Dynamical conservation of invariants by toroidal trajectories of guiding centres

The classical problem of calculating toroidal trajectories is treated here by comparing the results of two different methods in a given magnetic configuration, a standard divergence-free magnetic field model. The present work consists of adapting the analytical criteria of MERCIER et al. for classical toroidal trajectories, and to examine numerically the dynamical conservation of the toroidal invariant. The first method is based on the evolution equations for the guiding centres. These equations are then solved numerically (code TRATORIA) and the trajectories are drawn for different initial conditions. We use a modified standard model for the magnetic field, which insures a manifestly divergence-free field. Moreover we take into account the contribution of the poloidal field to the total strength of the magnetic field. These corrections contribute to the analytical expression of the conserved toroidal momentum. The latter is verified to be conserved by the present numerical simulation with a precision generally of the order of 10-14. The second method is based on the analytical treatment of the invariants to yield a semi-analytical (semi graphical) determination of the intersection point of a given trajectory with the equatorial plane. Both methods allows one to recover well-known toroidal trajectories with passing and trapped particles (bananas). The present analysis brings a clear description of some other, less well-known types of trajectories, namely the stagnation orbits, the smallest D-shape banana, some small circulating de-flated bananas, some huge classical bananas (potatoes), and the largest puffed bananas which exhibit only local mirroring, along with several kind of escaping or open trajectories which are of importance for fast ion losses and target damages in the machines