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

For an axisymmetric tokamak plasma, Hamiltonian theory predicts that the orbits of charged particles must stay on invariant tori of conserved energy in the moving frame of reference of a wave that propagates along the torus with fixed angular phase velocity, amplitude, and shape. The mode structure in the poloidal plane is arbitrary if the fluctuations are expressed in terms of potentials [Formula: see text] and [Formula: see text], which satisfy Faraday's law and the solenoidal condition by definition. Consequently, smoothing operations (such as gyroaveraging and noise suppression) do not violate the conservative laws. However, this is not guaranteed for models expressed in terms of the physical fields [Formula: see text] and [Formula: see text]. Here, we demonstrate that manipulations of [Formula: see text] and [Formula: see text] in the poloidal ( R, z) plane can cause spurious heating that is independent of time steps or numerical methods, but can be sensitive to geometry. In particular, we show that secular acceleration is enhanced when one imposes nonnormal modes that possess strong up–down asymmetry instead of the usual in–out asymmetry of normal toroidal (eigen)modes. We compare full gyro-orbit and guiding center models and find similar behavior. We also examine the effect of ad hoc N-point gyroaveraging in a guiding center model, as is done in some simulation codes. If one uses Faraday's law to (re)compute [Formula: see text] after gyroaveraging [Formula: see text], the guiding center motion remains conservative. Otherwise, spurious heating should be expected and monitored, but it may be tolerable when normal modes dominate.

Testing the conservative character of particle simulations: II. Spurious heating of guiding centers and full orbits subject to fluctuations expressed in terms of E and B

Bierwage

Shinohara

2022

Faithful guiding-center orbits in an axisymmetric magnetic field

Brizard

2023

The problem of the charged-particle motion in an axisymmetric magnetic geometry is used to assess the validity of higher-order Hamiltonian guiding-center theory, which includes higher-order corrections associated with gyrogauge invariance as well as guiding-center polarization induced by magnetic-field non-uniformity. Two axisymmetric magnetic geometries are considered: a magnetic mirror geometry and a simple tokamak geometry. When a magnetically confined charged-particle orbit is regular (i.e., its guiding-center magnetic moment is adiabatically invariant), the guiding-center approximation, which conserves both energy and azimuthal canonical angular momentum, is shown to be faithful to the particle orbit when higher-order corrections are taken into account.

Kinetic vs magnetic chaos in toroidal plasmas: A systematic quantitative comparison

Moges,

Antonenas,

Anastassiou

et al. 2024

Magnetic field line chaos occurs under the presence of non-axisymmetric perturbations of an axisymmetric equilibrium and is manifested by the destruction of smooth flux surfaces formed by the field lines. These perturbations also render the particle motion, as described by the guiding center dynamics, non-integrable and, therefore, chaotic. However, the chaoticities of the magnetic field lines and the particle orbits significantly differ in both strength and radial location in a toroidal configuration, except for the case of very low-energy particles whose orbits closely follow the magnetic field lines. The chaoticity of more energetic particles, undergoing large drifts with respect to the magnetic field lines, crucially determines the confinement properties of a toroidal device but cannot be inferred from that of the underlying magnetic field. In this work, we implement the smaller alignment index method for detecting and quantifying chaos, allowing for a systematic comparison between magnetic and kinetic chaos. The efficient quantification of chaos enables the assignment of a value characterizing the chaoticity of each orbit in the space of the three constants of the motion, namely, energy, magnetic moment, and toroidal momentum. The respective diagrams provide a unique overview of the different effects of a specific set of perturbations on the entire range of trapped and passing particles, as well as the radial location of the chaotic regions, offering a valuable tool for the study of particle energy and momentum transport and confinement properties of a toroidal fusion device.