Derivation of the quasi-linear theory equations for the axisymmetric toroidal systems

Proceeding from Vlasov's equation for the particle distribution function and Maxwell's equations for the electromagnetic fields the quasi-linear equations for the distribution functions averaged over the particle cyclotron gyration period tau B and bounce period tau b are derived. These equations describe the quasi-linear processes caused by the resonant and nonresonant interaction between waves and both trapped and untrapped particles in the axisymmetric toroidal systems. They take into account the finite gyroradius and the radial excursions of the particle guiding centres as well as the particle drift along the large torus azimuth. To derive the quasi-linear equations the multiple-time-scale method based on the small ratios tau B/ tau QL and tau b/ tau QL, tau QL being the characteristic time of the quasi-linear process, is applied.