Hamiltonian description of the topology of drift orbits of relativistic particles in a tokamak

Abstract

Using classical perturbation theory, a Hamiltonian description of the guiding center motion of relativistic electrons in a torus with an axially symmetric magnetic field is derived. The magnetic field itself generates concentric circular flux surfaces. This description enables us to assess the behavior of the drift orbits such that previously unknown details follow almost naturally due to the introduction of higher order terms in the description. The rotational transform of the orbit for a given particle is found to decrease when the particle energy increases and the radius of the particle orbit is almost constant during acceleration. As a consequence, the minor radius of a drift surface with a (fixed) rotational transform 1/qD is always smaller than the radius of the magnetic flux surface with q=qD from which the particle started with zero energy. At a critical energy, the drift surface has collapsed to a point in the poloidal cross section. Above this critical energy there is no particle orbit with the commensurate rotational transform 1/qD. When the symmetry of the magnetic field is broken by harmonic perturbations with amplitudes that increase towards the plasma edge, the resonant magnetic surfaces and drift surfaces break up, and form chains of islands. The Hamiltonian description was set up in such a way that it allows the computation of the width of the drift islands.