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

TL;DR

This paper investigates the conservative properties of particle simulations in guiding center models within Boozer coordinates, demonstrating that including certain terms preserves Hamiltonian properties and that nonnormal mode perturbations reveal deviations.

Contribution

It shows that the guiding center Lagrangian in Boozer coordinates can be canonicalized and that omitting small terms affects conservation laws in simulations.

Findings

Simulations with and without $B_{\Psi_{\rm P}}$ produce similar particle orbits.

The unabridged Lagrangian preserves phase space and energy conservation.

Perturbations resembling nonnormal modes better reveal conservation law violations.

Abstract

The guiding center (GC) Lagrangian in Boozer coordinates for toroidally confined plasmas can be cast into canonical form by eliminating a term containing the covariant component $B_{\Psi_{\rm P}}$ of the magnetic field vector with respect to the poloidal flux function $\Psi_{\rm P}$. Considering fast ions in the presence of a shear Alfv\'{e}n 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 $B_{\Psi_{\rm P}}$ 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 $B_{\Psi_{\rm P}}$ 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) & (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.