Resonant Interactions between Particles and Normal Modes in a Cylindrical Plasma

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Abstract
A plasma configuration with cylindrical symmetry is studied, containing axial and azimuthal magnetic fields and radial electric field, with arbitrary radial variation. The particle motion is parameterized by three exact action invariants: radial action, canonical angular momentum, and canonical axial momentum; in the limit of small gyroradius they are equivalent to magnetic moment, radial guiding-center position, and parallel velocity. The perturbed Vlasov-Maxwell equations lead to a set of normal modes, which can interact resonantly with the particles. The quantum rate equations for this interaction, together with the laws for conservation of energy, angular momentum, and axial momentum, lead (in the classical limit) to a Fokker-Planck equation in action space for the particles, and to an equation of evolution for mode energy. These coupled kinetic equations satisfy an H theorem, which implies a monotonic approach to a canonical distribution: a rigid-rotor distribution for particles, and a generalized Rayleigh-Jeans distribution for the modes. This asymptotic state may, however, be unconfined. The quantum transition probability is deduced from a classical calculation of emissivity. Explicit expressions are obtained for the mode growth rate and for the particle diffusion tensor. Finally, the Vlasov conductivity kernel is deduced from the growth rates, by the use of the Kramers-Kronig relations.