Asymptotic analysis of the electrostatic wave equation in a toroidal plasma

An analytical study is presented of the wave equation that describes the propagation of an electrostatic pulse in a cold plasma in a general magnetic equilibrium by means of a multiple spatial scale approach. This technique is strictly related with that discussed earlier by Zonca and Chen [Phys. Fluids B 5, 3668 (1993)], and, when applied to plasma instabilities, reduces to the well-known 'ballooning formalism' [J. W. Connor, R. J. Hastie, and J. B. Taylor, Phys. Rev. Lett. 40, 396 (1978)]. A simplified equation for the scalar potential in the cold plasma limit will be derived and studied by applying the WKB asymptotic technique to describe the slow radial dependencies of the wave envelope, while the full-wave equation will be considered along the magnetic field lines. This ansatz can be entirely justified on the basis of spatial scale separation in the radial direction and for waves that have parallel group velocity faster than in the perpendicular direction. Thus, this approach could be viewed as a mixed WKB-full-wave technique