Modelling RF heating in toroidal geometry: on the discrete bounce spectrum and the continuous bounce average

Abstract

Starting from a more general formalism due to Lamalle [Plasma Phys. Contr. Fusion32, 1409 (1997)], the dielectric response of a tokamak plasma to a radiofrequency (RF) perturbation is evaluated using a simple decorrelation model, assuming the poloidal cross-section of the magnetic surfaces to be circular and retaining leading-order terms in the drift parameter. Toroidicity, poloidal magnetic field and particle trapping are included in the model. Constants of the motion are used as independent variables. A semi-analytical method is adopted to evaluate the bounce spectrum of the dielectric response (needed to solve the wave equation) and the associated RF diffusion operator (required for solving the Fokker–Planck equation). The accent is on a study of this bounce spectrum. In particular, the link between the discrete bounce spectrum and the continuous bounce integral is highlighted. The critical parameter for the global justification of the replacement of the sum on the bounce spectrum by a bounce average is the relative magnitude of the decorrelation and bounce times. That for the local justification is the second derivative of the periodic part of the relative wave–particle phase. Numerical examples are provided to give a qualitative feeling for the justification of this substitution. The relation between the results presented here and those based on Hamiltonian models or on a simplified geometry is discussed.