THE PARKER INSTABILITY IN DISKS WITH DIFFERENTIAL ROTATION

We present a detailed study of the growth of the Parker instability in a differentially rotating disk embedded in an azimuthal equilibrium m agnetic field, such as the interstellar gas or an accretion disk. Basi c properties of the instability without shear are first recalled. Diff erential rotation is modeled in the shearing sheet approximation, clas sical in the theory of spiral density waves, and the linearized proble m is transformed into a second order differential equation. The action of differential rotation is reduced to two different effects, (i) a l inear time-dependence of the radial wavenumber, and (ii) a radial diff erential force. We present both exact numerical solutions, and approxi mate analytical ones based on the WKB approximation in the limit of we ak differential rotation. This allows us to discuss the mathematical a nd physical nature of new behaviours obtained numerically in cases wit h realistic differential rotation. Most important are (i) a transient natural stabilization of the Parker mode due to the radial differentia l force (ii) the generation of magnetosonic waves (i.e. spiral density waves if we had included self-gravity) and Alfvenic ones, and (iii) i n a certain parameter range a possible ''turn-over'' of the perturbati on whereby, quite surprisingly, matter which had started being elevate d by the instability may end up dropping towards the disk midplane. A simplified model shows the possible observable effects of this turn-ov er.