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.
Compact formulas for trapped-particle and passing-particle guiding-center orbits in axisymmetric tokamak geometry are given in terms of the Jacobi elliptic functions and complete elliptic integrals. These formulas can find applications in bounce-center kinetic theory as well as neoclassical transport theory.
The generating function for the canonical transformation from the parallel canonical coordinates $(p_{\|},s)$ to the action-angle coordinates $(J,ζ)$ for trapped/passing guiding-center orbits in axisymmetric tokamak geometry is presented. Drawing on the analogy between the phase-space portraits of the librating/rotating pendulum and the trapped/passing guiding-center orbits, the generating function is expressed in terms of the Jacobi zeta function, which can then readily be used to obtain an explicit expression for the bounce-center transformation for trapped/passing-particles in axisymmetric …
To identify under what conditions guiding-centre or full-orbit tracing should be used, an estimation of the spatial variation of the magnetic field is proposed, not only taking into account gradient and curvature terms but also parallel currents and the local shearing of field-lines. The criterion is derived for general three-dimensional magnetic equilibria including stellarator plasmas. Details are provided on how to implement it in cylindrical coordinates, and in flux coordinates that rely on the geometric toroidal angle. A means of switching between guiding-centre and full-orbit equations a…
The presence of non-axisymmetric perturbations in an axisymmetric magnetic field equilibrium renders the Guiding Center (GC) particle motion non-integrable and may result in particle, energy and momentum redistribution, due to resonance mechanisms. We analyse these perturbations in terms of their spectrum, as observed by the particles in the frame of unperturbed GC motion. We calculate semi-analytically the exact locations and strength of resonant spectral components of multiple perturbations. The presented Orbital Spectrum Analysis (OSA) method is based on an exact Action-Angle transform that…
Charged particle motion in axisymmetric toroidal magnetic fields is analyzed within the context of the canonical Hamiltonian Guiding Center 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 f…
A general orbit-averaged guiding-center Fokker-Planck operator suitable for the numerical analysis of transport processes in axisymmetric magnetized plasmas is presented. The orbit-averaged guiding-center operator describes transport processes in a three-dimensional guiding-center invariant space: the orbit-averaged magnetic-flux invariant $\ovψ$, the minimum-B pitch-angle coordinate $ξ_{0}$, and the momentum magnitude $p$.
The problem of the charged-particle motion in an axisymmetric magnetic-dipole geometry is used to assess the validity of Hamiltonian guiding-center theory, which includes higher-order corrections associated with guiding-center polarization induced by magnetic-field nonuniformity. 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 guiding-center polarization ef…
A consistent guiding-center Hamiltonian theory is derived by Lie-transform perturbation method, with terms up to second order in magnetic-field nonuniformity. Consistency is demonstrated by showing that the guiding-center transformation presented here satisfies separate Jacobian and Lagrangian constraints that have not been explored before. A new first-order term appearing in the guiding-center phase-space Lagrangian is identified through a calculation of the guiding-center polarization. It is shown that this new polarization term also yields a simpler expression of the guiding-center toroidal…
Hamiltonian Particle Dynamics in Fusion Plasmas: Orbital Tomography and Spectrum Analysis for Energy and Momentum Transport under Resonant Non-Axisymmetric Perturbations
This work proposes a parametric equilibrium distribution function F_eq to be applied to the gyrokinetic studies of the Finite Orbit Width behavior of guiding centers representing several species encountered in axisymmetric tokamak plasma, as fusion products, thermal bulk and energetic particles from Ion Cyclotron Radiation Heating and Negative Neutral Beam Injections. After the analysis of the basic results of orbit theory obtained with a particularly convenient orbit coordinates set, it is shown how the proposed F_eq satisfies the two conditions that make it an equilibrium distribution functi…
A numerical integration method for guiding-center orbits of charged particles in toroidal fusion devices with three-dimensional field geometry is described. Here, high order interpolation of electromagnetic fields in space is replaced by a special linear interpolation, leading to locally linear Hamiltonian equations of motion with piecewise constant coefficients. This approach reduces computational effort and noise sensitivity while the conservation of total energy, magnetic moment and phase space volume is retained. The underlying formulation treats motion in piecewise linear fields exactly a…
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 a fixed angular phase velocity. In principle, this is true for arbitrary mode structures in the poloidal plane, but only if the fluctuations are expressed in terms of potentials $Φ$ and ${\bf A}$, which satisfy Faraday's law by definition. Here, we use the physical fields ${\bf E}$ and ${\bf B}$, where Faraday's law may be violated by errors introduced during the proce…
Experimental diagnostics, analysis tools and simulations represent particle distributions in various forms and coordinates. Algorithms to manage these data are needed on platforms like the ITER Integrated Modelling & Analysis Suite (IMAS), performing tasks such as archiving, modeling, conversion and visualization. A method that accomplishes some of the required tasks for distributions of charged particles with arbitrarily large magnetic drifts in axisymmetric tokamak geometry is described here. Given a magnetic configuration, we first construct a database of guiding center orbits, which serves…
