Evolution of initially localized perturbations in stratified ionized discs

A detailed solution of an initial value problem of a vertically localized initial perturbation in rotating magnetized vertically stratified disc is presented. The appropriate linearized magnetohydrodynamics equations are solved by employing the Wentzel–Kramers–Brillouin (WKB) approximation and the results are verified numerically. The eigenfrequencies as well as eigenfunctions are explicitly obtained. It is demonstrated that the initial perturbation remains confined within the disc. It is further shown that thin enough discs are stable but as their thickness grows increasing number of unstable modes participate in the solution of the initial value problem. However, it is demonstrated that due to the localization of the initial perturbation, the growth time of the instability is significantly longer than the calculated inverse growth rate of the individual unstable eigenfunctions.