Non-Axisymmetric g-Mode and p-Mode Instability in a Thin Accretion Disk

It has been suggested that quasi-periodic oscillations of accreting X-ray sources may relate to the modes named in the title. We idealize a thin accretion disk by means of the shearing sheet model and derive a wave equation for nonaxisymmetric perturbations. We show that the equation can be transformed into the Whittaker equation, which enables us to write down analytical solutions. For modes with vertical wave number n > 0, there are four distinct permitted regions for waves, separated by two forbidden regions and a corotation resonance. We show that each forbidden region behaves like a wave amplifier so that any wave incident on it is reflected with a larger amplitude. The corotation resonance, however, behaves like a strong wave absorber. Following Kato (2001b, 2002), we look for internally trapped growing g-modes in which most of the wave action is concentrated in the region between the two forbidden zones. We find that there are no such modes in the shearing sheet. We discuss whether such modes may be present in more general shear flows that deviate from the idealized shearing sheet. We conclude that instability, though not ruled out, is likely to be rare. We also analyse externally trapped p-modes in which the wave action is concentrated in an external cavity between one of the forbidden zones and the inner edge of the disk. Unstable modes are certainly possible in this case. Moreover, the wave amplification can be substantially larger for n = 1 than for the previously studied case of n = 0. Therefore, n = 1 p-modes might have significant growth rates in realistic disks, provided that the disk inner edge reflects waves reasonably well. Subject headings: accretion, accretion disks — instabilities — waves — X-rays: binaries