Curvature and Acoustic Instabilities in Rotating Fluid Disks

Abstract

The stability of a rotating fluid disk to the formation of spiral arms is studied in the tightwinding approximation in the linear regime. The dispersion relation for spirals that was derived by Bertin et al. is shown to contain a new, acoustic instability beyond the Lindblad resonances that depends only on pressure and rotation. In this regime, pressure and gravity exchange roles as drivers and inhibitors of spiral wave structures. Other instabilities that are enhanced by pressure are also found in the general dispersion relation by including higher order terms in the small parameter 1/kr for wavenumber k and radius r. These instabilities are present even for large values of Toomre's parameter Q. Unstable growth rates are determined in four cases: a self-gravitating disk with a flat rotation curve, a self-gravitating disk with solid body rotation, a non-self-gravitating disk with solid body rotation, and a non-self-gravitating disk with Keplerian rotation. The most important application appears to be as a source of spiral structure, possibly leading to accretion in non-self-gravitating disks, such as some galactic nuclear disks, disks around black holes, and proto-planetary disks. All of these examples have short orbital times so the unstable growth time can be small.