# 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 tight-winding 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. We identify two important dimensionless physical parameters: ∈ = 2πGσ0/(rκ2), which is essentially the ratio of disk mass to total mass (disk and halo), and a/(κr), which is the ratio of epicyclic radius to disk radius (σ0 is the mass column density, κ is the epicyclic frequency, and a is the sound speed). The small term ζ = (k2r2 + m2)-1/2 is an additional parameter that is purely geometrical for number of arms m. When these terms are included in the dispersion relation, the oscillation frequency becomes complex, leading to the growth of perturbations even for large values of Toomre's parameter Q. The growth rate is proportional to a linear combination of terms that depend on ∈ and a/(κr). Instabilities that arise from ∈ are termed gravitational-curvature instabilities because ∈ depends on the disk mass and is largest when the radius is small, i.e., when the orbital curvature is large. Instabilities that arise from a/(κr) are termed acoustic-curvature instabilities because they arise from only the pressure terms at small r. Unstable growth rates are determined for these instabilities 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 protoplanetary disks. All of these examples have short orbital times so the unstable growth time can be small, even when only terms of order ∈ contribute.