagrangian with BΨP retained yields equations of motion that possess two key properties of Hamiltonian flows: (i) phase space conservation and (ii) energy conservation. As counter-examples, we also show cases where energy conservation (ii) or both conservation laws (i) and (ii) are broken by omitting certain small terms. When testing the conservative character of the simulation code, it is found to be beneficial to apply perturbations that do not resemble normal (eigen) modes of the plasma. The deviations are enhanced and, thus, more easily spotted when one inspects wave-particle interactions using nonnormal modes.