The computer algebra evaluation of the permeability tensor and bounce resonance effects in toroidal plasmas

Radio-frequency heating problems and current drive studies are becoming a very important part of the tokamak experimental programs. Usually, the current drive is created by means of the absorption of travelling waves. In these tokamak plasmas, collisionless wave dissipation has a peculiarity connected to the periodic motion of trapped and untrapped particles over magnetic surfaces. The theoretical and computational calculations done showed a strong reduction in the contribution of untrapped electrons, and a growth in the contribution of the trapped ones, to the dissipated power for Alfven waves with phase velocity smaller than the thermal velocity. The problem will be discussed, using the computer algebra Maple system, concentrating the attention on an analysis of untrapped electron dissipation in tokamak plasmas that have magnetic surfaces with circular cross-section, for waves with phase velocity greater than the thermal velocity (vph = Ω/K > vTe). Under these conditions, most of the dissipated power is related to the untrapped electrons. These collisionless wave dissipation problems are well described by the of Vlasov-Maxwell equations, which can be tackled by finding analytic solutions for Vlasov's equation separately through a linear approximation over the wave field amplitude. (author). 7 refs, 2 figs