Stability of Power-Law Disks II – The Global Spiral Modes

This paper reports on the in-plane normal modes in the self-consistent and the cut-out power-law disks. Although the cut-out disks are remarkably stable to bisymmetric perturbations, they are very susceptible to one-armed modes. For this harmonic, there is no inner Lindblad resonance, thus removing a powerful stabilising influence. A physical mechanism for the generation of the one-armed instabilities is put forward. Incoming trailing waves are reflected as leading waves at the inner cut-out, thus completing the feedback for the swing-amplifier. Growing three-armed and four-armed modes occur only at very low temperatures. However, neutral m = 3 and m = 4 modes are possible at higher temperatures for some disks. The rotation curve index β has a marked effect on stability. For all azimuthal wavenumbers, any unstable modes persist to higher temperatures and grow more vigorously if the rotation curve is rising (β < 0) than if the rotation curve is falling (β > 0). If the central regions or outer parts of the disk are carved out more abruptly, any instabilities become more virulent. The self-consistent power-law disks possess a number of unusual stability properties. There is no natural time-scale in the self-consistent disk. If a mode is admitted at some pattern speed and growth rate, then it must be present at all pattern speeds and growth rates. Our analysis – although falling short of a complete proof – suggests that such a two-dimensional continuum of non-axisymmetric modes does not occur and that the self-consistent power-law disks admit no global non-axisymmetric normal modes whatsoever. Without reflecting boundaries or cut-outs, there is no resonant cavity and no possibility of unstable growing modes. The self-consistent power-law disks certainly admit equiangular spirals as neutral modes, together with a one-dimensional continuum of growing axisymmetric modes.