Conditions for the stability of the dissipative trapped ion mode in axisymmetric toroidal confinement systems are investigated, and the ranges of temperature and magnetic field for which the plasma is expected to be unstable are calculated numerically for several proposed Tokamaks. The collisional damping is studied as a boundary layer problem in the velocity space of the trapped ions. The Fokker-Planck equation is solved by means of a variational form with the ordering νiR/r ≪ ω ≪2π/τi, where R/r is the aspect ratio, νiR/r is the effective collision frequency, and τi is the bounce time. The relative damping rate is found to be proportional to (νiR/rω) 1/2 [In (rω/νiR)1/2]−3/2 rather than νiR/rω. This damping term is compared with the electron driving term, which is proportional to rω/Rνe, and with the Landau damping caused by resonant untrapped ions. The presence of a temperature gradient is found to be destabilizing. The Landau term provides relatively strong damping when dT/dr=0 but changes sign when d ln T/d In n > 2/3. In the latter case it is concluded that ion collisions are insufficient to stabilize the mode under typical operating conditions of the proposed machines. Finally, if d ln T/d In n > 1.75 the ion collisional damping itself changes to growth, and the mode cannot be stabilized.
Using numerical methods, the authors investigate plasma diffusion in tokamaks when impurity ions of light elements are present. The model considered makes use of neoclassical expressions for diffusion fluxes and takes ionization and recombination processes into account. The question of the boundary conditions for the initial system of equations is discussed. The new boundary conditions for the proton and impurity ion densities are zero partial ion-ion fluxes at the boundary of the plasma column. The computational results show that the presence of impurity ions in the plasma column of a tokamak – even in relatively small amounts – strongly influences the spatial distribution of the protons. The entire diffusion process can be broken down into a number of stages which differ one from the other with regard to the amount of impurities in the system. If the proportion of impurities ξim does not exceed 0.01 of the electron density, the highly ionized ions collect at the centre of the plasma. If ξim > 0.01, the proton profile becomes flat within a short time and the impurities become more evenly distributed over the plasma cross-section.
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