On the global stability of magnetized accretion disks. 1: Axisymmetric modes

We investigate the global stability of differentially rotating fluid shell threaded by a constant vertical magnetic field to linear, axisymmetric perturbations. Such a system is known to be unstable due to the interaction between the magnetic field and fluid shear, and should mimic the behavior of an accretion disk far from its vertical boundaries. The magnetic field exterior to the fluid has an effect on the radial boundary motion, and we derive the appropiate boundary conditions. The growth rates of the instability, associated radial eigenfunctions, and critical field strengths for stability are presented for a wide variety of radii, rotation profiles, and magnetic field strengths. The growth rates of unstable global modes are always less than, but comparable to, the corresponding local growth rates. For near critical field strengths, nonrigid boundary conditions lead to much higher growth rates than do (physically unrealistic) rigid boundary conditions. Also, we find that the critical Alfven speed for stability is large (of order the sound speed) for realistic disk sizes, thus implying that the instability is always present. Finally, the connection between the global and local characters of the instability is elucidated.