Chaotic motion of charged particles in toroidal magnetic configurations

Abstract

We study the motion of a charged particle in a tokamak magnetic field and discuss its chaotic nature. Contrary to most of recent studies, we do not make any assumption on any constant of the motion and solve numerically the cyclotron gyration using Hamiltonian formalism. We take advantage of a symplectic integrator allowing us to make long-time simulations. First considering an idealized magnetic configuration, we add a non generic perturbation corresponding to a magnetic ripple, breaking one of the invariant of the motion. Chaotic motion is then observed and opens questions about the link between chaos of magnetic field lines and chaos of particle trajectories. Second, we return to an axisymmetric configuration and tune the safety factor (magnetic configuration) in order to recover chaotic motion. In this last setting with two constants of the motion, the presence of chaos implies that no third global constant exists, we highlight this fact by looking at variations of the first order of the magnetic moment in this chaotic setting. We are facing a mixed phase space with both regular and chaotic regions and point out the difficulties in performing a global reduction such as gyrokinetics.PACS numbers: 05.45.Ac, 52.25.Gj 1 Contrary to old studies, most of recent research dealing with hot magnetized fusion plasmas rely on numerical simulations. In the special case of a tokamak magnetic configuration, the majority of the numerical codes are based on the gyrokinetic theory. One of the assumptions made by this theory is to consider the magnetic moment of the particles moving in the magnetic field as an exact invariant of the motion. The straight consequence is to consider that all particle trajectories are integrable for axisymmetric configuration and no electric field.This assumption enables to make global reduction of the phase space and allows for faster numerical simulations. In fact, the magnetic moment is often an adiabatic invariant and it can present variations over very large-time scales. These remarks lay the ground for possible presence of Hamiltonian chaos in particle trajectory and a non-constant magnetic moment. In this paper, we solve numerically the motion of charged particles including the cyclotron gyration using Hamiltonian formalism in the sixth dimensional phase space, without using any assumption. We take advantage of a symplectic integrator allowing us to make long-time simulations. First, considering an axisymmetric magnetic configuration, we add a non-generic perturbation corresponding to a specific magnetic ripple, breaking one of the invariant of the motion. Chaotic motion is then observed whereas magnetic field lines are still integrable. So, we underline that the link between the two notions is not automatic. Second, we observe chaos of particle trajectories even in an axisymmetric configuration of the magnetic field. For this purpose, we study the limit case in which the major radius of the tokamak is infinite. The geometry becomes cylindrical. We tune the winding profile of magnetic field...

Influence of Toroidal Flow on Stationary Density of Collisionless Plasmas

“…Given the conclusions found in (author?) [7], we thus expect that these configurations should lead as well to regular trajectories when going to a toroidal configuration, at least in the large aspect ratio limit and this hints towards some stability of these solutions. On the contrary when a ≤ 0 it is possible to find self-consistent configurations that lead to the presence of a separatrix, and thus radial chaotic motion of particles can be expected if such configurations locally appear in the plasma, the phenomena are illustrated in Fig.…”

“…We shall come back to this statement, as this feature was shown to break the magnetic moment in some regions of the full phase space (author?) [7]. Let us now built a passive particle equilibrium distribution from the previous considerations.…”

Study on the creation and destruction of transport barriers via the effective safety factors for energetic particles

“…If we assume the magnetic momentum µ is the third integral of the motion, the particle system can be integrable when ǫ = 0, 21 and the present result can be generalized for the tokamak. However, it should be noted that the assumption of the third integral is sometimes broken, 16 but the generalized theory may work in some local region apart from the separatrix of H eff .…”

“…To determine the time average, we need an orbit associated with an effective Hamiltonian H eff defined on (r, p r ) plane, 9,16 H eff (r, p r ) = p 2 r /2 + V eff (r ),…”

“…Meanwhile, a) Electronic mail: shun.ogawa@cpt.univ-mrs.fr for energetic particles, it was found by computing full particle orbits numerically that these structures are quite different from the magnetic ones. 9,15,16 It is for instance possible that for a given class of particles with a range of energies and pitch angles, an ITB appears in the chaotic region of the magnetic field lines. 9 Conversely it is also possible that the magnetic ITB is essentially useless from the particles point of view.…”